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c Copyright 1998Andreas Wiegmann

The Explicit Jump Immersed Interface Method and InterfaceProblems for Di�erential EquationsbyAndreas WiegmannA dissertation submitted in partial ful�llment ofthe requirements for the degree ofDoctor of PhilosophyUniversity of Washington1998Approved by (Chairperson of Supervisory Committee)Program Authorizedto O�er DegreeDate

In presenting this dissertation in partial ful�llment of the requirements for the Doctoraldegree at the University of Washington, I agree that the library shall make its copies freelyavailable for inspection. I further agree that extensive copying of this dissertation is allowedonly for scholastic purposes, consistent with fair use as prescribed in the U.S. CopyrightLaw. Requests for copying or reproduction of this dissertation may be referred to UniversityMicro�lms, 1490 Eisenhower Place, P.O. Box 975, Ann Arbor, Michigan 48106, to whomthe author has granted \the right to reproduce and sell (a) copies of the manuscript inmicroform and/or (b) printed copies of the manuscript made from microform."SignatureDate

University of WashingtonAbstractThe Explicit Jump Immersed Interface Method and Interface Problems forDi�erential Equationsby Andreas WiegmannChairperson of Supervisory Committee: Professor Kenneth P. BubeDepartment of MathematicsWe study and numerically solve elliptic di�erential equations in the presence of interfaceswhere the solution is not smooth. We use uniform Cartesian grids and do not require theinterfaces to be aligned with the grid. We develop a one-dimensional theory for the newExplicit Jump Immersed Interface Method (EJIIM), which culminates in a proof of second-order convergence for piecewise-constant coe�cients for single-point interfaces. The proofis interesting in not requiring the numerical scheme to satisfy a discrete maximum principle,the usual means by which such results are proved, and in providing error bounds that areindependent of the geometry and the contrast in the coe�cients. EJIIM works by focusingon the jumps in the solutions and their derivatives, rather than on �nding coe�cients ofnew �nite di�erence schemes, like the original Immersed Interface Method (IIM). In ourformulation, the jump conditions for many di�erent problems all turn out to depend onlyon limits from one side of the interface. This view allows the use of fast solvers for theresulting large sparse systems, and easy incorporation of multiple interfaces. The one-dimensional corrections to �nite di�erence schemes in the presence of discontinuities couldprove useful far beyond the scope of EJIIM. We move on to prove second-order convergencefor singular source problems in two dimensions, and �nd bounds on the coe�cients ofthe scheme for elliptic equations with discontinuous coe�cients. New jump conditionsfor irregular domain problems are found, and the Liouville transformation is extended todiscontinuous coe�cients. Numerical examples demonstrate the second-order behavior ofEJIIM and improved performance in the presence of large contrasts in the coe�cients ona variety of boundary value problems and interface problems including multiply connecteddomains, and improved smooth dependence of the solutions for smoothly varying interfacelocations. Finally, we present a method to recover perturbations of a circular interface frommeasurements taken at the boundary of a circular domain, by linearizing about the circularinterface and using Fourier series.

TABLE OF CONTENTSList of Figures ivList of Tables viChapter 1: Introduction 11.1 Historical background of the IIM . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Other advances with the IIM . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Experience with nonlinear problems . . . . . . . . . . . . . . . . . . . . . . 81.4 The EJIIM approach to elliptic partial di�erential equations . . . . . . . . . 8Chapter 2: Analysis of the Explicit Jump IIM in 1D 122.1 Basic de�nitions and problems . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Correcting �nite di�erences with known jumps . . . . . . . . . . . . . . . . 152.2.1 Taylor Expansions for piecewise smooth functions . . . . . . . . . . 152.2.2 Correcting centered di�erences . . . . . . . . . . . . . . . . . . . . . 172.3 EJIIM for \elliptic" equations in 1D . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Discretization of the di�erential equation . . . . . . . . . . . . . . . 192.3.2 Discretization of the jump conditions . . . . . . . . . . . . . . . . . . 212.3.3 One system combining the di�erential equation and jump conditions 262.4 EJIIM and discrete deltas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 Convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.1 Convergence for the Poisson problem . . . . . . . . . . . . . . . . . . 292.5.2 Convergence for singular sources with third order corrections . . . . 302.5.3 Convergence for piecewise constant coe�cients . . . . . . . . . . . . 302.6 Schur complements and integral equations . . . . . . . . . . . . . . . . . . . 352.6.1 Solving the discrete systems . . . . . . . . . . . . . . . . . . . . . . . 352.6.2 EJIIM and integral equations . . . . . . . . . . . . . . . . . . . . . . 362.6.3 Discretization of the Dirichlet problem and EJIIM . . . . . . . . . . 372.7 General boundary conditions and Liouville transformation . . . . . . . . . . 392.7.1 EJIIM for general boundary conditions . . . . . . . . . . . . . . . . 392.7.2 Liouville transformation for discontinuous coe�cients . . . . . . . . 40Chapter 3: EJIIM in 2D 423.1 EJIIM for the Laplacian and for cross derivatives . . . . . . . . . . . . . . . 423.2 Interface properties and conversion of jumps to Cartesian coordinates . . . 453.3 The three basic elliptic problems . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Remarks on numerical examples . . . . . . . . . . . . . . . . . . . . . . . . 473.4.1 Example: In uence of approximate interfaces . . . . . . . . . . . . . 473.5 Poisson problems with singular sources . . . . . . . . . . . . . . . . . . . . . 483.5.1 Jump conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5.2 Convergence proof for singular sources . . . . . . . . . . . . . . . . . 513.5.3 Example: EJIIM for singular sources . . . . . . . . . . . . . . . . . . 543.6 Irregular domain and discontinuous coe�cient problems . . . . . . . . . . . 553.6.1 Irregular domain problems . . . . . . . . . . . . . . . . . . . . . . . . 553.6.2 Discontinuous coe�cient problems . . . . . . . . . . . . . . . . . . . 563.6.3 Discretization of solution{dependent jumps . . . . . . . . . . . . . . 583.7 Connection between EJIIM and potential theory . . . . . . . . . . . . . . . 633.7.1 Solving the linear systems . . . . . . . . . . . . . . . . . . . . . . . . 633.7.2 Fundamental solution and Green's function . . . . . . . . . . . . . . 633.7.3 Single and double layer potentials . . . . . . . . . . . . . . . . . . . 643.7.4 Boundary Integral Methods . . . . . . . . . . . . . . . . . . . . . . . 643.7.5 EJIIM as a Boundary Integral Method . . . . . . . . . . . . . . . . . 653.8 Neumann boundary conditions and �xed values at interior grid points . . . 673.8.1 Example: \Exterior" Dirichlet BVP . . . . . . . . . . . . . . . . . . 683.8.2 Example: Groundwater Flow Problem . . . . . . . . . . . . . . . . . 683.8.3 Example: Local Re�nement . . . . . . . . . . . . . . . . . . . . . . . 713.9 Liouville transformation for variable discontinuous coe�cient problems . . . 743.9.1 Jump conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.9.2 Example: Liouville transformation for discontinuous coe�cients . . 753.10 Application to crack problems . . . . . . . . . . . . . . . . . . . . . . . . . . 783.10.1 Jump conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.10.2 Example: Inclusions that are almost cracks . . . . . . . . . . . . . . 82Chapter 4: Numerical methods for Electrical Impedance Tomography 854.1 Experience with the IIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2 Numerical stability of EJIIM and FIIIM . . . . . . . . . . . . . . . . . . . . 884.2.1 Example: Composite material problem . . . . . . . . . . . . . . . . . 884.2.2 Example: Stability of EJIIM as a forward solver for inverse problems. 904.3 Recovery of perturbations of an interface in EIT via linearization . . . . . . 934.3.1 The mathematical formulation of the problem . . . . . . . . . . . . . 944.3.2 The linearized integral relation formula . . . . . . . . . . . . . . . . 944.3.3 Analytic solutions for circular interfaces . . . . . . . . . . . . . . . . 984.3.4 Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.3.5 Remarks on numerical results . . . . . . . . . . . . . . . . . . . . . . 1024.3.6 Solving the forward problem with a Boundary Integral Method . . . 1024.3.7 Examples: Pure frequency input . . . . . . . . . . . . . . . . . . . . 1034.3.8 Examples: Dipole input . . . . . . . . . . . . . . . . . . . . . . . . . 106ii

Chapter 5: Conclusion and Outlook 1105.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Bibliography 113

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LIST OF FIGURES2.1 One-sided approximation of u by a quadratic polynomial. . . . . . . . . . . . 242.2 Two discrete deltas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Interface and mesh geometry near a lattice point (xi; yj). . . . . . . . . . . . 443.2 a) Solution and b) error for Example 3.4.1 computed on a 40� 40 mesh. . . 483.3 Exact solution for Example 3.5.3 on a 40� 40 mesh. . . . . . . . . . . . . . 533.4 a) Truncation error and b) error without third order corrections. . . . . . . 533.5 a) Truncation error and b) error including third order corrections. . . . . . 543.6 Anchor points and selected complete stencils. . . . . . . . . . . . . . . . . . 593.7 Possible stencils permitted by the algorithm. . . . . . . . . . . . . . . . . . . 603.8 Solution for Example 3.8.1, computed on a 80� 80 mesh. . . . . . . . . . . 693.9 The geometry of the �rst groundwater ow problem in Example 3.8.2. . . . 693.10 Computed solution and sparsity structure of the system in Example 3.8.2. . 703.11 a) Computed solution and b) mesh, interfaces and contour lines for a problemwith 25 objects of very high contrast (1000). . . . . . . . . . . . . . . . . . . 713.12 Solution for Example 3.8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.13 Error on a) the partially re�ned grid and b) the uniformly �ne grid. . . . . 733.14 Solution ~u for Example 3.9.2 with b = 10, computed on a 40� 40 mesh. . . 763.15 a) Solution u with b = 10 and b) u with b = �3. . . . . . . . . . . . . . . . . 773.16 The crack � between the regions � and +. . . . . . . . . . . . . . . . . . 793.17 Contour lines for a thin, resistive ellipse. . . . . . . . . . . . . . . . . . . . 833.18 More contour lines for a thin, resistive ellipse. . . . . . . . . . . . . . . . . 833.19 Contour lines for a thin, conductive ellipse. . . . . . . . . . . . . . . . . . . 843.20 More contour lines for a thin, conductive ellipse. . . . . . . . . . . . . . . . 844.1 Solutions found with the IIM on a uniform grid with mesh-width h = 0:1. . 874.2 Solution and di�erence between solutions found with the IIM. . . . . . . . . 874.3 Exact solutions for Example 4.2.1. . . . . . . . . . . . . . . . . . . . . . . . 894.4 Shape of the interface and solution. . . . . . . . . . . . . . . . . . . . . . . . 914.5 Response surfaces found with a) EJIIM and b) FIIIM. . . . . . . . . . . . . 914.6 Cross sections through the response surfaces. . . . . . . . . . . . . . . . . . 924.7 The interface �0 and the perturbed interface �. . . . . . . . . . . . . . . . . 934.8 Geometry for Example 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.9 Harmonic resulting from Neumann input cos(3�) and potential for � = 5. . 994.10 Potential and responses for several values of �. . . . . . . . . . . . . . . . . 1004.11 True and recovered shifted circle. . . . . . . . . . . . . . . . . . . . . . . . . 104iv

4.12 True and recovered perturbation as a function of �. . . . . . . . . . . . . . . 1044.13 True and recovered perturbation for a rosetta interface. . . . . . . . . . . . . 1054.14 True and recovered perturbation for single dipole input. . . . . . . . . . . . . 1074.15 True and recovered perturbation for 8 dipole input. . . . . . . . . . . . . . . 1084.16 Comparison of single dipole recovery with 8 dipole least squares recovery. . . 109

v

LIST OF TABLES3.1 Numerical results for \interior" Dirichlet BVP. . . . . . . . . . . . . . . . . 483.2 Numerical results for a singular source term. . . . . . . . . . . . . . . . . . 553.3 Numerical results for \exterior" Dirichlet BVP. . . . . . . . . . . . . . . . . 683.4 Results for Example 3.9.2 with b = 10, C = 0:1. . . . . . . . . . . . . . . . . 783.5 Results for Example 3.9.2 with b = �3, C = 0:1. . . . . . . . . . . . . . . . 784.1 Results for Example 4.2.1 with � = 5000. . . . . . . . . . . . . . . . . . . . . 904.2 Results for Example 4.2.1 with � = 1=5000. . . . . . . . . . . . . . . . . . . 904.3 Comparing an analytic solution with the approximation computed with Bryan'sBoundary Integral code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4 Circles of radius 1=3 centered at various points near the origin, computedwith pure frequency input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.5 Various input (k) and rosetta (l) frequencies, for � = 0:02 . . . . . . . . . . 1064.6 Various input (k) and rosetta (l) frequencies, for � = 0:05 . . . . . . . . . . 1064.7 Circles of radius 1=3 centered at various points near the origin, computedwith dipole input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.8 Results using measurments for one and eight dipoles for various rosetta fre-quencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109vi

ACKNOWLEDGMENTSI would like to thank my advisor, Professor Ken Bube, for his encouragement andhelp throughout my �ve years at the University of Washington, and in particularhis meticulous reading of this dissertation. I am also grateful to Professor RandyLeVeque for much helpful and enlightening interaction, which sometimes providedthe all-important view from a di�erent angle, and for helpful suggestions about thepresentation of the material in this dissertation.Professor LeVeque and his former student Zhilin Li developed the original Im-mersed Interface Method and encouraged me to participate in its development. Car-los Tolmasky is largely responsible for my interest in inverse problems, and hascollaborated with me on using the original IIM as a forward solver (x4.1) and on therecovery of perturbations of a circular interface via linearization (x4.3).This work was supported in part by National Science Foundation Grants DMS-9303404 and DMS-9626645 and Department of Energy Grants DE-FG06-93ER25181and DE-FG03-96ER25292, and by a McFarlan Fellowship granted for outstandingperformance as a graduate student by the Department of Mathematics, Universityof Washington.I thank Emad Alfar, Jos�e Mart��nez-Morales and Carlos Tolmasky, for being thefriends that they are. Most important for my success has been the never{doubtingsupport and belief in me by my family and my �anc�ee Barbara.vii

Chapter 1INTRODUCTIONStandard �nite di�erence approximations fail when applied to non-smooth functionsbecause the Taylor expansions that they are based on are not valid. But many applicationslead to non-smooth solutions, for example in electrostatics problems on composite materials,or uid ow through inhomogeneous materials. We will call the location of abrupt changesin material properties interfaces. Such problems can be dealt with by adaptive methods,for example by �nite element methods, where element boundaries may be chosen to coincidewith the interface, or by boundary-�tted �nite di�erence schemes. Considerable e�ort inconstructing a grid has to be spent in these cases.Finite di�erence methods on Cartesian grids on rectangular domains and in particularthe Immersed Interface Method (IIM)1 are concerned with problems where this grid con-struction is not a�ordable or is impractical due to complex geometry, especially in threedimensions (3D), or where the location of the discontinuity is either not �xed, for exam-ple in moving interface problems, or to be determined, as in inverse problems or in designproblems. Another reason to use Cartesian grids is the availability of fast solvers on arectangular domain R, which one would like to take advantage of even on some irregulardomain � R. A non-smooth extension of u from to R makes the boundary of theirregular domain @ an interface inside R.The most general problem we consider in the �rst part of this thesis is the numericalsolution of the elliptic (forward) problem, where is a bounded, possibly multiply connecteddomain with smooth boundary2. Given � (� � � > 0), �, f , �, � and �, �nd u, where thesolution u satis�es r � (�ru) + �u = f in ; (1.1)�u + �u� = � on @: (1.2)By U we denote the solution to the discrete approximation to this problem. Possiblecomplications that we allow are:� f may be singular;� � and � may be discontinuous;� boundaries and interfaces need not be aligned with the uniform Cartesian grid (ofmesh-width h) on which we �nd the solution U .1 The IIM was introduced by Li and LeVeque [21] in 1994.2 The outward normal direction on the boundary @ is denoted by �.

2In the presence of these complications, we describe how to discretize the elliptic partialdi�erential equation (1.1) with boundary conditions (1.2), and how to solve the resultingsparse systems of linear equations. We call our modi�cations and extension (to generalboundary conditions) of the IIM the Explicit Jump Immersed Interface Method (EJIIM).Problems where � is discontinuous, � and f are continuous (or zero) arise for exampleat the interface between two materials with di�erent di�usion parameters in steady stateheat di�usion or electrostatic problems. Other potential applications are time-dependentparabolic equations where the elliptic part of the equation has the form (1.1). The coe�-cients might be �xed but discontinuous, for example for heat conduction in inhomogeneousmaterials, or the movement of a free boundary might be governed by other equations cou-pled to the parabolic equation. A very good overview of applications where these problemsarise and also other approaches to solve these problems can be found in the introductionof LeVeque and Li [21]. Our own presentation will stress the similarities between di�erentproblems of type (1.1) { (1.2), and take an abstract \black box" point of view, where we donot make any assumptions on the original application the elliptic problem came from, oron how the solution will be used. On the other hand, the \output" of our method is rich inthe sense that we can not only provide the solution on the grid, but also the jumps alonginterfaces and estimates for values of the solution and derivatives of the solution all alongthe boundaries and interfaces, which can be very useful for applications.The design of EJIIM wants to achieve the following qualities.� The solution U is found with high accuracy (second order, error O�h2� as we re�nethe grid) for a �xed elliptic problem. This is simply inherited from the IIM.� The solution U depends in a stable way on changes of geometry and coe�cients for�xed mesh-width h (for use in moving interface problems, inverse problems, etc.), evendown to sub-grid length scales.� The solution U can be found e�ciently even for variable coe�cients (we may want tosolve the problem many times).When making design choices, we are guided by connections with integral equations andkeep in the back of our mind the following issues:� the discretization of parabolic and hyperbolic problems;� the discretization of nonlinear problems;� the discretization of systems of equations.All together, the extension of the IIM to this list would permit the treatment of the Navier-Stokes equations. In earlier work with Bube [41], we described the extension of the originalIIM to nonlinear problems in one dimension (1D), and show how it can be used to solveparabolic problems with time-independent interfaces in 1D.Solutions to (1.1) and (1.2) can also be used to solve elliptic inverse problems. The basicproblem of electrical impedance tomography, for example, is to infer electrical properties

3inside a domain from measurements of electrical properties on the boundary of that domain.The following short description of the problem is excerpted from Bryan's thesis [5]. When acurrent ux is applied on the boundary of a domain, the current ux density J inside thatdomain must satisfy r � J = 0 in ; (1.3)when there are no current sources inside . If the material satis�es Ohm's law J = �E,where E is the electrical �eld inside , and � is the conductivity of the material that ismade of, then r � (�E) = 0. We can write E = ru, where u is called the potential, andwe will assume that we can also measure u, the voltage on the boundary; that is, we haveknowledge of the input current ux (�u�) and output voltage (u) on @.An example of an elliptic inverse problem is the following: given all pairs of \measure-ments" (�u(i); u(i)� ), where there exist u(i) that solve r � (�ru(i)) = 0 and satisfy the pairsof boundary conditions3 u(i) = �u(i) on @ and u(i)� = u(i)� on @ (assuming we know �on @), �nd the conductivity � in the interior of . This is a notoriously di�cult problem,even when � is smooth. Usually, only a �nite set of M measurements (and technically, onlyon a �nite set of points on the boundary) are available, and the problem can be formulatedas follows: �nd � such that@@�u(�; �u(i)) = u(i)� on @; for i = 1; 2; : : : ;M; (1.4)where u(�; �u(i)) is the solution of the Dirichlet problem with coe�cient � and u = �u(i) on@. In the case that � is discontinuous, the IIM can be used to �nd an approximation Ufor the given �. By using di�erent trial � and applying �nite di�erences to U , the IIM inprinciple could be used to �nd approximations to the solution � of this inverse problem.In the second part of this thesis we investigate the applicability of EJIIM as a forwardsolver for a particular inverse problem that requires �nding the location of discontinuitiesof the coe�cient �. Such inverse problems arise for example when trying to locate objectsof high conductivity \buried" in material of lower conductivity. Instead of focusing onthe solution to one elliptic problem, we now have to compare solutions to similar ellipticproblems. Instead of re�ning the mesh, we are now interested in the worst case behaviorfor a particular �xed grid, and that grid should be as coarse as possible.Suppose we have determined an initial guess for the location of an object by usingEJIIM on a coarse grid. Instead of using �ner and �ner grids to improve on this guess,which requires more and more computations for the solution of each forward problem, it isuseful to have another method that can do the \�ne tuning". Using computer code writtenby Bryan [5] to \simulate" the problem, i.e. to create measurement data, and following amethod by Kaup and Santosa [18], we assume we have a good initial guess for the objectlocation. We determine its precise shape by linearizing about the initial guess and then�nding the Fourier coe�cients of the perturbation. Joint work with Tolmasky [39] onthis problem, for the special case of a circular domain with linearization about a circularinterface, is described in x4.3.3 both special cases of (1.2)

41.1 Historical background of the IIMFoundations relevant for our own work were laid already more than twenty years ago bycapacitance matrix methods, that extend the solution from an arbitrary domain to arectangular domain R using arti�cial values, and introducing jump conditions. This allowsthe use of fast Poisson solvers (surveyed in [37]) for the problem4 �u = f in R, u = � on@R for arbitrary regions in the plane; see for example the papers by Buzbee, Dorr, Georgeand Golub [6] and Proskurowski and Widlund [35]. Concus and Golub [9] use the Liouvilletransformation in fast solvers for variable coe�cient problems, including an application topiecewise constant coe�cients.Another predecessor is Peskin's Immersed Boundary Method (IBM) [33, 34, 40]. TheIBM is used originally to solve the Navier-Stokes equations, where a boundary models theheart wall, with uid contained inside. By extending the uid to a rectangle containingthe original domain in its interior, Peskin can model the boundary conditions with forcesexerted on the uid. The heart wall becomes an interface. Peskin's key idea that we wantto mention here is his modeling of these (at that point known) forces with a \discrete deltafunction", a nice way to distribute the appropriate amount of force from an interface tothe grid, only \correcting" the �nite di�erences for points in a small neighborhood of theinterface. By a \singular sources problem" we will mean a Poisson equation with singularsources, which may be deltas or dipoles. For a delta with strength c(s), along the interfaceX(s), parameterized (with positive orientation) by arclength s 2 [0; T ], u is a distributionthat satis�es5 �u = Z T0 c(s)�(x�X(s)) ds in R; (1.5)u = � on @R: (1.6)This explicitly expresses the fact that u is continuous across X , but has a discontinuity ofmagnitude c in the normal derivative across X . The drawbacks of Peskin's discrete deltaapproach are that it is limited to delta singularities (i.e. continuous solutions) and that it isonly �rst order accurate. For more details on the IBM and discrete deltas in relation withtruncation errors and the IIM, see x2.4, [4], [21] and [42].Mayo [29] extends the solution to the Dirichlet problem on an irregular domain �u = 0 in ; (1.7)u = � on @; (1.8)to a rectangle R so that it has a jump across the embedded boundary. The jump isdetermined by solving an integral equation on the boundary to express the solution asa so-called double-layer potential, ensuring that the limit of u approaching the originalboundary @ from inside is the desired value. Another integral equation is used to �ndthe appropriate boundary values �R to be used on @R. For a dipole with strength c(s)4Other authors also write r2u = f in R.5 See page 13 for the de�nition of �(x�X(s)).

5along the interface @ = X(s), parameterized by arclength s 2 [0; T ], where �(s) is theoutward unit normal to X at s, u is a distribution that satis�es6�u = Z T0 c(s)��(s)(x�X(s)) ds in R; (1.9)u = �R on @R: (1.10)This explicitly expresses the fact that u has a jump of magnitude c across X , and thenormal derivative is continuous across X . With known jumps, Mayo corrects the �nitedi�erences for the Laplacian in a neighborhood of the interface by adding corrections on theright hand side of the linear system, so that the resulting discrete solution has the correctdiscontinuities, and the solution is second order accurate. She also uses these solutionsto solve the Poisson problem on the irregular domain. We will refer to (1.7){(1.8) as the\irregular domain" problem. Mayo e�ectively converts the irregular domain problem to asingular source problem by �nding the correct values for the jumps, corresponding to herextension of u to R.Beyer and LeVeque [4] use Mayo's truncation error point of view to analyze Peskin'sdiscrete delta approach. In contrast to our earlier examples, they also consider the casewhen the source strength is not known when setting up the �nite di�erences, but dependson the solution. This leads them to view the discrete delta also as an interpolation operator,that can be used to determine the jump strengths at the interface (a function it also hasin Peskin's IBM). Beyer and LeVeque's main interest is in parabolic equations. They �ndthat even if the source strength and location depend on time as well, they are able to �ndcorrections to make the computed solutions second-order accurate. All results in [4] arefound in 1D.Li and LeVeque [21] originally set out to �nd a second-order accurate version of Pe-skin's �rst-order accurate discrete delta approach in higher dimensions. To this day, to ourknowledge this can not be accomplished. Li and LeVeque's method, the Immersed InterfaceMethod (IIM, after Peskin's IBM), can treat variable discontinuous coe�cient problemsr � (�ru) + �u = f in R; (1.11)u = � on @R: (1.12)We will call problems of this type \discontinuous coe�cient problems". The IIM followsMayo's approach in that the truncation error point of view provides the insight. TheIIM improves on Peskin's approach both in accuracy and scope of problems; discontinuouscoe�cients may be combined with singular sources (f as in (1.5) or (1.9)), i.e. sources maybe deltas and/or dipoles, and on all these problems the IIM is numerically seen to be secondorder convergent as the mesh is re�ned. As in the IBM, for singular sources the IIM onlyrequires corrections on the right hand side of the linear systems, hence fast Poisson solverscan be used7.6 See page 13 for the de�nition of ��(x�X(s)).7 In x2.4, we will see that in this case in 1D EJIIM agrees with the IIM in using the discrete delta resultingfrom the \hat" function that gives second order accuracy, but we could not �nd such an interpretation in2D.

6 For discontinuous coe�cients, the physics provide two jump conditions (potential u andconductive current ux �u� are continuous across the interface), the equation another (theequation holds on both sides of the interface in the traditional sense), and the remainingthree jump conditions (in 2D) needed for O(h) truncation error can be found taking tangen-tial derivatives. However, Mayo's way of computing the jumps with an independent integralequation does not apply (at least for not piecewise constant discontinuous coe�cients). Liand LeVeque build the local properties of the solution, as expressed in the jump conditions,into the method. They compute modi�ed �nite di�erence coe�cients near the interface, theso-called s. The idea of the IIM for discontinuous coe�cients is essentially di�erent fromthe corrections on the right hand side used by Peskin and Mayo, or the IIM for \singularsources", in treating variable coe�cients, jumps that depend on the solution (and on thecontrast in the coe�cients), and in using di�erent divided di�erences from the standardones for the Laplacian. This last fact prohibits the use of fast Poisson solvers8.The strong dependence of the error on the relative position of interface and grid foreven moderate contrast in the coe�cients (at least in the conductive case), and the lackof fast solvers, turn out to be slight problems su�ered by this \original" (discontinuouscoe�cient) IIM. In [21], the discontinuities are always \mild" in the sense that the contrast(quotient between limits of � on the sides of the interface) is always less than 10. Bothof these limitations are obstacles for the use in inverse problems and for the extension ofthe discontinuous coe�cient IIM to 3D, where �ne grids are needed for good quality ofsolutions, and the problems would become enormous.Li applies the IIM idea in his thesis [24] to 3D (with an implementation for analyticallyknown spherical interfaces; see also [26]), heat equations in 2D with �xed interfaces, Stokes ow with moving interfaces in 2D (see also [22]) and heat equations in 1D with movinginterfaces (extended to the nonlinear case in [27] in a di�erent fashion from our own work[41]).A breakthrough occurred with the advent of Li's paper A Fast Iterative Algorithm forElliptic Interface Problems [25], when he made the observation that for piecewise constantcoe�cients, the equation can be written as the Laplacian by dividing through by the co-e�cient, if in addition one observes the jump conditions across the interface. This leadsnaturally to the idea of splitting the s into the standard di�erences and corrections(still to be derived from the jump conditions) to the standard di�erences, which brings thediscontinuous coe�cient IIM closer to the IIM for singular sources, the IBM and Mayo'sapproach. The jumps in the function and its derivatives (and not jumps like [�un] as con-sidered earlier) turn out to be the crucial quantities. For known jumps Li �nds himself inthe same situation as Mayo and Peskin: he simply needs to �nd corrections to the righthand side of the linear system. He chooses to do this in the spirit of the original IIM,by selecting a point on the interface and developing Taylor expansions about it on bothsides of the interface. The other question is how to �nd the jumps. Lacking an integralequation like Mayo, Li uses the equations for the jumps based on local properties of the8At least for piecewise constant coe�cients, one could divide the equation by � and treat the s as in acapacitance matrix method, as Laplacian + perturbation, which would allow fast Poisson solvers to beemployed | this is one of two major insights in [25].

7solution derived in [21]. However, he decides to compute the jumps only at a �xed set ofcontrol points and to interpolate intermediate values, and is able to express jumps in secondderivatives as derivatives of jumps in the solution and �rst-order derivatives. The big ad-vantage of this is that it keeps the number of auxiliary unknowns low and �xed, even as themesh is re�ned. Ultimately, a �ner set of control points is needed to represent the interfacemore accurately, but for smooth interfaces cubic splines do a terri�c job in (fourth order)interpolation. This results in a linear system in the original unknowns (solution values onthe grid) and auxiliary unknowns (jumps across the interface), with the standard �ve-pointdiscretization of the Laplacian as the biggest of four blocks in the matrix. Eliminating theoriginal unknowns results in a small (non-symmetric) system for the auxiliary unknowns, aSchur complement, that can be solved quite e�ciently with an iterative method (GMRES;see [36]). Each iteration requires the application of a fast Poisson solver on the rectangle,but only a few iterations are needed. A fast version of the IIM for discontinuous coe�cientsis born! But in addition to that, by making the coe�cients of the corrections to the stan-dard di�erences small (essentially signi�cantly widening the original six-point stencil, andensuring that coe�cients decay in magnitude away from the diagonal by using a weightedleast squares method), Li �nds a much more stable version of the IIM that does better thanthe IIM on large contrast problems both in the resistive and conductive case. We will referto Li's method from [25] as the Fast Iterative IIM (FIIIM). In our numerical examples inChapter 3, we will compare our own method against this improved version of the IIM.However, there still remain many questions. Can the FIIIM be extended to variablecoe�cients? What were the exact connections with earlier work? Is it possible to comeup with a convergence theory? The main obstacle there is that the new systems (derivedvia IIM) fail to satisfy a discrete maximum principle, a basic standard ingredient for con-vergence proofs for �nite di�erence schemes for elliptic problems. On the other hand, innumerical examples the FIIIM and IIM never fail to converge ultimately with second order,albeit sometimes requiring very �ne meshes before this can be observed. Also open are thequestions of extension to nonlinear equations, and to systems of equations. Some of thesequestions will be answered in the following chapters.1.2 Other advances with the IIMLi went on to replace his original spline interface with a level set method, which allowedhim to compute moving interface problems successfully even in the presence of changes inthe topology of the interface [17]. The slow solvers for the original IIM were studied andimproved by Adams with a multigrid approach [1]. Yang [42] extended the IIM \back" toMayo's earlier problem on irregular domains and combined her IIM for Boundary ValueProblems (IIMB) with the original IIM to treat uid ow problems in complicated geome-tries with discontinuous permeabilities, using Adams' multigrid [1] as the fast solver forvariable coe�cient problems. LeVeque and Zhang [23] and Zhang [43] showed how to usethe IIM for hyperbolic systems of partial di�erential equations with discontinuous coe�-cients arising from acoustic or elastic problems in heterogeneous media. The latter twouses of the IIM were successfully combined with LeVeque's CLAWPACK software package[20]. Calhoun is currently extending the IIM to a stream-function vorticity formulation of

8incompressible ow in 2D.1.3 Experience with nonlinear problemsThe �rst question we became interested in was the extension of the IIM to nonlinear equa-tions. We decided to focus on the simplest case, \elliptic" and parabolic problems in 1Dwith �xed interfaces, and applied our results to tra�c ow with ramps, both in the steadystate and time-dependent case. The main conclusion is that even though the equations arenonlinear, the coe�cients ( s in the spirit of the original IIM) can still be found by solvingsmall linear systems|at least if the solution is continuous. This work together with Bubeis published as [41]. At the same time, Li [27] also extended the IIM to nonlinear problemsin 1D, with a focus on moving interfaces and applications in ice-melting and glaciation(Stefan problem). Toward the end of our work in [41] it became apparent that ideas fromthe FIIIM could possibly lift the restrictions on the jump conditions in the nonlinear case.Simultaneously we became interested in inverse problems, so that the nonlinear case stillawaits its extension to general jumps, 2D and 3D.1.4 The EJIIM approach to elliptic partial di�erential equationsNumerical solvers for elliptic interface problems are fairly complex, and so our �rst step isto break the IIM into its major components, and work on them individually.1. Jump conditions { Partial di�erential equations, Di�erential Geometry.2. The interface and mesh geometry { Splines.3. The �nite di�erences { EJIIM.4. The solution of the linear systems { Numerical linear algebra.An IIM is de�ned by working with jump conditions on a Cartesian grid, based on truncationerror analysis, i.e. the �rst item. In principle, the choice made for each of the other partsdoes not in uence the choice of methods for the other components. For example, forthe geometry, both splines and level set methods have been used successfully in 2D, andpiecewise linear approximations are conceivably the easiest to use in 3D (at the cost ofrequiring very many points on the interface). For the linear systems, a variety of SOR, ADI,fast Poisson solvers and multigrid methods have been used, and could even be combined.Our changes to the IIM lie in the �nite di�erences.The Explicit Jump IIM is an idea to derive �nite di�erence equations which can be solvede�ciently via a Schur-complement method. The basic, quickly solvable problem of choiceis the Dirichlet problem on a rectangle, or Poisson problem on a rectangle. Any problemunder consideration is rewritten as a perturbation of this problem, then discretized (withthe possibility of discontinuities in the solution and its derivatives) via EJIIM. Usually, we�rst solve by a Schur-complement for the auxiliary variables introduced for the jumps. Thenwe use the values for these auxiliary variables to �nd the solution on the grid. Peculiar to

9our approach is the fact that the variables are jumps in the coordinate directions, and thatfor problems on irregular domains we extend the solution to be uniformly zero outside thedomain.In Chapter 2, we strip o� all the di�culties with the di�erential geometry of the interfaceby looking at the 1D problem. We develop a theory for our �nite di�erences for the irregulardomain and discontinuous coe�cient problems in 1D. One key insight is Li's idea of reducingthese problems to sequences of \singular source" problems, with jump strengths to bedetermined together with and from the solution. Viewed like this, the �nite di�erencesactually decouple into the questions of distributing singular sources from the interface tothe mesh, and determining what the source strengths (jumps) are. The jumps in a functionand its derivatives are identi�ed as the essential quantities that need to be known across aninterface where the coe�cients are discontinuous (compare Li and LeVeque's [�un] conditionin [21], for example), in order to correct �nite di�erences. The approach of writing separateequations for these jumps is similar to Yang's approach for boundary value problems onirregular domains by embedding them in rectangular domains [42]. EJIIM works by focusingon the jumps in the solutions and its derivatives, rather than on �nding coe�cients of new�nite di�erences, like the IIM. In our formulation, the jump conditions for many di�erentproblems all turn out to only depend on limits on one side of the interface, and using thesewe do not need to compute s that are the trademark of the IIM [1, 21, 24, 25, 42, 43]. Inour method, s exist only implicitly, but could be constructed by eliminating the equationsfor the jumps from the extended linear systems9.The new contributions in Chapter 2 are� In x2.2, we derive the fundamental lemmas on how to correct �nite di�erences withknown jumps in 1D.� Li's division by � can be carried through for piecewise smooth coe�cients. The mainpoints in x2.3 are (i) the O�h2� discretization away from the interface, and O(h)discretization near the interface (Proposition 2.8, Lemma 2.17, needed in x2.5), (ii) theinterpolation qualities of the one-sided di�erence operators and (iii) the introductionof the jumps as auxiliary variables, leading to the extended system (2.52). Our ansatzfor the irregular domain problem, introducing jumps that extend the solution by zero,is di�erent from all the literature on this problem known to us.� The jump conditions we use require one-sided limits, necessitating di�erent operatorsfor the distribution of the singularity than for the estimation of the jump strength.The main point in x2.4 is the high quality of our interpolation operators when viewedin the framework of Beyer and LeVeque [4].� A \brute force" proof of second order convergence for the Poisson problem is given inx2.5.1 and then adapted for the piecewise constant coe�cient case (Theorem 2.23).Special care has to be taken because the truncation error near the interface is only9 As in the original IIM, our implicit s are void of intuitive meaning, while the factors we construct allhave such meaning, so we prefer to work with the extended system.

10 O(h). The most interesting fact here is that the bounds in the proof for piecewiseconstant coe�cients are independent of the geometry and the contrast in the coef-�cients, indicating that solutions on a �xed grid should have comparable errors fordi�erent interface locations and even di�erent coe�cients. This property is essentialfor the treatment of inverse problems.We also give a simpler proof of second order convergence for the singular sourcesproblem based on including one higher order correction term than usually done in theIIM (Theorem 2.22).� The jumps as auxiliary variables, together with the division by � introduced by Li in[25], allow several options in the numerical linear algebra, which are hinted at in x2.6.1.The manipulations on the linear algebra in turn allow viewing the IIM as discretiz-ing integro-di�erential equations, rather than di�erential equations (Lemma 2.40).Both \continuous" and discrete Green's kernels are of fundamental importance, andallow a new look at the standard way of solving a Dirichlet problem on a rectangleas discretizing the integral of the Green's kernel against a dipole supported on theboundary.� In x2.7 we treat the extension of the IIM to general boundary conditions. We also showhow the Liouville transformation (also known as Stefanesco transformation, a well-known transformation that changes a di�usion equation into a Schr�odinger equation;see [8]) can be extended to discontinuous coe�cients. This reduces the order of theperturbation of the Laplacian by one further order compared to the division by �described earlier in the chapter.In Chapter 3 we extend these ideas to elliptic equations in 2D. Now the geometry of theinterface becomes very important.� Based on the corrections found in 1D in x2.2, we describe how to correct di�erencesapproximations to the Laplacian and cross derivatives to second order in 2D in x3.1.This approach is essentially following Mayo [29], rather than the IIM.� Instead of deriving jump conditions based solely on local coordinates like in [21],we �nd it convenient to use the arclength of the interface as the basic underlyingparameter, and derive all needed interface properties from it in x3.2. This section alsohas the formulas to convert jumps in the normal and tangential directions into jumpsin the Cartesian coordinate directions.� In x3.3, we formulate the basic elliptic problems | singular sources, irregular domainand discontinuous coe�cient | and follow up in x3.4 with some remarks that applyto all numerical examples in 2D, including the in uence of using a cubic spline toapproximate the interfaces.� Poisson problems with singular sources are treated in x3.5. Jumps up to third deriva-tives allow the standard convergence proof based on a discrete maximum principle to

11carry through. A numerical example illustrates that the gain using these correctionsis not in error magnitude, but a smoother error distribution.� The next section x3.6 has jump conditions for embedding irregular domains and dis-continuous coe�cients. All jumps are either known or can be written as one-sidedlimits of the solution10. We derive operators that estimate such limits when appliedto a grid function. The idea then is that the di�erential equation can be discretizedjust as in the singular source case, but that the source strengths are related to thesolution via the above operators. The introduction of the jumps as auxiliary variablesmakes the connection transparent.� The double layer density and the single layer density of potential theory have the inter-pretation of being the jump in a function and its normal derivative. This is frequentlyused in interpreting or designing domain embedding approaches and, after some ma-nipulation with Green's kernels in x3.7, applies also to our non-standard extension ofthe equation and solution from some irregular domain to an enclosing rectangle. TheSchur-complements of the discretization correspond to integro-di�erential equations,either on the boundary or the whole domain.� In x3.8, Neumann boundary conditions and �xed values at interior grid points areput in a form that allows the use of fast Poisson solvers. They do not allow theinterpretation as jumps as in the previous cases. An \exterior" Dirichlet problem,an application to ground water ow, and a grid re�nement example illustrate whereNeumann boundary conditions and �xed values at interior points can be necessary.� In x3.9 we derive the new jump conditions after the Liouville transformation is ap-plied. Two examples illustrate that this is a very e�cient way of computing solutions,because the work performed is O(n lnn) just as for the Poisson problem.� In x3.10 we consider crack problems. The jumps across two almost parallel veryclose interfaces are treated as a jump across a single interface. Numerical examplessimulating cracks with narrow ellipses con�rm our derived jumps.10 For general jump conditions, one can always replace limits on the \+" side using u+ = u�+[u] etc. Thisresults in possibly implicit equations for the jumps, but they always use only limits from one side.

Chapter 2ANALYSIS OF THE EXPLICIT JUMP IIM IN 1DThe strong dependence of the error on the relative position of the interface for largecontrast in the coe�cients for the original IIM led to Li's idea of reducing the piecewiseconstant coe�cient problem to the Laplacian (with jump conditions across the interface)and correcting the standard �nite di�erence approximations in [25]. In this and thenext chapter we follow this newer approach rather than �nding new �nite di�erences nearthe interface ( s) as proposed for the original IIM [21] and also used in our extension tononlinear equations in [41] as well as in [1, 24, 42, 43]. Li uses s in [25] in order to �ndthe corrections, whereas we replace them with one-sided interpolation operators.Two basic questions arise. First, for known jumps across an interface, how should onecorrect the �nite di�erences for stencils with support on both sides of the interface? Thisshould depend only on the geometry and the jump values. And second, for particularinterface conditions, what are those jump values?For known jumps the original IIM reduces to introducing a discretization of a deltaand dipole on the right hand side. The basic idea of the Explicit Jump IIM (EJIIM) isto carry this further and treat the jumps in the coordinate directions in the solutionand its derivatives as additional unknowns, in order to reduce general interface conditionswith unknown jumps to the case of known jumps, similar to [25]. For given values of theseunknowns, a solution can be computed by applying the inverse of the standard matrix for thePoisson problem to a nonstandard right hand side. These subproblems can be solved quicklywith fast Poisson solvers. Our choice of jumps in the coordinate directions as variables isthe same as Mayo's in [29], but di�erent from [25]. It enables us to correct the di�erencessimply one dimension at a time, and gives better accuracy, since more information is used.Our approach is di�erent from Mayo's (for boundary value problems on irregular domains)and Li's (for discontinuous coe�cients) in how we �nd the jumps.In case the jumps are not known a priori, some relation can be derived between thejumps and the behavior of the solution near the interface. Our particular, very successfulapproach is to write explicit equations for the jumps using one-sided limits of solution andderivative values. The jump equations are then discretized to complete the discrete systemin the solution variables and jump variables. Using one{sided limits allows viewing Schur{complement equations as discretizations of integral equations. This provides more insightthan just the truncation error analysis that was the only analytic tool available for the IIM.Even though EJIIM is a general method to derive �nite di�erences for di�erential equa-tions with piecewise-smooth solutions, the examples we give in this chapter are linear ellipticproblems in one dimension, i.e. second order ODEs. This is not because numerical methods,or EJIIM in particular, are needed to solve these problems, but because the ideas are mosteasily explained in this setting. Since EJIIM corrects \one dimension at a time" in higherdimensions, where the di�erential geometry of the interfaces in uences the magnitudes of

13the jumps in the coordinate directions, a good understanding of this case is essential. Onthe other hand, in one dimension the geometry of the interface(s) (just points) is the sim-plest imaginable and puts the focus on the �nite di�erences. It also allows convergenceproofs for all three jump conditions that we consider here.The remainder of this chapter is organized as follows. x2.1 has the basic notation andspeci�es three sample problems by which we will explain the method. x2.2 gives the basicformulas to do �nite di�erences in the presence of discontinuities. In x2.3, we provide detailson how to discretize the di�erential equation and one-sided jump conditions. The nextsection x2.4 relates EJIIM to the discrete delta methodology of Peskin, as treated by Beyerand LeVeque in [4]. We present a convergence result for piecewise constant coe�cients forsingle-point interfaces in x2.5. The proof is interesting in not requiring the numerical schemeto satisfy a discrete maximum principle, the usual means by which such results are proved,and in providing error bounds that are independent of the geometry. x2.6 relates the Schur-complements that we form for e�cient computation with integral equations, and �nallyx2.7 has details on general boundary conditions and presents a way to reduce the generaldiscontinuous coe�cient di�usion equation to a Schr�odinger-type equation, by derivingjump conditions to perform the Liouville transformation for discontinuous coe�cients.2.1 Basic de�nitions and problemsWe consider an interface � 2 (0; 1) and two sub-domains � = (0; �) and + = (�; 1). Forour purposes, functions de�ned on � are actually smooth on [0; �], and functions on +are smooth on [�; 1].De�nition 2.1 (1D delta, dipole) Let � 2 (0; 1). The delta �(x� �) (centered at �) isthe distribution that satis�es R 10 �(x)�(x� �)dx = �(�) for any function � 2 C1([0; 1]).The dipole �0(x� �) (centered at �) is the distribution that satis�es R 10 �(x)�0(x� �)dx =��0(�) for any function � 2 C1([0; 1]).De�nition 2.2 (2D delta, dipole) We will need the 2D analogue as well, where the oneadditional issue is the orientation of a dipole. Let � 2 � R2, where is open and itsclosure compact. The 2-dimensional delta �(x� �) (centered at �) is the distribution thatsatis�es R �(x)�(x��)dx = �(�) for any function � 2 C1(). The 2-dimensional dipole��(x� �) (in the direction �, centered at �) is the distribution that satis�es R �(x)��(x��)dx = �� � r�(�) for any function � 2 C1().De�nition 2.3 (Jumps) By x! �� we mean that x! � with x < �, or more generallyx 2 �. We use the notations�� � limx!�� �(x); �+ � limx!�+ �(x);with the obvious extensions, for example u+x . To denote jumps in a function u and itsderivatives at �, we write [u(m)]� = (u+)(m)(x)� (u�)(m)(x); if � is understood, [:] = [:]�,and also [uxx] = [u(2)] etc. As usual, by u(0) we mean the function u itself.

14 To have speci�c problems in mind, we distinguish three boundary value problems (BVP)with Dirichlet boundary conditions:Example 2.1 Poisson problems with singular sources (named for analogy with higher di-mensions): uxx = f(x) + v�(x� �) + w�0(x� �) (2.1)u(0) = 0 (2.2)u(1) = 0: (2.3)By the above notation we indicate that u satis�es Poisson's equation on � and + and hasknown jumps [u] = w, [ux] = v. The values for w and v are arbitrary; from the equationwe derive [uxx] = [f ], [uxxx] = [fx] etc.This is essentially a steady state of a model like Peskin's as considered by Beyer and LeVeque[4]. We include it as the basic problem to which the others are reduced, and to point out thefundamental di�erence between this problem and the next two examples: here the equationsfor [u], [ux], [uxx] and [uxxx] are independent of u, which means that the corrections tothe equation are constants. Hence we don't need any further equations to determine thesource strengths. If we do not include third order corrections, i.e. the [uxxx] term, EJIIMagrees with the IIM for this problem. However, this coincidence occurs only in one spacedimension.Example 2.2 Problems with discontinuous coe�cients:ddx ��(x)du(x)dx �+�(x)u(x) = f(x) (2.4)u(0) = ul (2.5)u(1) = ur: (2.6)�, � and f may be discontinuous at �. We assume � is discontinuous at �, and (to coverthe general case) [u] = w, [�ux] = v for some known w and v, i.e. f may be singular. Forphysical problems, usually w = v = 0, i.e. u and �ux are continuous. From the equation wederive [�uxx] = [f ]� [�xux]� [�u], [�uxxx] = [fx]� 2 [�xuxx]� [�xxux]� [�ux]� [�xu] etc.The 2D analogue of this problem is considered by Li [25] and (combined with singularsources) by Li and LeVeque [21] and by Li [24]. For the many important applications whereit arises (in 2D and 3D), see the excellent introduction of [21].Example 2.3 BVP on irregular domains:uxx = f; (2.7)u(�) = u�; (2.8)u(1) = ur; (2.9)

15where the grid is not aligned with the left endpoint �. This generalizes to complex geometriesin higher dimensions. Instead, we make the ansatz that u � 0 and f � 0 on �, whichde�nes the jump conditions. We assume that f is smooth on +, with [u] = ul, [ux] = u+x ,[uxx] = u+xx and [uxxx] = u+xxx (also possible, from the equation, [uxx] = [f ] = f+ and[uxxx] = [fx] = f+x ).This is essentially the problem treated by Yang's IIM extension IIMB [42], but we extend udi�erently and consequently need to discretize di�erent jump conditions. Again, there aremany applications where the analogue of this problem in higher dimensions arises.We discretize all of the above problems on a grid on � [0; 1] with (n � 1) uniformlyspaced interior grid points. Unless noted otherwise, i will be the spatial discretization index;for example, xi = ih (i = 0; : : : ; n), where h = 1=n is the mesh width. Let u(x) denote thetrue solution of the di�erential equation evaluated at x, ui = u(xi) its restriction and Uithe value of the numerical approximation at xi (i 2 f1; 2; : : : ; n � 1g). E = Eh restricts afunction de�ned on to its values on interior grid points for a given h; U and Eu denote thevectors with components Ui and ui, respectively. We de�ne hi = xi � � (i = 1; : : : ; n� 1),and choose j so that xj � � < xj+1. For similarity with higher dimensions we will alsowrite h� for hj and h+ for hj+1. Note that for �xed � the index j depends on h, andjhj+lj < 2h for l = �1; 0; 1; 2, which gives hj+l = O(h) as h ! 0. By O(h) we denote anyquantity q(h) such that jq(h)j � Kh (as h goes to zero) for some constant K independentof h.2.2 Correcting �nite di�erences with known jumpsHow can we modify standard di�erences, for example the O�h2� approximationsux(xi) � u(xi+1)� u(xi�1)2h ; (2.10)uxx(xi) � u(xi+1)� 2u(xi) + u(xi�1)h2 ; (2.11)in the case that an interface cuts through the stencil used by the di�erences?2.2.1 Taylor Expansions for piecewise smooth functionsIn the case of nonzero jumps, across the interface � the approximations (2.10) and (2.11) arenot O�h2�. Similarly to an observation by Beyer and LeVeque [4] and following formulasin Mayo [29], we expand u in a Taylor series about a grid point on the other side of theinterface. We start with a lemma where the interface occurs at � = 0, which lies betweentwo arbitrarily located grid points (labeled here h� and h+) with grid spacing h.Lemma 2.1 Assume u� 2 Cl+1[�1; 0] and u+ 2 Cl+1[0; 1]. Let � = 0 andu = (u�(x) for x � 0;u+(x) for x > 0:

16Then for h < 1 and any h� and h+ = h� + h satisfying �h < h� � 0 < h+,�����u(h+)� lXk=0 hkk! u(k)(h�)� lXk=0 (h+)kk! [u(k)]����� � K(l+ 1)!hl+1;where K = max� maxx2[�1;0] ���(u�)(l+1)(x)��� ; maxx2[0;1] ���(u+)(l+1)(x)����is sharp.Proof. A Taylor expansion for u+ at 0 yieldsu(h+) = lXm=0 (h+)mm! (u+)(m)(0) + (u+)(l+1)(�l+1)(l+ 1)! (h+)l+1;for some �l+1 2 (0; h+). Also, (u+)(m)(0) = (u�)(m)(0) + [u(m)], and henceu(h+) = lXm=0 (h+)mm! n(u�)(m)(0) + [u(m)]o+ (u+)(l+1)(�l+1)(l+ 1)! (h+)l+1:Taylor expansions for u� at h� yield for m = 0; 1; : : : ; l(u�)(m)(0) = l�mXj=0 (�h�)jj! (u�)(j+m)(h�) + (u�)(l+1)(�m)(l�m+ 1)! (�h�)l�m+1;for some �m 2 (h�; 0). Since u agrees with u� in a (left-sided) neighborhood of zero,u(h+) = lXm=0 (h+)mm! 8<:[u(m)] + l�mXj=0 (�h�)jj! u(j+m)(h�)9=;+ (u+)(l+1)(�l+1)(l+ 1)! (h+)l+1 + lXm=0 (h+)mm! (u�)(l+1)(�m)(l�m+ 1)! (�h�)l�m+1:Changing the order of summation (k = m+ j, so j = k �m),u(h+)� lXk=0((h+)kk! [u(k)] +( kXm=0 (�h�)k�m(k �m)! (h+)mm! )u(k)(h�)) =l+1Xm=0 (h+)mm! (u�)(l+1)(�m)(l�m+ 1)! (�h�)l�m+1;where u� = u� for m = 0; 1; : : : ; l and u� = u+ for m = l + 1. Finally, the desired resultfollows from the binomial theorem, the facts that h = h+ � h�, 0 � �h� < h, 0 < h+ � h,and simple estimates. The piecewise linear example u = jxj with l = 0 shows that K issharp.

17Remark 2.2 On the other side of the interface, the expansion formula isu(h�) = lXk=0 (�h)kk! u(k)(h+)� lXk=0 (h�)kk! [u(k)] + O�hl+1� :The \�" for the second sum is introduced by the \non-symmetric" de�nition of [u(k)].If [u] 6= 0 and h� = 0, i.e. � = xj , we have to make a choice whether the discrete variableUj should approximate u� or u+. By the convention that xj � � < xj+1, this choice ismade as Uj � u�. In higher dimensions, we will allow a choice of which limit should beapproximated. It is easiest to think of the interface � = xj to either lie in the interval[xj�1; xj ] or [xj ; xj + 1], and give the coordinate of the interface relative to the interval.For example, if � = xj and we want Uj � u�, then the interval is [xj ; xj+1] with h� = 0,h+ = h. If we want Uj � u+, then the interval is [xj�1; xj ] with h� = �h, h+ = 0.2.2.2 Correcting centered di�erencesFrom the expansions in Lemma 2.1 we derive the important explicit jump �nite di�erenceapproximations in correction form for derivatives of u. The corrected forms for secondderivatives need jumps in the third derivatives of u, while in the usual centered di�erencesthe third derivatives cancel out by symmetry. We apply Lemma 2.1 to the situation wherethe interface lies at an arbitrary location between two grid points xj and xj+1 = xj + h.Lemma 2.3 (jump-corrected di�erences) Let xj � � < xj+1, h� = xj � � and h+ =xj+1��. Suppose u 2 C4[xj�h; �)\C4(�; xj+1+h], with derivatives extending continuouslyup to the boundary �. Then the following approximations hold to O�h2�:ux(xj) � u(xj+1)� u(xj�1)2h � 12h 2Xm=0 (h+)mm! [u(m)]; (2.12)ux(xj+1) � u(xj+2)� u(xj)2h � 12h 2Xm=0 (h�)mm! [u(m)]; (2.13)uxx(xj) � u(xj+1)� 2u(xj) + u(xj�1)h2 � 1h2 3Xm=0 (h+)mm! [u(m)]; (2.14)uxx(xj+1) � u(xj+2)� 2u(xj+1) + u(xj)h2 + 1h2 3Xm=0 (h�)mm! [u(m)]: (2.15)Proof. (i) Formulas (2.12), (2.14): By shifting the coordinate system by �� we are in theregime of Lemma 2.1. The formulas follow by using Lemma 2.1 for u(xj+1) = u(h+) andthe usual Taylor expansions for u(xj) = u(h�) and u(xj�1) = u(h� � h). The signs onthe corrections are the same because u(xj+1) occurs with a positive sign in both di�erenceapproximations.(ii) Formulas (2.13), (2.15): The formulas follow by using Remark 2.2 for u(xj) = u(h�)and the usual Taylor expansions for u(xj+1) = u(h+) and u(xj+2) = u(h+ + h). The signs

18on the corrections di�er because u(xj) occurs in (2.15) with a positive sign but with anegative sign in (2.13).The same ideas extend to multiple interfaces.Lemma 2.4 (multiple corrections) a) Let xj�1 � �1 < xj � �2 < xj+1, h�1 = xj�1 ��1, h+1 = xj � �1, h�2 = xj � �2 and h+2 = xj+1 � �2. Suppose u 2 C4[xj�1; �1) \C4(�1; �2) \ C4(�2; xj+1], with derivatives extending continuously up to the boundaries ofthe subintervals. Then the following approximations hold to O�h2�:ux(xj) � u(xj+1)� u(xj�1)2h � 12h 2Xm=0 (h�1 )m[u(m)]�1 + (h+2 )m[u(m)]�2m! ; (2.16)uxx(xj) � u(xj+1)� 2u(xj) + u(xj�1)h2 + 1h2 3Xm=0 (h�1 )m[u(m)]�1 � (h+2 )m[u(m)]�2m! : (2.17)b) Let xj � �1 < �2 < xj+1, h�1 = xj��1, h+1 = xj+1��1, h�2 = xj��2 and h+2 = xj+1��2.Suppose u 2 C4[xj ; �1)\C4(�1; �2)\C4(�2; xj+1], with derivatives extending continuouslyup to the boundaries. Then the following approximations hold to O�h2�:ux(xj) � u(xj+1)� u(xj�1)2h � 12h 2Xm=0 (h+1 )m[u(m)]�1 + (h+2 )m[u(m)]�2m! ;ux(xj+1) � u(xj+2)� u(xj)2h � 12h 2Xm=0 (h�1 )m[u(m)]�1 + (h�2 )m[u(m)]�2m! ;uxx(xj) � u(xj+1)� 2u(xj) + u(xj�1)h2 � 1h2 3Xm=0 (h+1 )m[u(m)]�1 + (h+2 )m[u(m)]�2m! ;uxx(xj+1) � u(xj+2)� 2u(xj+1) + u(xj)h2 + 1h2 3Xm=0 (h�1 )m[u(m)]�1 + (h�2 )m[u(m)]�2m! :Proof. In either case a) or b) apply Lemma 2.1 twice.Remark 2.5 If � = xj in Lemma 2.3 or �1 = xj�1 or �2 = xj in Lemma 2.4, values of uand derivatives on u at these points indicate left sided limits.These approximations are extremely important because they can be used not just inone, but also in two and three space dimensions, where the di�erences are corrected \onedimension at a time". Each of the jumps can be approximated separately, using the jumpconditions depending on the problem as given in the examples. This simpli�es the analysisand implementation of the method.

192.3 EJIIM for \elliptic" equations in 1DLi derived a faster and more stable version of the IIM for elliptic equations with piecewiseconstant coe�cients in two space dimensions in [25]. We present two ways to extendthis approach to variable coe�cients, based on the Taylor expansions with jumps fromLemma 2.1. One of Li's two main changes is to divide the equation (2.4) by � to give theequivalent equation d2 udx2 + 1� d �dx d udx + �� u = f� : (2.18)In Li's case, the equation simpli�ed since for piecewise constant coe�cients d�=dx � 0 (awayfrom �). By equivalent, we mean that if u satis�es (2.4) on �[+, the boundary conditions(2.5)-(2.6) and the jump conditions stated in Example 2.2, then u also satis�es (2.18) with(2.5)-(2.6) and the same boundary and jump conditions, and vice versa. Examples 2.1 and2.3 are already in this form.The form (2.18) is important because it shows that we essentially treat a lower orderperturbation of a Laplace-type operator | which we may exploit theoretically and fornumerical purposes | and because it changes the derived jump condition from Example 2.2to [uxx] = �f��� ��x� ux�� ���u� ; (2.19)which reduces the order of the right hand side of the jump equation.The second way to extend Li's method is presented in x2.7.2. It can be treated numeri-cally in a similar way to (2.18), but has the advantage of reducing the perturbation of theLaplacian even one order further. This is particularly interesting when the coe�cient � isnot piecewise constant.2.3.1 Discretization of the di�erential equationWe discretize (2.18) near � using a discretization of the jumps given in x2.1 or as mod-i�ed in Lemma 2.10. We use auxiliary variables g � �g0; g1; g2; g3�T as place holders for�[u]; [ux]; [uxx]; [uxxx]�T and replace all derivatives on u in the ith equation according toLemma 2.3 if i 2 fj; j + 1g or by the standard centered di�erences (2.10) and (2.11) ifi 62 fj; j + 1g. We write (A1 + BA2 + K)U �g = F1; (2.20)where we use additional notation for second and �rst divided di�erences:A1 � 1h2 2666664 �2 11 �2 1.. . . . . . . .1 �2 11 �2 3777775 , A2 � 12h 2666664 0 1�1 0 1. . . . . .�1 0 1�1 0 3777775 , (2.21)

20K � diag (�(xi)=�(xi)), B � diag (bi), where bi � �x(xi)=�(xi). Also, F1 = F + B and = [0;1;2;3], whereF = �f(x1)�(x1) ; f(x2)�(x2) ; : : : ; f(xn�1)�(xn�1)�T ; (2.22)B = ��ulh2 + b1 ul2h; 0; : : : ; 0; �urh2 � bn�1 ur2h�T ; (2.23)0 = �0; : : : ; 0; 1h2 + bj2h;� 1h2 + bj+12h ; 0; : : : ; 0�T ; (2.24)1 = �0; : : : ; 0; h+h2 + bjh+2h ;�h�h2 + bj+1h�2h ; 0; : : : ; 0�T ; (2.25)2 = "0; : : : ; 0; h+22h2 + bjh+24h ;�h�22h2 + bj+1h�24h ; 0; : : : ; 0#T ; (2.26)3 = "0; : : : ; 0; h+36h2 + bjh+312h ;�h�36h2 + bj+1h�312h ; 0; : : : ; 0#T : (2.27)The indices of the two non-zero entries in all l (l = 0; 1; 2; 3) are j and j+1, i.e. the indicesof the two gridpoints closest to the interface; the product �g is the vector representationof the corrections to the centered di�erences in (2.12){(2.15).Remark 2.6 Our notation is a convenient way to group the terms corresponding to theirformal order in the di�erential equation, or magnitude in the approximation. The g termsubtracts out \just the right singularities".Remark 2.7 For Examples 2.1 and 2.3, the above matrices and vectors simplify consider-ably, since �i = bi = 0 for all i.Proposition 2.8 (Local Truncation Error) Away from �, that is for i 62 fj; j + 1g, itsu�ces to discretize bi to O�h2� to reduce the local truncation error of the ith equation in(2.20) to the same magnitude,j((A1 + BA2 + K)Eu�g � F1)ij = O�h2� :Near the interface, that is for i 2 fj; j + 1g, it su�ces to discretize bi to O(h), g0 =[u] +O(h3), g1 = [ux] +O�h2�, g2 = [uxx] +O(h) and g3 = [uxxx] +O(1) to reduce the localtruncation error of the ith equation in (2.20) to the same magnitude,j((A1 + BA2 + K)Eu�g � F1)ij = O(h) :Proof. For i 62 fj; j + 1g the standard di�erences for d2=dx2 and d=dx are O�h2�, and(g)i = 0. So if (d�=dx)=� is discretized to O�h2�, the restriction Eu to the grid of asolution u of (2.18) satis�es (2.20) to O�h2�, since (A1Eu)i = uxx(xi)+O�h2�, (BA2Eu)i =(�x(xi)=�(xi)) ux(xi) + O�h2� and uxx(xi) + (�x(xi)=�(xi)) ux(xi) + (�(xi)=�(xi)) u(xi) =f(xi)=�(xi).

21For i = j, the proposed accuracy of g0; g1; g2 and g3 when replacing [u], [ux], [uxx] and[uxxx] in (2.14) makes (2.14) an O(h) approximation to uxx(xj). Using g0, g1 and g2 in(2.12) yields an O�h2� approximation to ux(xj). But then the claim follows frombj u(xj+1 � u(xj�12h � 12h 2Xm=0 (h+)mm! gm! = ��x(xj)�(xj) +O(h)��ux(xj) + O�h2��= �x(xj)�(xj) ux(xj) +O(h)as in the i 62 fj; j + 1g case. The same argument can be used for i = j + 1.Remark 2.9 Next higher order aproximations for bj, bj+1, g0, g1, g2 and g3 give overallO�h2� truncation error.2.3.2 Discretization of the jump conditionsFormulas for the jumps [u], [ux], [uxx] and [uxxx] appearing in the correction terms inLemma 2.3 are already available in case of Examples 2.1 and 2.3. In order to take advantageof the jump-corrected �nite di�erence approximations in the case of Example 2.2, we rewritethe jump conditions at the interface. For convenience of notation, we write� � ��=�+: (2.28)Lemma 2.10 The jumps in (2.19) can be rewritten in � as a)[u] = w; (2.29)[ux] = (�� 1)u�x + v�+ ; (2.30)[uxx] = �f��� ��x� �u�x � �+x�+ [ux]� ���� u� � �+�+ [u]; (2.31)[uxxx] = �fx� �� 2 ��x� � u�xx � 2�+x�+ [uxx]� ��xx� �u�x � �+xx�+ [ux]� ����u�x � �+�+ [ux]� ��x� �u� � �+x�+ [u]; (2.32)and in + as b)[u] = w; (2.33)[ux] = �1� 1�� u+x + v�� ; (2.34)[uxx] = �f��� ��x� �u+x � ��x�� [ux]� ���� u+ � ���� [u]; (2.35)[uxxx] = �fx� �� 2 ��x� � u+xx � 2��x�� [uxx]� ��xx� �u+x � ��xx�� [ux]� ����u+x � ���� [ux]� ��x� �u+ � ��x�� [u]: (2.36)

22Proof. Purely algebraic manipulations.Lemma 2.11 Discretization of the equations for [�uxx] and [�uxxx] in Example 2.2 resultsin [uxx] = (�� 1)u�xx + [f ]� [�x] u�x � [�] u� � �+x [ux]� �+ [u]�+ ; (2.37)[uxxx] = (�� 1)u�xxx + [f ]� 2[�x]u�xx � 2�+x [uxx]� [�xx]u�x � �+xx[ux]�++�[�]u�x � �+[ux]� [�x]u� � �+x [u]�+ : (2.38)Proof. Purely algebraic manipulations.Remark 2.12 The two versions of the jump equations result from dividing �uxx = f by �+on both sides of the interface as done by Li and LeVeque in [21] or by �� on the left side ofthe interface and �+ on the right side of the interface as done by Li in [25]. Lemmas 2.10and 2.11 extend to higher derivatives in obvious (tedious) ways. Equation (2.31) expresses[uxx] in terms of jumps and one-sided limits up to �rst order. One can also express [uxxx]purely in terms up to �rst order, by using (2.37) to eliminate u�xx and then using (2.31) toeliminate [uxx] from (2.32). This pattern continues for higher derivatives as well.Lemma 2.13[uxxx] = �fx� �� 2� 1�� 1 + �+x�+��f��+ 2�� � �+ [f ]+�2� 1�� 1 + �+x�+���x� �+ 2�� � �+ [�x]� ��xx� �� �����u�x+�2� 1�� 1 + �+x�+� �+x�+ + 2�� � �+ �+x � �+xx�+ � �+�+� [ux] (2.39)+�2� 1�� 1 + �+x�+�����+ 2�� � �+ [�]� ��x� ��u�+�2� 1�� 1 + �+x�+� �+�+ + 2�� � �+ �+ � �+x�+� [u]Proof. As outlined in Remark 2.12Remark 2.14 By allowing one-sided limits in the jump conditions, we are able to derivediscrete jump conditions in a way suited for use with Lemma 2.3 for a very broad range ofproblems, in particular as needed in the general nonlinear case in [41].The condition [u] = w is simply discretized by g0 = w (g = 0, g = ul for examples 2.1,and 2.3, respectively), which is obviously not introducing any error, and hence su�cient forusing Proposition 2.8. But we also have to discretize u�x in (2.30) and (2.31), or u+x in (2.34)and (2.35), respectively, and possibly u�xx for Example 2.3 or if we were to use the jump

23derived from the original equation in Lemma 2.11. We have to compute these one-sidedlimits with enough accuracy so that the jumps are discretized to the accuracies requiredin Proposition 2.8. For completeness and for use in more general cases, e.g. Neumannboundary conditions, we list also functionals for approximating u� and u+.De�nition 2.4 (One sided approximating functionals) Let xj � � < xj+1 and h,h�, h+ be as de�ned before. For approximation in +, de�ne vectors in Rn�1:DT�+ � 12h2 [0; : : : ; 0; h+ �h+ + 3h�+ 2h2| {z }j + 1st entry ;�2h+ �h+ + 2h� ; h+ �h+ + h� ; 0; : : : ; 0];DTx;�+ � 12h2 [0; : : : ; 0; �2h+ � 3h| {z }j+ 1st entry; 4h+ + 4h;�2h+ � h; 0; : : : ; 0];DTxx;�+ � 12h2 [0; : : : ; 0; 2|{z}j+ 1st entry;�4; 2; 0; : : : ; 0]:Similarly, in �:DT�� � 12h2 [0; : : : ; 0; h� �h� � h� ;�2h� �h� � 2h� ; h� �h� � 3h�+ 2h2| {z }jth entry ; 0; : : : ; 0];DTx;�� � 12h2 [0; : : : ; 0;�2h� + h; 4h� � 4h;�2h� + 3h| {z }jth entry ; 0; : : : ; 0];DTxx;�� � 12h2 [0; : : : ; 0; 2;�4; 2|{z}jth entry; 0; : : : ; 0]:The main idea in De�nition 2.4 is to �t a quadratic ~u through the three points (xj+1; u(xj+1)),(xj+2; u(xj+2)), (xj+3; u(xj+3)) and then evaluate ~u(�), ~u0(�) and ~u00(�). This view willturn out to be useful in higher dimensions, where we also need to �nd one-sided limits of uand its �rst and second derivatives.Lemma 2.15 Suppose u 2 C4[xj � 2h; �) \ C4(�; xj+1 + 2h], with derivatives extendingcontinuously up to the boundary � 2 [xj ; xj+1). ThenDT��Eu = u� +O(h3); DT�+Eu = u+ + O(h3);DTx;��Eu = u�x +O�h2� ; DTx;�+Eu = u+x +O�h2� ;DTxx;��Eu = u�xx +O(h) ; DTxx;�+Eu = u+xx +O(h) :Proof. Taylor Expansions.

24−0.2 0 0.2 0.4 0.6 0.8 1 1.2

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Figure 2.1: One-sided approximation of u. Fat dots mark Eu; small dots represent u, dash-dots the quadratic polynomial ~u interpolating u on the three interpolation points \+". \�"marks the interpolated value DT�+Eu approximating u+(�). Here � = 0:33 and h = 0:1.Figure 2.1 illustrates an instance of Example 2.3, whereu = (0 0 � x < 0:33;sin(�x) 0:33 � x � 1;so that ur = sin(�), u� = [u] = sin(:33�), [ux] = u+x = � cos(:33�) and [uxx] = u+xx =��2 sin(:33�). For h = 0:1,��DT�+Eu� u+(�)�� = ��u+(�)�� = 0:0025;���DTx;�+Eu� u+x (�)��� = ��u+x (�)�� = 0:041;���DTxx;�+Eu� u+xx(�)��� = ��u+xx(�)�� = 0:15:This demonstrates the growth of the relative error by roughly a factor 1=h per order ofderivative, in accordance with Lemma 2.15.Using the functionals in De�nition 2.4, the jump conditions are discretized by replacingany one-sided limits of u, ux or uxx by the corresponding discrete operator. For Example 2.1the functionals are not needed because the jumpsg0 = 0; (2.40)g1 = v; (2.41)g2 = [f ]; (2.42)g3 = [fx]; (2.43)

25do not depend on u, while for Example 2.2 (from Lemma 2.10 a) and Lemma 2.13)g0 =w; (2.44)g1 = v�+ � (1� �)DTx;��U; (2.45)g2 = [f� ]� �+�+w � �+x v(�+)2 � ����DT��U � ���+x�+ � ��x���DTx;��U; (2.46)g3 = �fx� �� 2� 1�� 1 + �+x�+��f��+ 2�� � �+ [f ] (2.47)+ �2� 1�� 1 + �+x�+���x� �+ 2�� � �+ [�x]� ��xx� �� �����DTx;��U+ �2� 1�� 1 + �+x�+� �+x�+ + 2�� � �+ �+x � �+xx�+ � �+�+�� v�+ � (1� �)DTx;��U�+ �2� 1�� 1 + �+x�+�����+ 2�� � �+ [�]� ��x� ��DT��U+ �2� 1�� 1 + �+x�+� �+�+ + 2�� � �+ �+ � �+x�+�w;and for Example 2.3 g0 = u�; (2.48)g1 = DTx;�+U; (2.49)g2 = DTxx;�+U (or = [f ]); (2.50)g3 = [fx]: (2.51)Remark 2.16 Writing discretizations of the jump conditions completes the discrete systemin the variables U and g.If we have analytic formulas for �� etc., it is clear from Lemma 2.15 that the jumps arediscretized with the accuracy required in the hypothesis of Proposition 2.8. If we have onlydiscrete values of �, the following Lemma states that we can also satisfy the hypothesiswith approximations.Lemma 2.17 Using approximations for �� etc. in (2.45) and (2.46), gives su�cient orderof the truncation error1:[ux] = vDT�+E� � (1� �)DTx;��U + c(h�)(1� �)u�xxxh2 + O�h3� ;1 The O�h2� term for [ux] is explicitly included for use in the proof of Theorem 2.23.

26where �1=3 > c(h�) = h�=h� 1=3� (h�=h)2=2 > �2, and[uxx] = �DT�+ �DT���Ef� � wDT�+E�� � vDTx;�+E��DT�+E��2� �DT�+ �DT���E��DT��U � �DTx;�+E�DT�+E� � DTx;��E�DT��E� !DTx;��U + O�h2� :Proof. Taylor expansion (as in the proof of Lemma 2.15) and variations of11 + � = 1� �+ O(�2)for j�j < 1.Remark 2.18 The lower order of the jump equations after division by � leads to a betterorder for the truncation order as well. If we had discretized the original jump, we wouldhave gotten only O(h) in the second estimate.2.3.3 One system combining the di�erential equation and jump conditionsThe general structure of the discrete system derived with EJIIM will always be combining(2.20) with a set of equations like (2.40){(2.43), (2.44){(2.47) or (2.48){(2.51) into thefollowing system (we emphasize the block structure, with DT and F2 depending on thejump conditions): � A1 + BA2 +K �DT I � � Ug � = � F1F2 � : (2.52)Working with this type of extended system is a common technique in domain embeddingmethods and was Li's second improvement in [25]. The same methodology is used by Yangin [42]. The extended system allows for the use of e�cient solvers, which becomes veryimportant in higher dimensions. One may want to work with this system directly, chooseto eliminate some or all of g0, g1, g2 and g3, or to eliminate U . A \natural" candidate forelimination in all examples above is g0 since it is independent of U . And for Example 2.1one should eliminate g1, g2 and g3 for the same reason.If one chooses to eliminate g in the general case, one arrives at an (n � 1) � (n � 1)system similar to the original IIM:�A1 + BA2 + K� DT�U = F1 � F2: (2.53)Remark 2.19 Equation (2.53) may be viewed as a way to de�ne s similar to the originalIIM. However, there result more than three s, because D widens the stencil.

272.4 EJIIM and discrete deltasThe two essential operations of EJIIM are distribution of functions supported on theinterface to the grid and interpolation of grid values to the interface. In [4], Beyer andLeVeque developed a theory of discrete deltas in 1D to accomplish both of these operations.EJIIM may be viewed in the light of this theory as introducing di�erent operators forthe interpolation and distribution operations. The distribution is accomplished with operators. For simplicity, consider the case � � 1. In this case1 = �0; : : : ; 0; h+h2 ;�h�h2 ; 0; : : : ; 0�Tis used to distribute force from the interface to the neighboring gridpoints in accordancewith d(1)h from [4]: d(1)h = (h�jxjh2 x 2 [�h; h];0 otherwise;via T1 U = Pj ujd(1)h (xj � �): So the transpose (adjoint) operation of h is just linearinterpolation between grid points. The fundamental di�erence between EJIIM's distributionoperation and the discrete delta approach is that the discrete delta only models kinks,i.e. jumps in the �rst derivative. EJIIM can model jumps in u (0) as well as jumps inhigher derivatives (2, 3, and the approximations can be carried further). Thus the localtruncation error near the interface can be made to be the same order as for any smoothscheme away from the interface.The interpolation operator D�� from De�nition 2.4 corresponds to the discrete deltad(5)h = 12h 8>>>><>>>>:�xh�2 � 3xh + 2 x 2 (�h; 0];�2 �x+hh �2 + 4x+hh x 2 (�2h;�h];�x+2hh �2 � x+2hh x 2 (�3h;�2h];0 otherwise;via 1hDT��U =Xj ujd(5)h (xj � �):This discrete delta satis�es the conditions in Lemma 4.1 in [4] for m = 0; 1; 2 (3rd orderinterpolation) and the conditions in Lemma 4.2 in [4] for m = 1; 2 (3rd order interpolationfor functions with kinks). Thus it has better interpolation properties than \the best"discrete delta d(4)h found in [4] which only satis�es the conditions in the Lemmas 4.1 and4.2 with m = 0; 1 and m = 1 respectively, and it approximates u(��) even when [u] 6= 0.EJIIM uses one-sided interpolation operators for several reasons. The theory in [4]had at least continuity of the function to work with. But we do not assume continuity.For discontinuous functions there are two (a priori) independent limits of every interestingquantity at the interface. Assuming speci�c knowlege about the jumps ([�ux] = v in theIIM for example) allows the derivation of interpolation operators across the interface, but

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a) b)Figure 2.2: a) Standard \hat function" discrete delta d(1)1 . b) The left-sided discrete deltad(5)1 . Its unsymmetric shape results from the fact that it approximates the \�" limit of apossibly discontinuous quantity.destroys the generality of the method. New coe�cients need to be rederived for every newproblem, and the method is very hard to analyze | the bad stencil problem for the IIM in2D is well observed, but still not understood theoretically, after several years of investigationby several researchers. Hence, we do not assume any speci�c knowlege about the jumps, butinstead use the interpolation (now neccessarily on one side) to �nd the jumps. This assumesonly the existence of some formula that relates a quantity on one side of the interface withthe same quantity on the other. The approximation is then simply used in this formula,which becomes a new equation, written into the second block in the discrete system (2.52).Simply using the di�erence between the two one-sided limits as the jump creates an (at leastasymptotically) underdetermined system; some independent condition is always needed atthe interface.The price to pay with EJIIM is that it is somewhat more complicated to �nd thecorrections and to get the corrections on the mesh, but the bene�t is a general and insome cases provably second order method. This compares for example with Peskin's IBM(which uses a discrete delta) which has local truncation error O(1) near the interface andis numerically seen to be converging with �rst order only.2.5 Convergence resultsIn previous work, second order convergence of the IIM was demonstrated numerically. Here,for the �rst time, we give proofs of second order convergence for some special cases of one-point interfaces.

292.5.1 Convergence for the Poisson problemFirst, we recall the ingredients of a proof of second order convergence for the Poissonproblem (with smooth forcing term f), which is a special case of (2.4-2.6). It may bediscretized to A1U = F1;with A1 de�ned in (2.21), F de�ned in (2.22) (reducing to (2.56)), B de�ned in (2.23),(reducing to (2.57)), and F1 = F + B. NowA1U = F1;A1Eu = F1 + T;where the truncation error T = O(h2) in the 1-norm. So we �ndEu� U = A�11 T; andkEu� Uk1 = kA�11 Tk1 � kA�11 k1kTk1 = kA�11 k1O(h2): (2.54)To get second order convergence, we need kA�11 k1 � K, where the bound K has to beindependent of n. Lemma 2.21 below gives such a bound.Lemma 2.20 The (i,j) entry of A�11 2 R(n�1)�(n�1)(for n > 3) isaij = min(i; j) (max(i; j)� n)n3 :Proof. Check that the proposed \inverse" does the job.Lemma 2.21 kA�11 k1 = 18 for n even and kA�11 k1 = 1�n�28 for n odd.Proof. For the two cases n even and n odd, �rst we show that Pj jaij j has the desiredvalue for some i, then we show that this value is the maximum. We rely heavily on theexplicit formulas given in Lemma 2.20.For n even, row n2 has entries12n2 ��1;�2;�3; : : : ; �n2 ; 2� n2 ; 4� n2 ; : : : ; �22 � :The absolute values sum up asn�1Xj=1 ���an2 j ��� = 1n3 8<:n2 n2�1Xj=1 2j + n24 9=; = 1n3 �n38 � n24 + n24 � = 18 :Consider row i, with i < n2 . For columns j = 1; 2; : : : ; i� 1, we �nd jai+1;j j = jai;j j� j. Butfor columns j = i; i+ 1; : : : ; n � 1, we �nd jai+1;j j = jai;jj + n � j. Since there are morecolumns of the latter type, the absolute row sums increase:n�1Xj=1 jai+1;j j � n�1Xj=1 jai;j j � n� i > 0:

30For row i with i � n2 , there are more columns of the �rst type, which yields that the absoluterow sum decreases from row i to row i + 1, �nishing the argument for the case where n iseven.For n odd, rows n�12 and n+12 have the entries (for row n+12 up to permutation):1�n2n3 ��n+11�n;�2�n+11�n� ;�3�n+11�n� ; : : : ; n+12 ; n�12 ; n�32 ; : : : ; 1� :The absolute values sum up asn�1Xj=1 ���an�12 j��� = 1n3 n�12Xj=1 �n + 12 + n � 12 � j = 1n3 n3 � n8 = 1� n�28 :The proof that these two rows attain the maximum absolute row sum is analogous to thecase where n is even, with the exception that here two rows attain that maximum and weneed to consider the cases i < n�12 and i � n+12 .We believe that analogous proofs can be found in two and three space dimensions, butpoint out that convergence is more commonly proved with a discrete maximum principle;c.f. Theorem 3.5 for an example of the latter type of proof in 2D.2.5.2 Convergence for singular sources with third order correctionsFor problem (2.1), the discretization (2.20) reduces toA1U = F1 � gas de�ned in (2.21), (2.22) { (2.27) and (2.40) { (2.43). The truncation error is O�h2� inthe maximum norm and the same proof as for the Poisson problem yields the result.Theorem 2.22 With the third order EJIIM discretization the discrete approximation tothe solution of problem (2.1) satis�eskEu� Uk1 = O�h2� :Proof. (2.54) holds.2.5.3 Convergence for piecewise constant coe�cientsNow we consider the special case of (2.4) where � is piecewise constant and � � 0. Wewish to show quadratic convergence even without the third order corrections, as observednumerically already in [21], and set g3 = 0. Eliminating g0 and g2 as suggested in x2.3, wederive the system � A1 �DT 1 �� Ug1 � = " F +B + h f�i2 + w0v�+ # ; (2.55)

31where F = �f(x1)�(x1) ; f(x2)�(x2) ; : : : ; f(xn�1)�(xn�1)�T ; (2.56)B = ��ulh2 ; 0; 0; : : : ; 0; �urh2 �T ; (2.57)DT = �����+ � 1�DTx;��; (2.58)and 0 = �0; 0; : : : ; 0; 1h2 ;� 1h2 ; 0; 0; : : : ; 0�T ; (2.59)1 = �0; 0; : : : ; 0; h+h2 ;�h�h2 ; 0; 0; : : : ; 0�T ; (2.60)2 = "0; 0; : : : ; 0; h+22h2 ;�h�22h2 ; 0; 0; : : : ; 0#T : (2.61)This assumes 1 > � � 3h. In case 3h > � > 0, use lower left block DT = �(1� 1=�)DTx;�+and right hand side v=�� in the lower block equation of (2.55), with corresponding modi-�cations in what follows.Our goal is to prove the following analogue of the convergence result for the Poissonproblem:Theorem 2.23 (EJIIM convergence for piecewise constant coe�cients) Considerthe BVP (2.4-2.6), where � is piecewise constant, i.e.� = (�� for 0 � x < �;�+ for � < x � 1:With second order corrections, EJIIM �nds approximate solutions U so that kEu�Uk1 =O(h2).The proof of Theorem 2.23 is parallel to the proof for the Poisson problem, basedon the second Schur-complement (compare the suggested methods of computation of thesolution via Schur-complements in x2.6.1). For convenience of notation, we write � 1,F1 � F +B + h f� i2 + w0 and F2 � v=�+. Then A1U � g1 = F1 and DTU + g1 = F2,and hence �A1 +DT �U = F1 + F2:This is a system only involving the approximate solution after eliminating the jump variable.Before we write the proof, we need some auxiliary results. The �rst of these, found forexample in Golub and Van Loan [15], is a very useful result with many applications.

32Lemma 2.24 (Sherman-Morrison-Woodbury)�A1 +DT ��1 = A�11 � A�11 �I +DTA�11 ��1DTA�11 : (2.62)Proof. Recall that A�11 exists by Lemma 2.20 and check that the proposed \inverse" doesthe job.Like in the proof for the Poisson problem we want k �A1 + DT��1 k1 � K, with Kindependent of n. First we de�ne the Green's kernel for the interval [0; 1].G(�; x) = ((�� 1)x 0 � x � �;(x� 1)� � < x � 1: (2.63)It satis�es @2G(�; x)=@x2 = �(x � �) with G(�; 0) = G(�; 1) = 0 for all � 2 (0; 1). TheGreen's kernel is symmetric and negative everywhere in � .The main insight is that 1 is a \discrete delta-function" supported in (�� h; � + h),and application of A�11 is \discrete integration" against a \discrete Green's kernel". Thisis made rigorous in the following Lemma:Lemma 2.25 (discrete Green's kernel) The kth entry of G� � A�11 1 satis�esG�k = ((�� 1)hk 0 � hk � �;(hk � 1)� � < hk � 1:Proof. Instead of using A�11 explicitly, it is easier to verify that A1G� = 1.Corollary 2.26 kA�11 k1 � (1� �)�:Proof. Use Lemma 2.25.The entries in G� are exactly the values of the Green's kernel on the grid. That is, thefollowing diagram commutes for all h and all �:�(x� �) \discretization"�������������! 1integration against G(�; y)??y ??yapplication of A�11G(�; y) ���������������!discretization via E G������� ���Remark 2.27 Furthermore, not surprising given the interpolation properties of (seex2.4), for � = xi, hGxik is just the kth entry in the ith row of A�11 . The h makes sense aswell, representing \integration". This is an observation similar to one made in the proof ofLemma 5.1 in [4].Remark 2.28 It is not needed here, but noteworthy for problems with [u] 6= 0, that 0allows interpretation as a \discrete dipole" supported in (�� h; �+ h).

33Lemma 2.29 (discrete integration of a di�erence functional) The kth entry of D �A�11 D satis�es Dk = (1� �)8><>:h2k k � j � 2;h2k + h2 � h� k = j � 1;h2k � h k � j;where xj � � < xj+1 de�nes j as usual.Proof. Instead of using A�11 explicitly, it is easier to verify that A1D = D.Corollary 2.30 kA�11 Dk1 � 52n j�� 1j:Proof. Take the maximum of the entries in D for all possible values of �.Corollary 2.31 DTA�11 = (�� 1)(1� �) (2 R) :Proof. Use the symmetry of A�11 (i.e. DTA�11 = DT ) and the de�nition of .Corollary 2.32 A1 +DT is invertible for all � > 0, and all � 2 (0; 1).Proof. By Lemma 2.24 is su�ces to show that 1 + DTA�11 6= 0. But for this term wemay use the previous Corollary. For 0 < � � 1, (� � 1)(1 � �) > �1, and for � > 1,(�� 1)(1� �) > 1: So 1 + (�� 1)(1� �) > 0 for both cases.Lemma 2.33 (sup norm bound for rank 1 matrices) Let u; v 2 Rn�1, thenkuvTk1 � (n� 1) kuk1 kvk1:Proof. In the absolute row sums, bound all entries ui, vi by kuk1, kvk1, respectively.Proposition 2.34 For any matrics A;B, and any vectors e1, e2, the following holds:k(A+B)(e1+e2)k1 � k(A+B)e1k1+k(A+B)e2k1� (kAk1+kBk1)ke1k1+kAe2k1+kBe2k1:Proof. Matrix and vector norm properties.Proof. (Theorem 2.23) By Proposition 2.8 and Lemma 2.15, the truncation error Tsatis�es �A1 +DT � (U � Eu) = T = T1 + T2;where kT1k1 = O(h2) and kT2k1 = O(h). The j and j + 1 entries of T1 are zero, and T2has only two nonzero entries: (h�=h � 1=3 � (h�=h)2=2)(1� �)u�xxxh� + O�h2�, with h+corresponding to xj and h� corresponding to xj+1. Using Lemma 2.24 and Proposition 2.34with A � A�11 and B � �A�11 �I +DTA�11 ��1DTA�11 ;

34we will show (i) k �A1 + DT ��1 T1k1 = O(h2) and (ii) k �A1 +DT ��1 T2k1 = O(h2);which proves the theorem.(i) From Lemma 2.21 we know that 18 � kAk1 � K1, and for B we see from Lemma2.33 and the preceeding corollaries thatkA�11 �I+DTA�11 ��1DTA�11 k1 � n(1��)�(1�(1��)(1��)) 52n j��1j=K2:Hence k �A1 + DT ��1 T1k1 � (K1 +K2)kT1k1 = O(h2):(ii) To be precise, T2 = O(h)ej +O(h)ej+1 where el is a unit vector with the lth entryequal to one. From Lemma 2.20 we �nd kA�11 elk1 � supi jailj = O(h). Similarly,kA�11 �I +DTA�11 ��1DTA�11 elk1 � kA�11 k1 ��I +DTA�11 ���1 ��DTA�11 el��� (1� �)�1� (1� �)(1� �)kDTA�11 k1� (1� �)�1� (1� �)(1� �) 52n j�� 1j= K2h = O(h) :Corollary 2.35 The numerical results found with EJIIM for varying � on a �xed grid canbe compared 2: kEu� Uk1 < �52 + 14� (kT1k1 + kT2k1h) :Proof. We only need to show that the bounds in Theorem 2.23 are uniform in the geometryand coe�cient. For use in both (i) and (ii) in the proof of Theorem 2.23, there exists auniform bound on K2 for all � > 0 and 0 < � < 1. For � > 1,5j�� 1j(1� �)�2� 2(1� �)(1� �) < 5(�� 1)(1� �)�2(1� �)(�� 1) < 5�=2 < 5=2and for 0 < � � 1 5j�� 1j(1� �)�2� 2(1� �)(1� �) < 5(1� �)�2� 2(1� �) < 5�=2 < 5=2:Using Proposition 2.34 as in the proof of Theorem 2.23, this yields the uniform error bounds.Remark 2.36 The same ideas give second order convergence results for the discretizationof the singular sources problem and the irregular domain problem with only up to secondorder jumps. For singular sources, the vector DT does not appear, and the proof only relieson (ii) in the proof of Theorem 2.23. For Example 2.3, the de�nition of DT in (2.58) hasto be replaced by DT = �DTx;�+, and the proof carries through replacing (1� �) by �1.2This is true also for varying �, as long as (�� 1)u�xxx is bounded.

35Remark 2.37 Comparing Theorem 2.22 and Theorem 2.23 shows that using third orderjumps does not improve the asymptotic order of the error. Their usefulness lies in theimproved smoothness of the error, and in allowing the construction of higher than secondorder methods, when a higher quality of the discretization near the interface is necessary.Remark 2.38 The truncation error T2 decides the quality of the solution in the followingsense: If the geometry is such that either [ux] = (� � 1)u�x or [ux] = (1=� � 1)u+x couldbe used, then one should do �nite di�erences after estimating and comparing j(�� 1)u�xxxjand j(1=� � 1)u+xxxj. If u�xxx and u+xxx are comparable, this comes down to discretizing[ux] = (�� 1)u�x if � < 1 and [ux] = (1=�� 1)u+x if � > 1.2.6 Schur complements and integral equations2.6.1 Solving the discrete systemsWe suggest the following methods to solve (2.52).� Work on (2.53) with an iterative method, since all matrices are sparse. Preferablythis is done with preconditioning by a fast Poisson solver, i.e. solve the equation�I + A�11 �BA2 +K +DT ��U = A�11 (F1 +F2) : (2.64)� Suppose A1 + BA2 + K can be inverted easily, or at least the product of the inversewith any vector can be computed e�ciently, e.g. for examples 2.1, 2.3. We write Uas a function of g: U = (A1 + BA2 +K)�1 (F1 +g) (2.65)and eliminate U from the second equation in (2.52)to get one equation for g:DT (A1 + BA2 + K)�1 (F1 +g) + Ig = F2;i.e. �I +DT (A1 + BA2 +K)�1� g = F2 �DT (A1 + BA2 + K)�1 F1: (2.66)The (small) matrix on the left in this system is the Schur complement of the I inthe lower right block of the matrix in (2.52) and is called the capacitance matrixin the domain embedding literature. In specialcases it can be computed e�ciently(see Proskurowski and Widlund [35]) and allows a direct solver. But in general, aniterative method is preferred. After g is found, one more solution of (2.65) is needed.We became aware of this possibility through Li's fast iterative method [25].� A1 + BA2 +K is costly to invert. Solve the system iteratively as follows: given U (k),�nd U (k+1) as the U component of the solution (U; g) of the systemA1U � g = F1 � (BA2 + K)U (k)DTU + Ig = F2

36 with Schur complement as above, and iterate. Since the entries in BA2 + K are oneorder (in 1=h) smaller than those in A1, it seems reasonable to expect convergence.In x2.7.2 we will see that it is possible to reduce the order even further by a changeof variables.Remark 2.39 For one-dimensional problems, in practice we never had any problem withthe inversion of the systems. The choices above are given mainly with higher-dimensionalproblems in mind.2.6.2 EJIIM and integral equationsAs usual in this chapter, we derive results in 1D mainly for the purpose of guiding us inhigher dimensions. Integral formulations that can be directly seen in 1D require insightsfrom potential theory in 2D.Consider the discretization of problem (2.7-2.9):AU = F + u�0 + C1 + f+(�)2 + f+x (�)3;C = �DTU = DTx;�+U:After absorbing known terms into ~F , the two Schur complements (2.64) and (2.66) in thiscase are �I +A�11DT�U = A�1 ~F ; (2.67)�I +DTA�11�C = �DTA�1 ~F : (2.68)Equation (2.68) follows from (2.67) by applying �DT and replacing �DTU by C, which isour notation for g1 in this section.Both of these equations have interpretations as discretizations of Fredholm integralequations of the second kind, (2.67) on [0; 1] and (2.68) at the interface; in 1D (2.68) isa single equation at the point �. We remind ourselves of the signi�cance of the operatorsin the equations above: ~F is a discretization of f(x) + [u]�0(x � �), DT corresponds toddx(�)��x=�+ and is a discretization of �(x��). Recall that the Green's kernel for uxx = f ,u(0) = 0, u(1) = 0 was given before Lemma 2.25 and satis�esu(x) = Z 10 G(x; y)f(y)dy: (2.69)We also remarked that A�1ij = hG(xi; yj), and so U = A�1F is a discretization of (2.69).With these correspondences, and the notationGf (x) = Z 10 G(x; y) �f(y) + (�u�)�0(y � �)� dy;we �nd that the Schur complements (2.67) and (2.68) are discretizations of the integralequations (2.70) and (2.71), respectively, where the discrepancy in signs results from ourchoice of D = �DTx;�.

37Lemma 2.40 The solution u to the problem (2.7) { (2.9) and its jump c = [ux] satisfyu(x)� Z 10 G(x; y)�(y� �) ddxu(x)����x=�+ dy = Gf (x); (2.70)c� ddx �Z 10 G(x; y)c�(y� �)dy�����x=�+ = ddxGf ����x=�+ : (2.71)Proof. (2.71) follows immediately from (2.70) by applying ddx(�)��x=�+ to both sides andrenaming c = ddx(u)��x=�+ ; just as in the discrete case. To see that u must satisfy (2.70),we decompose u = u0 + [ux]u1 + [u]u2, where�u0 = f with u0(0) = u0(1) = 0;�u1 = �(x� �) with u1(0) = u1(1) = 0;�u2 = �0(x� �) with u2(0) = u2(1) = 0:Using the Green's kernel solution for each of the three terms separately and keeping knownquantities on the right, this leads tou(x)� Z 10 G(x; y)[ux]�(y � �)� ddxu(x)����x=�+� dy = Z 10 G(x; y) �f + [u]�0(y � �)� dy:(2.72)But [ux] = �u�x and [u] = �u� so this is just (2.70).Remark 2.41 Since f is discontinuous at �, u0 is di�erentiable, but u0;xx is not continuousat �. This is where 2 and 3 are needed.Similarly, for problem (2.4-2.6) with � = 0 and � piecewise constant, we use [ux] =(1� 1=�)u+x in (2.72) and �ndu(x)� (1� 1=�) Z 10 G(x; y)�(y� �) ddxu(x)����x=�+ dy = G�(x);where G� = R 10 G(x; y)(f(y)=�(y))dy: Again, this corresponds to (2.67), now with DT =(1� 1=�)DTx;�+.2.6.3 Discretization of the Dirichlet problem and EJIIMRecall that the Poisson problem uxx = f;u(0) = 0;u(1) = 0;

38is solved by u(x) = Z 10 G(x; y)f(y)dy; (2.73)where the Green's kernel G was de�ned in (2.63). The discretization of this equation isU = A�11 F: (2.74)Now consider solving the Dirichlet problem,uxx = 0;u(0) = u0;u(1) = u1:The discrete solution takes again the form (2.74), if we de�ne F appropriately:F = [�u0=h2; 0; : : : ; 0;�u1=h2]T :In this case, U happens to be the the restriction of the exact solution on the grid. Isthere, and if so what is the integral equation discretized by (2.74) with this F? The anlyticsolution to the Dirichlet problem, for x 2 (0; 1), is:u(x) = (1� x)u0 + xu1= �u0(x� 1) + u1x= �u0 @@yG(x; y)����y=0 + u1 @@yG(x; y)����y=1 (2.75)= Z 10 G(x; y) �u0�0(y � 0)� u1�0(y � 1)�dy: (2.76)What about x = 0 or x = 1? The Green's kernel satis�es G(0; y) = G(1; y) = 0 for all y, soby (2.76), u(0) = u(1) = 0. But clearlylimx!0+ u(x) = u0;limx!1� u(x) = u1:So this means [u]0 = u0, and [u]1 = �u1. The standard way of solving the Dirichlet problemcan be viewed as discretizing the singular integral in (2.76)! The entries in F in this caseare exactly the quantities that the IIM would place there to introduce exactly the jumps atthose points | if the jumps occurred at interior grid points. The complete analogue involvesalso setting the value of the solution on the boundary grid points to zero, as explained inx2.7.If one uses (2.76) to extend the solution beyond [0; 1], by applying (2.63) also for x 62[0; 1], one sees that the resulting function has jumps at 0 and 1, but continuous derivatives.

39In higher dimensions, the analogue of this correspondence to an integral equation holds.For the Dirichlet problem �u = 0 in R;u = g on @R;on rectangular regions, the formulau(x) = Z@R g(t)@�(t);2G(x; y(t))dtgives the solution, where G(x; y) is the Green's kernel for R and @�;2G(x; y(t)) is called thePoisson kernel for the domain R (c.f. (2.75) in 1D). The �-derivative on G is applied in thesecond variable; the boundary of R is parametrized by arclength t 2 [0; T ], and � is the unitoutward normal to @R.Finite di�erences for the Dirichlet problem correspond to the integral equation3u(x) = ZRG(x; y) Z T0 ��(t)(y �X(t))g(t)dt dy:2.7 General boundary conditions and Liouville transformationAs in previous sections in this chapter, everything we describe here is mostly interesting asan introduction to ideas that are useful in higher-dimensional settings.2.7.1 EJIIM for general boundary conditionsSuppose we want to solve (2.7){(2.9) with a di�erent boundary condition at �, replacing(2.8) by c1u(�) + c2ux(�) = c3: (2.77)First we exclude the possibility � = 0, which is dealt with separately below. As usual, wemake the ansatz of extending u � 0 on [0; �), but if c1 6= 0 and c2 6= 0, we do not knoweither [u] or [ux]. Consequently, we introduce two auxiliary variables, g0 and g1 (it is knownthat [uxx] = f+ and [uxxx] = f+x ). By the ansatz u+ = [u] and u+x = [ux], the discreteversion of (2.9) becomes c1g0 + c2g1 = c3. The second equation to determine g0 and g1 isg1 = DTx;�+U . This guarantees that u�x = 0, and together with u(0) = 0 from the Green'skernel, this enforces u � 0 on [0; �), and hence [u] = u+ as required.Now suppose that � = 0, an important special case extending EJIIM to the possibilityof general boundary conditions on regular domain boundaries. The important featureis that � coincides with a lattice point, allowing us to make the value of u at thislattice point the single auxiliary variable g0 instead of the jumps. The idea of3 Recall that G(x;y) = 0 for all x 2 R and y 2 @R and see page 13 for the de�nition of ��(x�X(s)).

40using DTx;�+ is still useful, but now it is applied not to U alone, but to a combination of Uand g0. The equation (2.77) is discretized asc1g0 + c22h2 (�3hg0 + 4hU1 � hU2) = c3: (2.78)This comes from the formula for DTx;�+ with h+ = 0 in this case, and g0 is the right sidedlimit approximating u+(0).For c2 = 0, g0 = c3=c1 is enforced on the mesh by the standard method for Dirichletboundary values, by adding �g0=h2 to the right hand side of the �rst equation of the discretesystem. This may be interpreted as enforcing 0 = u�(0) = u+(0)� g0, a view fullyconsistent with the view of A�1 being a discretization of the Green's kernel for the Poissonproblem on [0; 1]; see x2.6.3. So the function value agrees with the jump, but only becausethe domain on the left shares a boundary with [0; 1], the domain for which we applythe Poisson kernel. If � agrees with some other grid point xi, this is exactly the waycapacitance matrix methods work: the equation discretizing the di�erential equation at thegrid point is replaced with an equation assigning a value �U to the variable Ui. By writingthis assignment as a perturbation of the discretization of the di�erential equation,Ui+1 � 2Ui + Ui�1h2 = �Uh2 + Ui+1 � 3Ui + Ui�1h2and introducing the auxiliary variable C = Ui+1� 3Ui+Ui�1 and the coe�cient = 1=h2,we arrive again at a system of the form considered in x2.6.1.If c2 6= 0 and � = 0, g0 is enforced in the usual way as a boundary value, but theadditional equation (2.78) is needed to determine its value. But again, we arrive at asystem of the form considered in x2.6.1.The reduction of the above boundary conditions to systems considered in x2.6.1 is usefulbecause fast solvers can then be applied. We will do this extensively for several examplesin the next chapter.2.7.2 Liouville transformation for discontinuous coe�cientsFor the problem (2.4){(2.6), a well-known change of variables (see for example [9]): ~u = p�uavoids the matrix B in (2.20). Instead, it adds a matrix like K that has entries only onthe diagonal that are (formally) two orders of magnitude smaller than those of A1. Thisreduces the order of the perturbation of the Laplacian from the approach by division by�, which led to the matrix BA2, with entries one order of magnitude smaller than those ofA1, leading possibly to faster convergence of an iterative method. Problem (2.4){(2.6) with� � 0 becomes ~uxx � �p��xxp� ~u = fp� ;~u(0) = p�(0)ul;~u(1) = p�(1)ur:

41What's new compared to [9] is our use of the jump conditions,� ~up�� = 0;�p�~ux � �x2p� ~u� = 0;"~uxx � �p��xxp� ~u� fp�# = 0:These jumps can be discretized using the functionals introduced in De�nition 2.4.

Chapter 3EJIIM IN 2DIn this chapter we describe EJIIM in 2D and apply it to the three basic elliptic interfaceproblems, singular sources, irregular domains, and discontinuous coe�cients.First, we need to answer the same basic questions as in 1D. How can we distribute asingular source from the interface to the grid? What is the nature of the singularity of thesolution at the interface? The �rst question is answered (for EJIIM) by simply correctingthe di�erences one direction at a time, carried through for two examples in x3.1. Next, itis convenient to summarize all di�erential geometry needed for the interface in x3.2.In x3.3, we formulate the basic elliptic problems | singular sources, irregular domainand discontinuous coe�cient | and follow up in x3.4 with some remarks that apply to allnumerical examples in 2D, including the in uence of the use of approximate interfaces. x3.5has the theory and an example for Poisson problems with singular sources in 2D. As in 1D,third order corrections allow a standard second-order convergence proof based on a discretemaximum principle. In x3.6 we derive the jumps for Dirichlet and Neumann boundaryconditions on irregular domains, and for discontinuous coe�cients. The most importantresult is Theorem 3.5, that implies uniform bounds for the truncation error in 2D (up tothe same dependence on higher derivatives of the solution as in 1D).x3.7 culminates in �nding the integro-di�erential equations that are discretized by thetwo Schur-complements for the solution and jumps. We conclude the chapter with detailsneeded for the irregular domain problems (x3.8), the fast treatment of the variable coe�cientcase via Liouville transformation (x3.9), and another example that it is always possible toexpress jumps as one-sided limits, now for crack problems (x3.10).3.1 EJIIM for the Laplacian and for cross derivativesHow can we modify standard di�erences, for example the O�h2� approximation�u(xi; yj) � u(xi+1; yj) + u(xi�1; yj) + u(xi; yj+1) + u(xi; yj�1)� 4u(xi; yj)h2 ; (3.1)in the case that an interface cuts through the stencil used by the di�erences? The 1DLemmas are also the basis for correcting �nite di�erences in higher dimensions. In 2D, leta smooth interface � = ~X(t) = (X(t); Y (t)) be parameterized by arclength t. In the caseof a closed curve, let � be oriented in the positive direction, with the bounded domain onthe left; in this case ~X(0) is an arbitrary but �xed point on �.To look at a non-trivial example, suppose there are three intersections in a stencil; fornotation refer to Figure 3.1. We show how to correct discretizations of �u and uxy = uyxat the boldface gridpoint (xi; yj) in Figure 3.1. Apply Lemma 2.4 a) in the y-direction and

43Lemma 2.3 in the x-direction and get�u(xi; yj) � u(xi+1; yj) + u(xi�1; yj) + u(xi; yj+1) + u(xi; yj�1)� 4u(xi; yj)h2� 1h2 3Xm=0 (k+2 )mm! �@mu@ym ��2 � 1h2 3Xm=0 (k+4 )mm! �@mu@ym ��4 (3.2)� 1h2 3Xm=0 (h+3 )mm! �@mu@xm��3 +O�h2� ;where we assume a uniform mesh with �x = �y = h. The k� variables indicate distancesin the y-direction and the h� variables indicate distances in the x-direction. The \�" onthe �rst correction term does not agree with the sign in (2.17) because in higher dimensions,it is convenient to de�ne limx!�+ to mean limx2+;x!�;which reverses the sign of the jump at �2. The \non-symmetry" of the one-dimensionalcase is absorbed into this de�nition of the jump. Note also the modi�ed de�nition of h�iand k�j . The \+" variables are found by subtracting coordinates of � from coordinates ofgrid points in + and the \�" variables are found by subtracting coordinates of � fromcoordinates of grid points in �.Cross derivatives are a little more tricky. Suppose we wish to correct the approximationuxy(xi; yj) � fu(xi+1; yj+1)� u(xi�1; yj+1)g � fu(xi+1; yj�1)� u(xi�1; yj�1)g4h2 ;which for smooth u is identical touyx(xi; yj) � fu(xi+1; yj+1)� u(xi+1; yj�1)g � fu(xi�1; yj+1)� u(xi�1; yj�1)g4h2 :With notation as in Figure 3.1 we see that these two expressions give rise to disctinctcorrected versions, but with truncation error of the same order. If we compute di�erencesin y �rst, then uy(xi+1; yj) � u(xi+1; yj+1)� u(xi+1; yj�1)2h ;uy(xi�1; yj) � u(xi�1; yj+1)� u(xi�1; yj�1)2h :Based on these and using Lemma 2.3 for uyuyx(xi; yj) � fu(xi+1; yj+1)� u(xi�1; yj+1)g � fu(xi+1; yj�1)� u(xi�1; yj�1)g4h2� 12h 2Xm=0 (h+3 )mm! �@m+1u@y@xm��3 :If we compute di�erences in x �rst, the approximations for ux(xi; yj+1) and ux(xi; yj�1)have to be corrected via Lemma 2.3 at �1 and �5, respectively, and the di�erences foruxy(xi; yj) need to be corrected according to Lemma 2.4 at �2 and �4.

44u BBBBBBBBBBBBM ������������1~� ~� ��

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Figure 3.1: Interface and mesh geometry near a lattice point (xi; yj). We show the orientedtangent (~�) and normal (~�) directions to the interface at a selected intersection with ahorizontal mesh line, and the angle (�) that the outward normal forms with the positivehorizontal direction. Five intersections of the interface and the mesh are of interest here;they are labeled �1; �2; : : : ; �5.Remark 3.1 In cases where the equality of uxy and uyx is also desired for the approxima-tions, one may wish to take the average of the two corrected forms.It should be clear how general �nite di�erences of arbitrary order can be corrected withthis approach, provided one knows the intersections of the interface with the mesh and thejumps there. As in 1D, we call the application of the corrections described in this sectionthe Explicit Jump Immersed Interface Method, EJIIM. It di�ers from the IIM [21] by� correcting standard di�erences instead of �nding completely new di�erences ( s);� being independent of the particular jump conditions;� separating the contributions from the jumps in u and its derivatives;� using jumps in the Cartesian coordinate directions, including introducing multiplecorrections for a grid point;

45� allowing the use of third and even higher order correction terms.3.2 Interface properties and conversion of jumps to Cartesian coordinatesFormulas for the jumps will be derived in x3.6.1 in the coordinate directions, or in x3.5,x3.6.2, x3.9.1 and x3.10.1 in the normal and tangential directions. They make use of thetangent and normal directions to the interface, and the curvature as well as the derivativeof the curvature. Since these quantities are independent of the particular jump conditions,we collect them here.Recall that � = ~X(t) = (X(t); Y (t)) is parametrized by arclength t. The normalizedtangent direction to � at t is ~�(t) = (X 0(t); Y 0(t)) = (� sin(�(t)); cos(�(t))), where �(t) isthe angle from the vector (1; 0) to the outward normal ~�(t) = (cos �(t); sin �(t)) to � at t.See Figure 3.1. The formulas are this simple because we parametrize by arclength. To useLemma 2.3 or Lemma 2.4, we change coordinates for the jumps. For a �xed intersection(X(t�); Y (t�)) of � with the mesh we introduce local coordinates as follows1. The originis placed at (X(t�); Y (t�)), and the local coordinate axes are parallel to the tangent andnormal to � at (X(t�); Y (t�)). In these coordinates, the interface is expressed as a functionof arclength by ��(t)�(t)� = � cos �(t�) sin �(t�)� sin �(t�) cos �(t�)��X(t)�X(t�)Y (t)� Y (t�)� :Note that �0 is the curvature, where primes denote derivatives with respect to t.�(t�) = 0;�(t�) = 0;�0(t�) = 0;�0(t�) = 1;�00(t�) = ��0(t�);�00(t�) = 0;�000(t�) = ��00(t�);�000(t�) = � ��0(t�)�2 ; (3.3)which come from the general formulasdn�dtn (t�) = cos �(t�)dnXdtn (t�) + sin �(t�)dnYdtn (t�);dn�dtn (t�) = � sin �(t�)dnXdtn (t�) + cos �(t�)dnYdtn (t�):The \-" on the curvature results from the left handed coordinate system (~�; ~�).1 The choice of coordinates is the same as in [21, 24], but the arclength formulation is new.

46 Writing for short c = cos �(t�) and s = sin �(t�), one �nds the following jumps at(X(t�); Y (t�))2: �[ux][uy ]� = �c �ss c ��[u�][u�]� ;0@[uxx][uxy ][uyy ]1A = 0@c2 �2cs s2cs c2 � s2 �css2 2cs c2 1A0@[u��][u��][u��]1A ;0BB@[uxxx][uxxy ][uxyy ][uyyy ]1CCA = 0BB@ c3 �3c2s 3cs2 �s3c2s c3 � 2cs2 s3 � 2c2s cs2cs2 2c2s � s3 c3 � 2cs2 �c2ss3 3s2c 3c2s c3 1CCA0BB@[u���][u���][u���][u���]1CCA : (3.4)For the inverse transformations simply replace s by �s.3.3 The three basic elliptic problemsThe jumps depend on the PDE, or actually on the problem that the PDE models. For sim-plicity and explicitness, we consider elliptic equations in 2D with three types of jump con-ditions. Below, R is the rectangular computational domain. We view the gradient as a rowvector, ru = (ux; uy), and use the notation from x3.2, for example ~�(t) = (cos �(t); sin �(t)).Poisson problems with known singular sources: f is smooth at least in R n �,�u(~x) = f(~x) + Z� v(t)� �~x� ~X(t)�+ w(t)��(t) �~x� ~X(t)� dt; (3.5)u = 0 on @R: (3.6)Later, in x3.5.3 we will see an instance of this problem where f and w vanish and v isconstant. The solution to (3.5) satis�es [u] = w and [u�] = v.BVP on irregular domains: � � R is a connected domain with smooth boundary � =@�, �u = f(x; y) in �; (3.7)u = �u on @�: (3.8)Later, in x3.8.1 and x3.4.1, we will see instances of this problem, for the cases where @�\@Ris empty and non-empty, for both \interior" and \exterior" BVPs.Composite material problems: f and � may be discontinuous across an interface � andr � (�ru) = f(x; y) in R; (3.9)u = �u on @R: (3.10)2The �rst order jumps are given in x4.2 in [24], and the jumps up to second order are also given in x6.6in [43].

47Later, in x4.2.1 and x3.9.2 we will see instances of this problem, with piecewise constantand piecewise smooth but not piecewise constant coe�cients.Considering only these cases is just for brevity of exposition. The use of EJIIM is by nomeans restricted to these simple cases. x3.8.2 shows instances of the more general problem(again � � R) r � (�ru) + �u = f(x; y) in �; (3.11)lu+ (1� l)u� = on @�; (3.12)with multiply connected � and various boundary conditions on several interfaces, whichdemonstrate the real strength of EJIIM.3.4 Remarks on numerical examplesIn all numerical examples, the computational domain R is a rectangle, and often a squareso that the number of grid points in the two directions is equal, nx = ny = n. En denotesthe vector of errors at the grid points, Tn is the usual truncation error (for (3.17) only).For discretization with n � 1 by n� 1 interior points, by the ratio � we meankEnk1kEn=2k1 or kTnk1kTn=2k1 :For singular sources, we include the third order jump corrections because this allows aproof of second order convergence; see Theorem 3.5. However, numerical experiments (ourown as well as those in [21, 24]) indicate that it is su�cient for second order convergence(in the max norm) for all three problems to have a truncation error that is O(h) near theinterface and O�h2� elsewhere, i.e. one needs to include only corrections up to second order.For computational purposes, the interfaces are assumed to be cubic spline interpolantsof given points on the interface. Forces at the interface are also assumed to be speci�ed atthese points. All other needed quantities are interpolated from these using periodic cubicsplines.In all examples but 3.4.1, we have chosen enough points on the interfaces so that thein uence of an inexact interface representation may be neglected.3.4.1 Example: In uence of approximate interfacesConsider the \interior" BVP: �u + �u = f where u(x; y) = 2(x � 0:7)2 + 2(y � 0:4)2,�(x; y) = 12((x � 0:2)2 + (y + 0:1)2)=(2 + (x � 0:2)4 + (y + 0:1)4), and f satis�es theequation. This variable coe�cient Helmholtz equation is solved with Dirichlet boundarycondition given by the exact solution on @�, which is given by r(�) = 0:395+ 0:05 sin(6�)in polar coordinates centered at (0:5; 0:5). The solution (see Figure 3.2 a)) is found on aCartesian grid with jump conditions of the form in problem x3.6.1 b), using GMRES on apreconditioned system similar to (3.46) below. The solution is a second-degree polynomialand should be reproduced exactly up to machine precision by our second-order method.This is not the case only because the representation of the interface is not exact. Figure 3.2

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xya) b)Figure 3.2: a) Solution for Example 3.4.1 computed on a 40� 40 mesh. Values outside �are part of the solution; they are all very close to zero as they should be by our ansatz. b)Error for 101 control points on the interface.Table 3.1: Numerical results for \interior" Dirichlet BVP.n n1 kEnk1 �40 51 4.5e-540 101 1.3e-6 3540 201 3.2e-8 4140 401 2.3e-9 14b) shows the error in the case that the interface is given by 101 points. Table 3.1 showshow the error behaves when the representation of the interface is improved; we see at leastfourth order behavior which should not come as a surprise given the quality of interpolationby cubic splines. In the table, n1 is the number of points on the interface. The points areequally spaced in � which gives a slight advantage over equal spacing in arclength sinceparts of the curve with high curvature receive more points.3.5 Poisson problems with singular sourcesThis problem is the easiest to solve after the jumps are derived, because they are indepen-dent of u. We derive the jumps in detail in this case and will be more cursory for the othertwo problems.

493.5.1 Jump conditionsThe jumps in the function and normal derivative across the interface, together with theequation, determine all other jumps as follows: the \natural" jump conditions are[u] =w; (3.13)[u�] =v; (3.14)�@n(�u)@�n � = �@nf@�n � for n � 0: (3.15)Equation (3.15) is a simple but powerful generalization of Li and LeVeque's way of usingthe equation to get a third jump condition.Equations (3.13){(3.15) hold along the (smooth) interface for su�ciently smooth func-tions of arclength t on the right hand side. The normal derivatives above are taken indirections depending on t according to ~�(t) = (cos �(t); sin �(t)). We derive more jumpconditions by taking derivatives with respect to t, which are indicated by primes below, on(3.13) and (3.14).w = [u];w0 = [u�]�0 + [u�]�0;w00 = [u��](�0)2 + 2[u��]�0�0 + [u��](�0)2 + [u�]�00 + [u�]�00;w000 = [u���](�0)3 + 3[u���](�0)2�0 + 3[u���]�0(�0)2 + [u���](�0)3 +3[u��]�00�0 + 3[u��] ��00�0 + �0�00�+ 3[u��]�00�0 + [u�]�000 + [u�]�000;v = [u~�(t)] = [ru ~�(t)];v0 = [~�(t)Tr2u ~�(t) + �0ru~�(t)];v00 = [��0~�(t)Tr2u ~�(t) + ~�(t)T � ddtr2u� ~�(t) + �0~�(t)Tr2u~�(t) +�00ru~�(t) + �0 � ddtru� ~�(t)� (�0)2ru ~�(t)]:At t = t�, using (3.3), they reduce tow = [u];w0 = [u�];w00 = [u��]� [u�]�0;w000 = [u���]� 3[u��]�0 � [u�]�00 � [u�](�0)2;v = [u�];v0 = [u��] + [u�]�0;v00 = [u���] + 2[u��]�0 � [u��]�0 + [u�]�00 � [u�] ��0�2 :

50And then there are the simple jumps (already at t = t�)[f ] = [u��] + [u��];[f ]0 = [u���] + [u���];[f�] = [u���] + [u���]:Substituting, this gives the following jumps in local coordinates:[u] = w;[u�] = w0;[u�] = v;[u��] = w00 + �0v;[u��] = v0 � �0w0;[u��] = [f ]� w00 � �0v;[u���] = w000 + 3�0v0 � 2 ��0�2w0 + �00v;[u���] = v00 � 3�0w00 � 2 ��0�2 v + [f ]�0 � �00w0;[u���] = [f ]0 � w000 � 3�0v0 + 2 ��0�2 w0 � �00v;[u���] = [f�]� v00 + 3�0w00 + 2 ��0�2 v � [f ]�0 + �00w0: (3.16)For completeness, we use (3.4) to derive the jumps in the coordinate directions, as they arerequired by EJIIM.[u] = w;[ux] = cv � sw0;[uy ] = sv + cw0;[uxx] = c2[f ] + �s2 � c2� ��0v + w00�� 2sc �v0 � �0w0� ;[uyy ] = s2[f ]� �s2 � c2� ��0v + w00�+ 2sc �v0 � �0w0� ;[uxxx] = c3[f�] + �c3 � 3cs2���v00 + 3�0w00 + 2 ��0�2 v � �0[f ] + �00w0��3c2s[f ]0 + �3c2s� s3��w000 + 3�0v0 � 2 ��0�2w0 + �00v� ;[uyyy ] = s3[f�] + �s3 � 3c2s� ��v00 + 3�0w00 + 2 ��0�2 v � �0[f ] + �00w0�+3cs2[f ]0 + �c3 � 3cs2� �w000 + 3�0v0 � 2 ��0�2w0 + �00v� :Here and from now on, we suppress the dependence on t in our notation, but equationsinvolving jumps ([:]) hold for all t, and the directions of normal and tangential derivativesdepend on t.Remark 3.2 By using n > 1 in (3.15), this process can be carried out to any desired order.Remark 3.3 Another important idea from [25] is that even when w and v depend on u,these formulas can be used by �tting a spline through the points on the interface where wand v are approximated, and then taking derivatives on the splines.

513.5.2 Convergence proof for singular sourcesThe Poisson problem on the rectangle R = (0; nxh)� (0; nyh) = (0; a)� (0; b) with smoothright hand side is discretized on the grid (xi; yj) = (ih; jh) via the standard 5-point stencil,(3.1), as �hU = F:Here and from now on, the vector U 2 R(nx�1)�(ny�1) denotes the numerical approximationto u at the grid points in the interior of R in the \natural order", i.e. the grid function thatsatis�es our discretization of the partial di�erential equation under consideration. F is thevector of values of f at these points.We order the intersections of � with the mesh with increasing arclength t. If an inter-section of � with the mesh agrees with a grid point, it may have to be treated as one, twoor three intersections. For example, if we shift the interface in Figure 3.1 to the left, so that�3 agrees with (xi; yj), then we have two possibilities, depending on whether we want theapproximation to u(xi; yj) to approximate the limit in � or the limit in +. In the formercase, the intersection with the horizontal line y = yj belongs in the interval [xi; xi+1] andthe intersection with the vertical line x = xi belongs in the interval [yj ; yj+1]. In the lattercase, the intervals are [xi�1; xi] and [yj�1; yj ], respectively. If the tangent to the interfaceat the grid point is parallel to one of the coordinate axes, the two cases to be distinguishedinvolve one or three intersections.The vector C 2 R4m of approximations of the jumps in u has four entries for each of them intersections, counting multiplicities as described in the previous paragraph. Dependingon the type of intersection (with a horizontal or vertical mesh line), the elements of Capproximate jumps in derivatives in the x- or y-direction. Given the �rst few intersectionsas in Figure 3.1,C = [[u]�1; [ux]�1 ; [uxx]�1 ; [uxxx]�1; [u]�2; [uy]�2 ; [uyy]�2 ; [uyyy ]�2; [u]�3; [ux]�3; : : : ]T :Each jump is used twice to correct the standard di�erences, namely for the two grid pointsneighboring the intersection. For example, jumps at �3 are used to correct di�erences(approximating x-derivatives) at (xi; yj) and (xi+1; yj). The coe�cients for the jump in themth derivative are �(h+3 )m=(m!h2) and (h�3 )m=(m!h2) respectively.We write all these coe�cients into the matrix , where rows are ordered like the gridpoints, and columns are ordered like the jumps. The two non-zero entries per column cor-respond to the two grid points where the di�erences are being corrected. Hence, equationslike (3.2) say how to correct the right hand side of the equation for (xi; yj). We combineall the needed corrections into the system�hU = F + C: (3.17)Remark 3.4 If we correct only to second order, then C 2 R3m. Also, it is convenient toeliminate known jumps and absorb them into F to further shorten C.The discrete system (3.17) mimics the partial di�erential equation (3.5) in that we onlyhave correction terms on the right hand side of the system, but the di�erential operator

52(di�erence operator) is the usual one. This feature is similar to Peskin's Immersed BoundaryMethod [34], a �rst order predecessor of the IIM. Peskin's view of a discrete delta functionon the right hand side of (3.17) was further explored in the context of EJIIM in 1D in x2.4.Eu denotes the restriction of u to the grid. Our discretization has O�h2� truncationerror, i.e. �hEu = F + C +O�h2� :Since the truncation error is O�h2� for all equations, the standard convergence proof(based on a discrete maximum principle, see for example x6.2 in Morton and Mayers [30])for the Poisson problem on rectangles carries over to this case.Theorem 3.5 (Second order convergence) For a Poisson Problem with singular sources,numerical solutions obtained by discretization with EJIIM including up to third order jumpssatisfy k��1h (F + C)� Euk1 = O�h2� :Proof. Let the error in the solution T = U �Eu, and denote the bound on the truncationerror �hT = F + C � �hEu on the rectangle (0; nxh) � (0; nyh) = (0; a) � (0; b) byk�hTk1 � Kh2 for h < h�.Now extend �h 2 R(nx�1)(ny�1)�(nx�1)(ny�1) to ~�h 2 R(nx�1)(ny�1)�(nx+1)(ny+1) byincluding the boundary contributions, i.e. using the standard approximation (3.1) for allinterior points. Extending U by the values of ~Eu on the boundary (all zero) in the appro-priate order, we see that ~�h ~U = �hU . Similarly, extend ~T (by zero, since the boundaryvalues are exact) and see that k ~�h ~Tk1 � Kh2.De�ne the entries of the vector � by �ij = (xi � a=2)2+ (yj � b=2)2 for i = 0; 1; : : : ; nxand j = 0; 1; : : :ny , then ~�h� = [4; 4; : : : ; 4]T . Let � = ~T + Kh2�=4, then ( ~�h�)ij � 0.~�h satis�es a discrete maximum principle, i.e. ui;j+1 + ui;j�1 + ui+1;j + ui�1;j � 4ui;j � 0implies ui;j � max(ui;j+1; ui;j�1; ui+1;j ; ui�1;j), so � assumes it's maximum at a boundarypoint. But ~T vanishes on the boundary, and � has its maximum at the corners, so 8i; j~Ti;j � �i;j � a2 + b216 Kh2:Repeating the argument with � = Kh2�=4� ~T , we arrive atkTk1 = k ~Tk1 � a2 + b216 Kh2:Remark 3.6 The special case where no singular source is present and f is simply discon-tinuous across the interface is also covered by our corrections, and consequently by Theo-rem 3.5.

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xya) b)Figure 3.4: a) Truncation error and b) error in the computed solution for EJIIM on Example3.5.3 without third order corrections. The truncation error is only de�ned in the interior,while the error on the boundary is zero due to the exact boundary conditions. The spikes ina) are terms of magnitude O(h), they get smoothed out to O(h2) in b).

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xya) b)Figure 3.5: Truncation error a) and error in the computed solution b) for EJIIM for Example3.5.3 including third order corrections. The spikes have disappeared, and the truncationerror is smooth in the tangential direction on either side of the interface.3.5.3 Example: EJIIM for singular sourcesThis numerical example demonstrates Theorem 3.5, the e�ects of including third ordercorrections and shows how EJIIM compares with the IIM [21]. Letu(x; y) = (1 for (r; �) 2 � = f(x; y) 2 [0; 2]� [0; 2]jr < sg;1� log(r=s) for (r; �) 2 + = f(x; y) 2 [0; 2]� [0; 2]jr > sg;where s is the radius of a circular interface. Here r = p(x� 1)2 + (y � 1)2 and � =arctan((y � 1)=(x � 1)) are polar coordinates centered at (1; 1). Then u satis�es �u = 0with [u] = 0 and [un] = �1=s. This example is adapted from [21].

55From the jumps, and since [�u] = 0 we specialize the general formulas to[ux] = � cos �s ;[uy ] = � sin �s ;[uxx] = cos2 � � sin2 �s2 ;[uyy ] = sin2 � � cos2 �s2 ;[uxxx] = 2(3 cos� sin2 � � cos3 �)s3 ;[uyyy ] = 2(3 sin � cos2 � � sin3 �)s3 :Table 3.2 shows the results of the computations for (3.5) with the above jumps ands = 0:5, compared with numbers from [21]. The boundary data are found by evaluating theanalytic solutions. We see that for a given mesh, our method without (EJIIM, columns 4,5) or with (EJIIM-2, columns 6, 7) the third order corrections gives more accurate answersthan the IIM (columns 2, 3). By including the third order corrections, we reduce thetruncation error by one order of magnitude | compare columns 9 and 11, which give theratios between consecutive truncation errors; also compare Figure 3.4 a) and Figure 3.5 a).This does not improve the error in the max norm, but yields a smoother error as can beseen comparing Figure 3.4 b) and Figure 3.5 b), which may be an interesting feature inapplications.All methods converge with O�h2� and perform better on the coarsest mesh than Peskin'sIBM on the �nest mesh as reported in [21] for this problem.Table 3.2: Numerical results for a singular source term.IIM EJIIM EJIIM-2 IIM EJIIM EJIIM-2n kEnk1 � kEnk1 � kEnk1 � kTnk1 � kTnk1 � kTnk1 �20 2.4e-3 1.4e-3 1.3e-3 2.8e-1 2.8e-1 1.1e-140 8.4e-4 2.9 1.8e-4 7.7 2.7e-4 4.9 1.7e-1 1.7 1.6e-1 1.8 3.0e-2 3.780 2.5e-4 3.4 6.6e-5 2.7 7.4e-5 3.7 8.3e-2 2.0 8.2e-2 1.9 8.9e-3 3.4160 6.7e-5 3.7 1.9e-5 3.4 1.9e-5 3.9 4.2e-2 2.0 4.2e-2 2.0 2.3e-3 3.9320 1.6e-5 4.3 3.4e-6 5.7 4.8e-6 4.0 2.3e-2 1.8 2.1e-2 2.0 6.6e-4 3.63.6 Irregular domain and discontinuous coe�cient problems3.6.1 Irregular domain problemsThe irregular domain problem (3.7){(3.10) is one of domain embedding. The idea is to takeadvantage of fast solvers on the larger, regular domain. Our own approach is di�erent from

56Yang's IIM for boundary value problems [42], Mayo's approach [29], and also Proskurowskiand Widlund's approach [35], in that we extend f and the solution u to be identically zeroin the complement + of �. Of course, this is only possible by allowing both [u] 6= 0and [u�] 6= 0. The two essential jump conditions are [u] = ��u and [u�] = �u�� . Instead offollowing Remark 3.3, i.e. deriving the rest of the jumps from those two, we see that thefollowing jump conditions hold: [u] = ��u;[ux] = �u�x ;[uy ] = �u�y ;[uxx] = �u�xx;[uyy ] = �u�yy : (3.18)Now let us investigate why our ansatz u � 0 in + makes sense. For components of +that share a boundary with @R, the jump conditions imply a mixed BVP (zero Dirichletboundary condition on @R, zero Neumann boundary condition on the boundary shared with�) which has exactly the trivial solution. On components of + whose boundaries do notpartially coincide with @R, we get a Neumann problem which has constants as solutions.Imposing that the numerical solution be zero on the grid points in + along the interfaceon these components creates a \corridor" with zero Neumann and zero Dirichlet boundaryconditions along the interface and a Dirichlet problem with zero boundary conditions inthe interior of these components of +. The solution to these is clearly zero. In x3.8 wedescribe how we can impose values at interior grid points with EJIIM.For a Neumann boundary condition u� = u� replacing (3.8) on an interior boundary,the jumps are derived from the same ansatz (extension by zero), but now with essentialjump conditions [u] = �u� and [u�] = �u�; we obtain[u] = �u�;[ux] = � cos �u� � sin �u�� ;[uy] = � sin �u� + cos �u�� ;[uxx] = �u�xx;[uyy ] = �u�yy :The distinction between components as in the Dirichlet problem case does not exist in thiscase. The limit u+ on @+ is zero, and these Dirichlet problems on each of the connectedcomponents of + have exactly the trivial solution.3.6.2 Discontinuous coe�cient problemsThe discontinuous coe�cient problem (3.9){(3.10) is the hardest to solve numerically andwas the motivation for our initial interest in developing EJIIM. When trying to �nd thelocation of interfaces between di�erent constant values for � (an inverse problem) with theImmersed Interface Method [21], we found that the method was not stable enough for largecontrast (� = ��=�+ � 1) problems. EJIIM was developed after many ideas from [4, 25, 29]

57and others to overcome this problem. Later we realized the much broader applicability ofthis approach, in particular to boundary value problems on irregular domains. Following[21], we assume that the essential jump conditions are[u] = w(t);[�u�] = ~v(t);where the physical conditions are usually w = ~v = 0. Equation (3.9) provides at least twoways to derive another jump condition. By taking limits on both sides of the interface weget (note that the equation is invariant under the change of coordinates; see [21])�+ ��u+�+r�+ � ru+ = �+ �u+�� + u+���+r�;��+ � r�;�u+ = f+; (3.19)�� ��u��+r�� � ru� = �� �u��� + u����+r�;��� � r�;�u� = f�: (3.20)Extending Li's ideas in [25] to piecewise smooth (but not piecewise constant) coe�cients,we can divide by � and then discretize the divided equation�u+ r� � ru� = f� ; (3.21)which holds everywhere except at points on �. The jumps at � are (see [21, 24], or derivethem from the formulas for singular sources):[u] =w;[u�] =w0;[u�] = ~v�+ + (�� 1)u�� ;[u��] =w00 + �0[u�];[u��] = ��� �+ � ���+�(�+)2 u�� + (�� 1)�u��� + �0u�� � � �0[u�] + ~v0�+ � �+� ~v(�+)2 ;[u��] = (�� 1)�u��� + u���� � [u��]+ 1�+ �[f ] + ��� u�� + ��� u�� � �+� �u�� + [u�]�� �+� �u�� + [u�]�� : (3.22)[u�], [u��] and [u��] are found by di�erentiating lower order jumps, while [u��] comes fromdividing both (3.19) and (3.20) by �+. In x4.2.1, this approach is used in the simpler caseof constant coe�cients.Again one could also use Li's technique in [25] (as suggested in Remark 3.3), and derivetwo di�erent jump conditions. Writing for shortv = ~v�+ + (�� 1)u�� ;Li [25] approximates v0 by �tting a spline and then taking its derivative with respect toarclength. Dividing (3.20) by �� instead of �+ provides more equations for second order

58jumps:��u+ r� � ru� � = �f�� ; [u��] = v0 � �0[u�] and [u��] = �f��� [u��]� ���u�� �� ���u�� � :After eliminating [u��], the last two equations have the advantage of not using one-sidedlimits of second derivatives. We do not currently use these latter forms of the jumps incomputations, but point out that [25] does and that we wish to experiment with them inthe future.3.6.3 Discretization of solution{dependent jumpsFor the irregular domain problem and the discontinuous coe�cient problem, some of thejumps depend on the solution, which in turn depends on the boundary conditions andcoe�cients. Suppose we knew the solution on the grid, just as we made the ansatz ofknowing the jumps before. Then we could get approximations to the quantities on theright hand side of the jump equations (3.16), (3.18), (3.22) or (3.39) by �tting a low degreepolynomial p(x; y) through six points (for O(h) truncation error near the interface) or tenpoints (for O�h2� truncation error), which have to lie on the same side of the interface.For irregular domains, that means the points lie in �, while for discontinuous coe�cientswe have a choice of either side. For the sake of explicitness, we will write one-sided limitswith superscript \�" to indicate the limiting operation in the chosen region, after possiblyrenaming the regions.The value of p on the interface is an approximation for u, and derivatives of p approx-imate derivatives of u with decreasing accuracy for higher derivatives. The points have tobe such that uniform bounds are possible for these approximations. The following Lemmasprovide such bounds.Given six points P0(h0; k0), P1(h1; k1), P2(h2; k2), P3(h3; k3), P4(h4; k4), P5(h5; k5). De-�ne the Vandermonde matrix of bivariate Lagrange interpolation by quadratic polynomialsin two variables on these six nodesP = 266666641 h0 k0 h20 h0k0 k201 h1 k1 h21 h1k1 k211 h2 k2 h22 h2k2 k221 h3 k3 h23 h3k3 k231 h4 k4 h24 h4k4 k241 h5 k5 h25 h5k5 k2537777775 ; (3.23)and the quantities lijm = (hi�hj)(km�kj)�(ki�kj)(hm�hj), for i; j;m 2 f0; 1; 2; 3; 4; 5g.Lemma 3.7 Interpolation of u by polynomials p(x; y) = p0+p1x+p2y+p3x2+p4xy+p5y2on the points P0, P1, P2, P3, P4 and P5 is always possible if and only if detP 6= 0.Proof. Let the vector of values of u at the six points be denoted by F = [f0; f1; f2; f3; f4; f5]Tand de�ne P = [p0; p1; p2; p3; p4; p5]T . The fact that p(x; y) satis�es p(hi; ki) = fi fori 2 f0; 1; 2; 3; 4; 5g can be written as PP = F , which has a solution for arbitrary F if andonly if detP 6= 0.

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ya) b)Figure 3.6: In a), we see all the anchor points near the interface, stars for vertical meshsegments, circles for horizontal mesh segments. Some are used for two di�erent intersec-tions. In b), we see selected complete stencils. The algorithm chooses a stencil for everypoint labeled by a circle or star in a).Lemma 3.8 (Lorentz [28], Theorem 11.2.1)detP = l015l325l034l124� l014l324l035l125:Corollary 3.9 If P0; P1, and P5 are on a line, none of P2; P3, and P4 is on that line, andP2; P3, and P4 are not on any line, then interpolation by quadratic polynomials is possibleon these points.Proof. Noting that lijm is twice the signed area of the triangle formed by the nodesPi; Pj; Pm, it is easy to see that if P0; P1, and P5 are on a line, then l015l325l034l124 = 0since l015 = 0. By hypothesis, none of the other triangles is degenerate. This means thatall of l014, l035, l125, and l324 are non-zero. So detP = �l014l324l035l125 6= 0.Remark 3.10 Lorentz [28] also provides a reference to a similar result needed for theextension of EJIIM to 3D.Given I, the set of intersections of the interface with the mesh lines, our algorithm isdesigned to �nd six points in � satisfying the hypothesis in Corollary 3.9, by making theclosest grid point in � the \anchor" point and selecting �ve more \neighbors" on the grid.Lemma 3.11 (Curve condition) If for every intersection in I, one of the two disks ofradius p5=2h and tangent to � at I lies entirely in �, then it is possible to �nd six gridpoints in � as required in Corollary 3.9.

60 r r rr rr4 5 61 32 r r rr rr4 5 61 32 rr r rr r64 1 52 3a) b) c)Figure 3.7: The three possible stencils the algorithm allows for interpolation. Numbersindicate the row in P that corresponds to this point, and may be permuted in each �gure.Proof. It su�ces to show that any closed disk of radius p5=2h contains six grid pointsin the position required by Corollary 3.9. By rotating the mesh by �=2, � or 3�=2, andshifting by multiples of h=2, we may assume that the mesh is of the form xi = ih + h=2and yj = jh + j=2 for i; j 2 Z, and that the center of the disk in the hypothesis liesin [0; h=2]� [0; h=2]. The six points (�h=2;�h=2), (�h=2; h=2), (h=2;�h=2), (h=2; h=2),(h=2; 3h=2), (3h=2; h=2) are in the position as required and all lie in the disk. This iseasily seen by observing that the disks of radius p5=2h centered at these points contain[0; h=2]�[0; h=2]. The constantp5=2h is sharp because the disks of radiusp5=2h centeredat (h=2;�h=2) and (h=2; h=2) contain only six grid points and have the point (0; 0) on theboundary, so that disks of smaller radius have less than six points in them.The algorithm succeeds provided that the mesh size h is chosen small enough dependingon the curvature of �, and that the curve does not get \too close to itself" as expressed inthe curve condition. If the program cannot �nd six points, it recommends a �ner mesh. Upto symmetry (rotation by angles �=2, �, or 3�=2), for closed curves only three constellationsof the 6 points are allowed by the algorithm, shown in Figure 3.7. We �nd P using Cartesiancoordinates centered at the anchor point, so that always h0 = k0 = 0 and all other hi andkj are integer multiples of h.Lemma 3.12 The entries of P�1 as found by our algorithm are bounded as follows: entriesin the �rst row (yield constant term in p) by O(1), entries in the second and third row (linearterms) by O�h�1� and entries in the last three rows (quadratic terms) by O�h�2�.Proof. We need to consider more than the three cases in Figure 3.7 since the anchorcan be almost any of the points in each of the three allowed stencils, except the pointlabeled 1 in case c). This gives 17 cases, up to row permutations. In each of the 17cases the matrix P = P1Dh factors into a matrix P1 which is independent of h withentries in f�2;�1; 0; 1; 2g and the same diagonal matrix Dh = diag(1; h; h; h2; h2; h2).P1 is invertible by Corollary 3.9, and the entries of the 17 possible inverses are boundedsimply by the maximum absolute value over the 36 � 17 entries of the inverses, whileD�1h = diag(1; h�1; h�1; h�2; h�2; h�2). Finally, P�1 = D�1h P�11 , which completes theproof.

61Now we describe how to compute one-sided limits: The goal is to estimate u�, u�x andu�xx on points in I that lie on horizontal mesh lines, and u�, u�y and u�yy on points in Ithat lie on vertical mesh lines.Suppose the anchor point S for a �xed point (Xt; Yt) 2 I has coordinates (�x; �y), and theselected �ve neighbors have relative coordinates (hj ; kj), j = 1; 2; 3; 4; 5 (here h0 = k0 = 0),where all six points lie on the \�" side of the interface. The restriction operator RS :R(nx�1)(ny�1) ! R6 depends on S and restricts the values of any grid function z to thesix points in the desired order. The quadratic polynomial p in shifted Cartesian coordinatevariables h = x� �x and k = y � �y interpolates z on these six points, i.e.p(hj ; kj) = p0 + p1hj + p2kj + p3h2j + p4hjkj + p5k2j = (RSz)j :As in Lemma 3.7 we know that [p0; p1; p2; p3; p4; p5]T = P�1S RSz.Theorem 3.13 If we set h� � �x�Xt and k� � �y � Yt, and let z = Eu, thenu�(Xt; Yt) = p(�h�;�k�)+O(h3)= p0�p1h��p2k�+p3(h�)2+p4h�k�+p5(k�)2+O(h3);u�x (Xt; Yt) = @p@h(�h�;�k�)+O(h2) = p1�2p3h��p4k�+O(h2);u�xx(Xt; Yt) = @2p@h2 (�h�;�k�)+O(h) = 2p3+O(h);u�y (Xt; Yt) = @p@k(�h�;�k�)+O(h2) = p2�p4h��2p5k�+O(h2);u�yy(Xt; Yt) = @2p@k2 (�h�;�k�)+O(h) = 2p5+O(h):Proof. Taylor expansions about the anchor point (�x; �y). Minus signs result from our con-vention to de�ne h� as subtracting an interface coordinate from a grid coordinate, i.e. theexpansion about (Xt; Yt) instead of (�x; �y).Remark 3.14 By Lemma 3.12 the constants in the order terms in Theorem 3.13 do notdepend on the interface geometry. Also, the constants do not depend on the co-e�cients of the equation, other than implicitly through the bounds on derivatives ofu.Written as linear operators applied to Eu,u� � �1;�h�;�k�; (h�)2; h�k�; (k�)2�P�1S RSEu;u�x � �0; 1; 0;�2h�;�k�; 0�P�1S RSEu;u�xx � [0; 0; 0; 2; 0; 0]P�1S RSEu;u�y � �0; 0; 1; 0;�h�;�2k��P�1S RSEu;u�yy � [0; 0; 0; 0; 0; 2]P�1S RSEu:By applying these discrete linear operators to U instead of Eu, composing with coordi-nate transformations (see x3.2) as needed and �nally taking linear combinations as imposed

62by the jump conditions (see x3.3), we know how to discretize the jump conditions. Forexample, we discretize [u�] = ~v�+ + (�� 1)u�� (3.24)as [u�] � f2 + (�� 1) �c s��0 1 0 �2h� �k� 00 0 1 0 �h� �2k��P�1S RSEu;where f2 = ~v=�+. All the linear operations are combined into one linear operator D, whosecolumns correspond to grid points (it has 6 non-zero entries per row), and whose rows areordered like the jump variables. The discretization of all jump conditions isC = F2 �DTU:We can combine the discretization of the PDE (3.17) and the discretization of the jumpconditions: �hU �C = F1;DTU + C = F2: (3.25)This notation covers the singular sources problem, the iregular domain problem, and thepiecewise constant coe�cient problem. For variable coe�cients, a slightly more generalequation is used in x3.9.2. F1, F2, and DT depend on the problem, but even P�1S andRS , two important factors of DT , depend only on the interface and mesh geometry. Mul-tiple interfaces are handled by simply stacking the DT matrices and jump vectors C andconcatenating the matrices, even if the jump conditions are di�erent for the di�erentinterfaces. For singular sources, simply DT = 0 and F2 contains the signed magnitudes ofthe jumps in the coordinate directions. We de�ne the jump truncation error to measurehow well the jumps in u and derivatives satisfy the discrete jump equations, and observethat it is trivially zero for the singular sources problem.For iregular domains and discontinuous coe�cients, with the operators we have givenabove (i.e. for a six point stencil), it is easy to see that for nonzero rows in DT the jumptruncation error is O�h3� for entries in C approximating [u], O�h2� for entries in C ap-proximating [ux] and [uy ], and O(h) for entries in C approximating [uxx] and [uyy ]. Check-ing carefully the magnitudes of the coe�cients in that multiply these terms (compareLemma 3.12) we see that after elimination of C the �rst equation has truncation error O�h2�except at neighbors of the interface, where the truncation error is O(h). For piecewise con-stant coe�cients the bounds in these estimates depend on the contrast in the coe�cient(the factor (��1) enters D for one-sided limits in �, while (1=��1) comes in for limits in+ | so we pick the smaller one), but not on the geometry; this property was not availablefor the original IIM [21]. For irregular domains, this contrast factor does not occur, butotherwise the bounds for the coe�cients are the same.Remark 3.15 By analogy with the independence of the bounds on � in 1D, Corollary 2.35,we suggest that the one-sided di�erences should be used on the side where u is expected tobe smoother. Numerical examples in x4.2.1 con�rm this in general, but not for all cases,especially on coarse grids.

633.7 Connection between EJIIM and potential theory3.7.1 Solving the linear systemsEquation (3.17) with known C can be e�ciently solved for U since ��1h is easily appliedto a vector via a fast Poisson solver; see for example [37]. We take advantage of this bylooking at the second Schur-complement of the 2� 2 block system (3.25),�I +DT��1h �C = F2 �DT��1h F1: (3.26)The similarity of this equation with the boundary integral equations of potential theorymay provide the insights needed to prove the stability of this approach in the future, whilehere we only verify convergence numerically. The entries in C correspond to double andsingle layer densities and also take care of possible jumps in f and f�. discretizes theirin uence on the domain, i.e. it resembles a delta or dipole. ��1h is a discrete Green's kernelfor the rectangle, and DT corresponds to restricting u and derivatives to the interface.Finally, (3.26) resembles a Fredholm equation of the second kind along the interface. OnceC is found, it takes one more solution of the �rst equation in (3.25) to �nd U .Equation (3.26) is best solved with an iterative method (we use GMRES from [19]; see[36]) to avoid forming the matrix on the left explicitly. The main cost per iteration liesin �nding ��1h (C) and is O(nxny log(nxny)). In order to use direct methods on (3.26)one would need an explicit formula for ��1h . One way to achieve this was suggested byProskurowski and Widlund in [35], where a di�erent choice of basic problem on the rect-angle, namely periodic boundary conditions, allows cheap computation of the replacementof ��1h in that setting. This in turn allows forming the \small" matrix on the left in (3.26)explicitly and brings down the cost per iteration to O(m2).3.7.2 Fundamental solution and Green's functionFollowing Folland [11], we de�ne the fundamental solution N(~x) of Laplace's equation,as satisfying �N = �. In 3D, N is forNewtonian potential, generated by a unit negativecharge at the origin3. It depends only on r = k~xk and the dimension. For example, in 2DN(~x) = 12� ln k~xk:It is convenient to regard N as a function of two variables via N(~x; ~y) = N(~x � ~y). Forexample, in 1D we had N(x; y) = jx� yj. We also de�ne the Green's kernel for an opendomain (the complete interior of its closure, with C1 boundary � = @) asG(~x; ~y) = N(~x; ~y)�H(~x; ~y);where H is harmonic in (satis�es Laplace's equation) for any �xed ~y 2 and matches Non @, so in particular G(~x; ~y) = 0 for ~x 2 @. In 1D, harmonics are of the form c1+c2x, andthe Green's kernel was found by choosing c1 and c2 so that G(x; y) = jx�yj+c1(y)+c2(y)xvanished at x = 0 and x = 1.3 In R3, the fundamental solution is characterized by the fact that it vanishes at in�nity. In R and R2this property cannot be achieved, so we choose N to have minimum zero and to vanish on the unit circle,respectively. Some authors prefer to de�ne the fundamental solution to be �N .

643.7.3 Single and double layer potentialsLet ~� be the outward normal to the interface �, which is as usual written as ~X(t) andparametrized by the arclength parameter t 2 [0; T ]. We indicate the variable of di�erentia-tion by a subscript on the derivative, e.g. @�;2N(~x; ~y) denotes the normal derivative in thesecond variable (i.e., ~y). Considerw(~x) = Z� �( ~X)@�;2N(~x; ~X)d�( ~X) = Z T0 �( ~X(t)) ~�(t) � r2N(~x; ~X)��� ~X= ~X(t) dt; (3.27)v(~x) = Z� �( ~X)N(~x; ~X)d�( ~X) = Z T0 �( ~X(t))N(~x; ~X(t))dt: (3.28)The second equation in (3.27) holds because k ~X 0(t)k = 1. These formulas de�ne harmonicfunctions w�; v�; w+ and v+ in � = and + = Rnf�[�g, respectively. The singularnature of the integrals is such that:1. w(~x) exists for ~x 2 �, but w is discontinuous at �. 8~x 2 � :w+(~x) + �(~x)=2 = w(~x) = w�(~x)� �(~x)=2;w+� (~x) = w�� (~x);i.e. [w�] = 0 and [w] = ��(~x).2. v(~x) is continuous but not di�erentiable across �. 8~x 2 �:v+(~x) = v�(~x);v+� (~x)� �(~x)=2 = Z T0 �( ~X(t)) ~�(t) � r2N(~x; ~X)��� ~X= ~X(t) dt = v�� (~x) + �(~x)=2;i.e. [v�] = ��(~x) and [v] = 0.In the literature, w is known as a double layer potential with density �, and v is known asa single layer potential with density �.3.7.4 Boundary Integral MethodsMethods that reduce di�erential equations to integral equations on the boundary are oftenreferred to as boundary integral methods (BIM). Both (3.27) and (3.28) can be used tosolve the Dirichlet problem on �, �u = 0 and u = �u on � = @�.1. Make a double layer ansatz: for ~x 2 �, tryu(~x) = w�(~x) = Z T0 �( ~X(t)) ~�(t) � r2N(~x; ~X)��� ~X= ~X(t) dt;

65for some suitable �, to be found. From the previous section and using the boundaryvalues, one �nds the following Fredholm Integral Equation of the second kind for �on �: �u(s) = �( ~X(s))2 + Z T0 �( ~X(t)) ~�(t) � r2N( ~X(s); ~X)��� ~X= ~X(t) dt:This equation is discretized for example by Mayo [29] with the Nystr�om method,which takes advantage of the explicit knowledge of the kernel in this equation. Inx4.3.6 we explain how Bryan [5] used the Nystr�om method for a Neumann problemwith discontinuous coe�cients (note that N means a di�erent kernel there), and wealso use Bryan's method for numerical computations.2. Similarly, a single layer ansatz is possible. The continuity of u up to the boundarygives directly: �u(s) = Z T0 �( ~X(t))N( ~X(s); ~X(t))dt:This approach results in a Fredholm equation of the �rst kind for �. The numericalinversion of these equations is known to be ill-behaved, so the single layer ansatz isless favorable than the double layer ansatz.Once � or � are found, the Dirichlet problem can be solved by simply evaluating the formulas(3.27) or (3.28) in �.3.7.5 EJIIM as a Boundary Integral MethodAgain, as simplest case we consider the Dirichlet problem on � � R2. Again � = @�.According to Folland [11, Thm. 2.23], any harmonic w 2 C1(�) \ C1() satis�esw(~x) = Z� w( ~X)@�;2N(~x; ~X)� w�( ~X)N(~x; ~X)d�( ~X): (3.29)Viewed as the sum of a double and a single layer potential, this extends w also for~x 62 �, where w and w� inside the integral in (3.29) need to be interpreted as w� andw�� . We see that [w] = �w� and [w�] = �w�� , and hence w+ = 0 and w+� = 0 on �, whichmeans that w+ is identically zero on R2 n f�[�g. Now let R be a rectangle with R � �and H be harmonic in the larger (but bounded) domain R. By Green's second identity,Z� w�( ~X)H�( ~X)�H( ~X)w�� ( ~X)d�( ~X) = Z� w�(~x)�H(~x)�H(~x)�w�(~x)d~x:Both terms under the integral on the right vanish since w and H are harmonic in �. Bychoosing H(~x; ~y) = N(~x; ~y)� G(~x; ~y), where G is the Green's kernel for R, we see that forall ~x 2 R also w(~x) = Z� w�( ~X)@�;2G(~x; ~X)� w�� ( ~X)G(~x; ~X)d�( ~X): (3.30)

66That is, the interpretation in terms of jumps carries over from N to G. The function@�;2G(~x; ~y) is known as the Poisson kernel for R. Our particular choice for R was therectangle, but we see that di�erent choices of R allow more e�cient methods. For example,making R a large enough disk has the advantage of a known Poisson kernel and could thusavoid the inversion of �h that we currently require to approximate G.The solution u to the Dirichlet problem satis�es the boundary condition, i.e.u(~x) = Z� �u( ~X)@�;2G(~x; ~X)� u�� ( ~X)G(~x; ~X)d�( ~X): (3.31)Remark 3.16 Replacing u by �u has to be done with care, since ~u = u+k satis�es [~u] = ��uand [~u�] = �u�� for any constant k. When using (3.31) to compute u, connected componentsof + that share a boundary with R (exterior regions) cause no problem, since the Green'skernel forces u to be identically zero on these components. For connected components of +that do not share a boundary with R, we need to ensure k = 0 by other means; see x3.8.Finally, we keep known terms on the right, di�erentiate (3.31) once again in the normaldirection on � (in our usual sense, meaning the limit in �) and evaluate on � to �ndu��(s)( ~X(s)) + �@�(s) Z T0 G(~x; ~X(t))u��(t)( ~X(t))dt�����~x= ~X(s) =�@�(s) Z T0 @�(t);2G(~x; ~X)��� ~X= ~X(t) �u( ~X(t))dt�����~x= ~X(s) : (3.32)As singular integrals over R, this becomes the following integro-di�erential equation on theboundary:c(s) + �@�(s) ZRG(~x; ~y) Z T0 �(~y � ~X(t))c(t)dt d~y�����~x= ~X(s) =0B@@�(s) ZRG(~x; ~y) Z T0 ��(t)(~y � ~X(t)) (��u(t))| {z }=[u] dt d~y1CA�������~x= ~X(s) ; (3.33)where c(s) = u�� ( ~X(s)). We write the equation in this particular form to point out thecorrespondence with (3.26). Here, F2 = 0 but F1 has absorbed the known jump [u] = ��u,i.e. F1 = �C�. The other approximations are c � C, @� � �DT , RRG( ~X(t); ~y) � ��1h andR T0 �(~y � ~X(t)) � 1, R T0 ��(t)(~y � ~X(t)) � �. Once c is determined, (3.31) is used to �ndu, just like the usual BIM.Under the same correspondences, the second Schur-complement of (3.25), when precon-ditioned by ��1h , �I +��1h DT �U = ��1h (F1 +F2); (3.34)turns out to be a discretization of (3.31).The correspondence between EJIIM and these integro di�erential equations may turnout very useful in the future analysis of EJIIM and is one of the main insights of this work.

67Remark 3.17 Di�erent from Lemma 2.40, the signs in the integral equations agree withthe signs in the discrete equations. This is due to the reversed signs on the jumps: inLemma 2.40, [ux] = u+x , [u] = u� and DT � �DTx;�+, while here [u�] = �u�� , [u] = ��u andDT � �DT�;��.Remark 3.18 Interestingly, our ansatz leads to a Fredholm Integral Equation of the secondkind for a single layer density. The essential di�erence between our approach and Mayo's[29] or capacitance matrix methods (see for example Proskurowski and Widlund, [35]) isthat we do not just allow a double layer (\capacitance") potential, but a combined singleand double layer potential, with one of the densities known.3.8 Neumann boundary conditions and �xed values at interior grid pointsIn x3.6.1, we found the need to enforce values at interior grid points; see also x3.8.1. Inx3.8.2 we need to discretize Neumann boundary conditions. These things are done exactlyas described in x2.7.1 for 1D.Consider the discretization of the Poisson problem, �hU = F , for the rectangle =(0; a)� (0; b) = (0; nxh)� (0; nyh).Suppose we want to enforce a Neumann boundary condition at the single grid point(nxh; jh) = (a; jh) on the right boundary. To be precise,@@xu(a; jh) = ureplaces u(a; jh) = 0. We introduce the auxiliary variable C to approximate u(a; jh).Thisvalue now enters the equation, as in AU = F + C, where 2 R(nx�1)(ny�1) has onlyone nonzero entry, with value �1=h2 in the row corresponding to the point (a; jh). Theadditional equation to determine C is the O�h2� approximation12h (3C � 4Unx�1;j + Unx�2;j) = u:Similarly, we enforce the value �u at the point (ih; jh). On the right hand side of theequation for Uij , include the correction term C=h2 + �u=h2 � Fij , so that that equationbecomes Ui+1;j + Ui�1;j + Ui;j+1 + Ui;j�1 � 4Ui;jh2 = �uh2 + Ch2 ; (3.35)with the additional equationC = Ui+1;j + Ui�1;j + Ui;j+1 + Ui;j�1 � 5Ui;j:Again, this can be written as �hU �C = ~F ; (3.36)DTU + C = 0; (3.37)

68where ~F is equal to F , except that the entry for (ih; jh) is replaced by �u=h2. Any solutionto this system must satisfy Uij = �u.Clearly, both of these techniques can be extended to more than just one point, combinedwith each other, and with other conditions on interior interfaces, as long as the corrections\do not interfere', or \know about each other" (In x3.8.1, a one-sided limit needs to bedistributed to one side of the interface, while a value is assigned on the other side of theinterface). In introducing values at grid points, the Neumann boundary conditions and\interior Dirichlet conditions" do not allow viewing the auxiliary variables as jumps acrossan interface. Our treatment of these conditions is rather similar to the way that capacitancematrix methods would do it.3.8.1 Example: \Exterior" Dirichlet BVPThis example is interesting because it demonstrates how EJIIM can handle Dirchlet BVPof \exterior" type. The IIM [21] did not cover BVP on irregular domains (at least Li didn'ttreat them via jump conditions in [24]) but was adapted by Yang [42] to this case.Consider the Dirichlet BVP, �u = 0, u = 0 on @, and u = 1 on � given in polarcoordinates centered at (0:5; 0:5) by r(�) = 0:25 + 0:05 sin(6�), � 2 [0; 2�]. The solutionis extended to be 1 in � (so [u] = 0) with di�erences enforcing this value on all interiorneighbors of �; see Figure 3.8. Table 3.3 shows how the number m of interior neighbors ofthe interface increases like O(n), where the number of grid points is n2, while the number ofGMRES iterations grows more slowly slowly. In the � column, we have divided the Flopsby the number of GMRES iterations and by n2 ln n). The quotients indicate that the workperformed per iteration is O(n2 lnn). They decrease slightly because we have not separatedsome work of O(n). For n = 20, the mesh did not resolve the interface well enough forone-sided di�erences. In the future we want to keep the number of iterations constant byreducing the number of auxiliary variables, similar to [25].Table 3.3: Numerical results for \exterior" Dirichlet BVP.n m its Flops �40 72 36 3.0e+07 3.3e+0280 146 47 1.7e+08 3.0e+02160 292 63 1.0e+09 2.8e+02320 586 83 5.9e+09 2.8e+023.8.2 Example: Groundwater Flow ProblemThese examples are of interest because they combine three (two) di�erent interface condi-tions. We consider an equation for the pressure in a problem of (steady) groundwater ow,as previously studied by Yang [42]. Included in the computational domain are one morepermeable object and one less permeable object, modeled by discontinuous � with interiordi�erences, and two impermeable objects, modeled by the Neumann boundary condition

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(0; 0) (1; 0)xyFigure 3.9: The geometry of the �rst groundwater ow problem in Example 3.8.2. Onehighly permeable region 4, one not very permeable region 2, and two impermeable regions1 and 3. Dirichlet conditions on x = 0, x = 1 and Neumann conditions at y = 0 andy = 1. Superscript \�" indicates that di�erences are done in the interior domain for theinterface given by the boundary of the object.

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nz = 10697a) b)Figure 3.10: Computed solution and sparsity structure of the system in Example 3.8.2. Ina), the plateau indicates a region of higher permeability, while the solution behaves verydi�erently around the impermeable and nearly impermeable regions. Inside the two imper-meable objects, the solution is very close to zero as it should be by our ansatz. In b), sixdi�erent interfaces can be distinguished: two Neumann conditions, two regions with inte-rior di�erences and two regions with exterior di�erences. Standard ordering of the meshvariables and sorting the intersections with arclength yields patterns that \resemble" theinterfaces.u�n = 0. See Figure 3.9 for the geometry of the interfaces. For the jumps across @+2 and@+4 , we use interior di�erences (motivated by the results in x4.2.1), while for jumps across@+1 and @+3 we have to use exterior di�erences. For exterior di�erences to work, theinterfaces have to be su�ciently far apart. At the top and bottom boundaries, we imposeuy = 0. These boundaries are treated via Schur-complements as well to make use of a fastPoisson solver. On the left and right boundaries, the pressure is at two di�erent constants.A numerical solution is plotted in Figure 3.10 a). The jump condition [u] = �u� setsup a Dirichlet problem inside the impermeable objects, with boundary condition zero, andhence the solution is zero there. Figure 3.10 a) shows that this approximately the case.In our setup di�erent conditions for di�erent interfaces can be incorporated by simplystacking the sparse matrices. In Figure 3.10 b), we see the sparsity pattern of the matrixin system (3.25). The bottom left contains the entries for D, with distinct entries forthe two boundaries with Neumann boundary conditions and the four objects. On the topright we see the entries of , the discretization of the deltas (for permeable objects) anddipoles (for impermeable objects). The stacking can be carried out to a much higher degreethan in Figure 3.10. Figure 3.11 shows an example with 25 included objects, and we havesuccessfully carried out experiments with up to 1985 circular objects on a 256� 256 mesh.

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ya) b)Figure 3.11: a) Computed solution and b) mesh, interfaces and contour lines for a problemwith 25 objects of very high contrast (1000). We impose Dirichlet boundary conditions onthe right and left boundaries, and Neumann boundary conditions on the top and bottomboundaries.3.8.3 Example: Local Re�nementTo further illustrate the power of the ideas from x3.8, we consider the following \localre�nement" problem. Given the function u, whereu�(r; �) = r2 cos��+ 1 + s4(�� 1) ;u+(r; �) = (r2(�+ 1)� s4r�2(�� 1)) cos�2(�+ 1 + s4(�� 1)) ;u(x; y) = (u�(r; �) for (r; �) 2 � = f(x; y) 2 [0; 1]� [0; 1]jr < sgu+(r; �) for (r; �) 2 + = f(x; y) 2 [0; 1]� [0; 1]jr > sg;where s = 0:0455 is the radius of the circular interface. Here r = p(x� 0:6)2 + (y � 0:6)2and � = arctan((y � 0:6)=(x� 0:6)) are polar coordinates centered at (0:6; 0:6). Then usatis�es �u = 0 with [u] = 0 and u+n = �u�n at the interface r = s. This is another specialcase of a general analytic solution for circular interfaces given in x4.3.3. Since the radius ofthe interface is small, we need a �ne mesh to resolve it. In order not to use a very �ne meshon all of the domain (0; 1)�(0; 1), we re�ne the mesh in a neighborhood of the interface. Thecoarse mesh has hc = 1=16 = 0:0625, and the �ne mesh has hf = 1=64 = 0:016625, whichresolves the interface well enough4. Figure 3.12 shows the solution (with � = 150) that we4 Subscripts c and f denote quantities related to the (global) coarse and (local) �ne grids.

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xyFigure 3.12: Solution for Example 3.8.3. The coarse grid with meshwidth hc = 1=16 isre�ned to meshwidth hf = 1=64 in a neighborhood of a small circular region of high con-ductivity. The correct values to put on the interface between grids are found by a discreteSteklov{Poincar�e operator and interpolation by splines.compute on the mesh where we have re�ned the square patch (6=16; 13=16)� (6=16; 13=16),whose boundary we view as an \arti�cial interface".We solve this problem using the following domain decomposition idea. Given values atcoarse grid points on the arti�cial interface, this de�nes an \exterior" Dirichlet problem.By interpolating these values to the rest of the boundary points on the �ne grid (we use acubic spline on each of the four sides), it also de�nes an \interior" Dirichlet problem, withan interface inside.The question of course is what values to use on the interior boundary of the coarse grid.One answer is to choose the values such that the normal derivative is continuous across thearti�cial interface. While the map uj@ ! u�j@ is known as the Dirichlet{to{Neumannmap, our closely related map u! PS(u) = u+� �u�� is called the Steklov{Poincar�e operator([7], usually written as u! u+n+u�n , where n is the respective outward normal, and points inopposite directions in the two terms). Our goal is to approximate the condition PS(u) = 0.Let the auxiliary values on the arti�cial boundary be denoted by �Up (which also absorbsthe usual factor �1=h2), where a subscript p will mean that a quantity is related to thecoarse grid points on the arti�cial boundary. By �p we denote the matrix that agrees with�hc in size and on the coarse grid points on the boundary, but has zero rows otherwise.Ip is a projection matrix of that same size which is the identity on the coarse grid pointson the arti�cial boundary, and zero otherwise. The matrix F in this case only containsthe coarse grid outer boundary values, which we take from the analytic solution. Then theexterior problem can be written as (c.f. (3.35))(�hc ��p + Ip)U = F + �Up;

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xya) b)Figure 3.13: a) The error in the solution computed on the partially re�ned grid, and b)the error on the uniformly �ne grid. The error magnitudes are almost the same, indicatingthat the dominating contribution to the error comes from the discretization of the interiorinterface, the boundary of the conductive region.with solution Uc = (�hc ��p + Ip)�1F + (�hc ��p + Ip)�1 �Up:If we write the interpolation from coarse to �ne grid points on the arti�cial boundary withthe operator S, and eliminate the jump variables from the interior problem, it may bewritten as (�hf � DT )U = S �Up;with solution Uf = (�hf � DT )�1S �Up:The discrete operators for the outward normal derivatives (note they point in oppositedirections) on the arti�cial interface (where from the inside, we only approximate the normalderivatives for points that are also coarse grid points) are written as DTc and DTf . Thediscretization of PS = 0 is then DTc Uc +DTf Uf = 0;or �DTc (�hc ��p + Ip)�1 +DTf (�hf � DT )�1S� �Up = �DTc (�hc ��p + Ip)�1F:This equation is now in a form that an iterative method can solve. We use GMRES, whereeach iteration in itself requires the inverses (�hc � �p + Ip)�1 and (�hf � DT )�1, that

74we also apply by solving systems iteratively. The e�ciency of this approach is clearly notgood, but our results indicate at least the feasibility of this kind of approach with EJIIM.Compare Figure 3.13, which shows the error on the partially re�ned grid, and on the all �negrid, which in the maximum norm are very comparable, 5:13e�4 and 5:11e�4, respectively.This approach to grid re�nement also obviously extends to Poisson problems.Remark 3.19 In order not to clutter the notation even further, we have oversimpli�edwhat our computer codes do. To round the awkward \reentrant corners" in the exteriordomain, these four corner grid points are actually treated as interior points of the exteriordomain (following a suggestion from Strikwerda and Scarbnick [38]). The interpolationoperator uses these values, and so the interior solution actually (weakly) depends on theexterior solution for a given �Up, Uf = (�hf � DT )�1SUc (with appropriately rede�ningS), which further complicates the above formulas.3.9 Liouville transformation for variable discontinuous coe�cient problemsWe also explore a di�erent approach to variable discontinuous coe�cient problems, knownfor smooth coe�cients as the Liouville Transformation (Stefanesco Transformation). Thistransformation is used for example by Concus and Golub in [9] mainly for smooth coe�cientson irregular domains, but they also include an example with piecewise constant coe�cients.However, they do not use jump conditions, and report rapidly deteriorating behavior oftheir method for non-smooth or discontinuous coe�cients. They suggest using domaindecomposition into regions with smooth coe�cients as a possible solution, but demonstratethe e�ect only for a problem where each subdomain is a rectangle.3.9.1 Jump conditionsWe change variables: ~u = p�u. Then�~u = ��p��u + 2r�p�� � ru+p��u= � �p��p� ~u+p��r� � ru� +�u�using (3.21)= � �p��p� ~u+ fp� ;and �nally �~u� � �p��p� ~u = fp� : (3.38)This turns the problem into a variable coe�cient Helmholtz equation, of the form of (3.11).Of course, this equation is only valid away from the interface. To use EJIIM, we also needto derive the jump conditions for ~u; again the derivations follow closely the steps executedin deriving the jump conditions for problem a). For example, the solution ~u inherits the

75discontinuity from �. Note that � is now a function of position. By ��, for example, wemean (@=@�)(��=�+), in the sense of taking the appropriate limits.[~u] = ���1=2 � 1� ~u�;[~u�] = ��1=2� 1� ~u�� + �+��+ [~u] + [(p�)�]~u�p�+ ;[�~u] = � fp� �+ ��(p�)p� � ~u� + �(p�)+p�+ [~u] ;[~u�] = � ��3=2��2 ~u� + ���1=2 � 1� ~u�� ;[~u��] = �0[~u�]� ��3=2��~u�� + 3��5=2�2� � 2��3=2 (��0�� + ���)4 ~u�+ ���1=2 � 1����0~u�� + ~u���� ;[~u��] = � �0[~u�] + ��1=2��2 ~u�� + ��1=2 � 1��~u��� + �0~u�� �+ ��+�� + �0�+� � �+ � �+� �+�(�+)2 [~u] + �+��+ [~u�]� � ��2p�� �+� (�+)�3=22 ~u�+ h2 ���� + �0�����1=2 � ������3=2i (�+)�1=24 ~u� + � ��2p�� ��+��1=2 ~u�� :(3.39)

The fourth and �fth jump equation are derived by taking derivatives of the �rst equation,and the sixth equation is derived from the second equation. The validity of these conditionsis (partially) numerically veri�ed below | many terms vanish in that example because ofthe special choice of �.3.9.2 Example: Liouville transformation for discontinuous coe�cientsWe compare EJIIM with results for the IIM on an example from [21]. Let r2 = x2+ y2 andr � (�ru) = f + C Z� �(~x� ~X(~y))d�(~y); (3.40)f(x; y) = 8r2 + 4; (3.41)�(x; y) = (r2 + 1 if r � 0:5;b if r > 0:5: (3.42)The Dirichlet boundary conditions are determined from the exact solution,u(x; y) = (r2 if r < 0:5;�1� 98b� =4 + �r4 + 2r2� =(2b) + C log(2r)=b if r > 0:5: (3.43)

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xyFigure 3.14: Solution ~u for Example 3.9.2 with b = 10, computed on a 40� 40 mesh.The two parameters b and C may be chosen arbitrarily. The jump conditions (3.39) simplifysigni�cantly, but we need to take care of the case C 6= 0, where the jumps look as follows:[~u] = 2pbp5 � 1! ~u�;[~u�] = p52pb � 1! ~u�� � p55pb ~u� + 2Cpb ;[~u�] = 2pbp5 � 1! ~u�� ;[~u��] = p5pb � 4pbp5 ! ~u�� � 2p55pb ~u� + 2pbp5 � 1! ~u��� + 4Cpb ;[~u��] = 6p5pb � 4pbp5 ! ~u�� + p52pb � 1! ~u���;[~u��] = 4pbp5 � p5pb! ~u�� + 2p55pb � 3625! ~u� � 2pbp5 � 1! ~u��� � 4C � 6pb � 12p55 :The behavior for extreme values of b is of interest, but [21] only provides data forthe cases b 2 f10;�3g and C = 0:1. The solutions for these two cases are shown inFigure 3.14 and Figure 3.15. For b < 0, the Liouville transform may also be applied,after multiplying the equation by �1 on � in order to be able to take p�. This re-quires the following replacements in the jump equations: C ! �C, �1=2 ! ��1=2, and[(p�)�]=p�+ ! �[(p�)�]=p�+. Finally, f+ changes sign and a�ects one of the constantterms. Table 3.4 shows the results from [21] and for our method for b = 10 and C = 0:1.Both methods converge quadratically, but EJIIM gives better results particularly on the

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ya) b)Figure 3.15: a) Solution u for Example 3.9.2 with b = 10, computed on a 40� 40 mesh. b)Solution u for b = �3.coarser grids. To demonstrate the e�ciency of our method, we also include the numberof iterations that GMRES needs to solve the large linear systems for this problem. Thealmost constant behavior is in accordance with the experience reported in [9].Similar to (3.25), now (�h +K) ~U = F1 + CC = F2 �DT ~U; (3.44)where K is a diagonal matrix with entries �(p�)(xi; yj)=p�(xi; yj). Instead of using a fastsolver for the variable Helmholtz equation, we eliminate the second equation and work withthe bigger Schur-complement,��h +K +DT � ~U = F1 + F2; (3.45)and after preconditioning�I +��1h �K +DT �� ~U = ��1h (F1 + F2) : (3.46)Using GMRES lets us avoid forming the matrix on the left. From the results for b = �3 inTable 3.5, we see again that the number of iterations is fairly independent of the problemsize and rather low.The simple geometry and radial shape of � and u may account for some of this goodperformance, and further testing on more general geometries and coe�cients is planned.

78 Table 3.4: Results for Example 3.9.2 with b = 10, C = 0:1.IIM EJIIM (u) Liouville (~u) Liouville GMRESn kEnk1 � kEnk1 � kEnk1 � kTnk1 � iterations20 3.5e-3 7.6e-4 1.4e-3 3.4e-1 1040 7.6e-4 4.7 2.4e-4 3.2 7.5e-4 1.8 4.0e-1 0.9 1080 1.7e-4 4.6 7.9e-5 3.0 2.5e-4 3.0 2.6e-1 1.5 9160 3.6e-5 4.6 2.2e-5 3.6 6.9e-5 3.6 1.4e-1 1.9 9320 8.4e-6 4.3 5.3e-6 4.2 1.7e-5 4.1 9.2e-2 1.5 8Table 3.5: Results for Example 3.9.2 with b = �3, C = 0:1.EJIIM (u) Liouville (~u) Liouville GMRESn kEnk1 � kEnk1 � kTnk1 � iterations20 8.6e-3 1.4e-2 4.0e-1 1040 3.0e-3 2.9 5.2e-3 2.7 5.1e-1 0.8 1380 8.8e-4 3.4 1.5e-3 3.5 3.4e-1 1.5 12160 2.4e-4 3.7 4.2e-4 3.6 1.9e-1 1.8 11320 6.3e-5 3.8 1.1e-4 3.8 1.1e-1 1.7 103.10 Application to crack problemsA crack is a \thin" structure where material properties are very di�erent from the sur-roundings. We assume that it is usually infeasible to use a grid that resolves the crack, andinstead derive jump conditions treating the crack as a lower dimensional interface, to beused with EJIIM. This problem arises for example in porous media ow.3.10.1 Jump conditionsThe crack � lies between the regions � and +. It is described by the curve � =~X(t) = (X(t); Y (t)), where t is arclength (by our convention, + lies on the right of �),and the width5 w(t), in the sense that the crack boundaries are (for t 2 [0; T ]) �+ =~X(t) +w(t)~�(t)=2 and �� = ~X(t)�w(t)~�(t)=2. See Figure 3.16 for the situation. This is adiscontinuous coe�cient problem; u satis�es r � (�ru) = 0 in = � [�[+. But nowwe consider the case w � h, so that in general both �+ and �� will intersect the mesh inthe same intervals. The goal is to �nd jump conditions across the single interface � that aresatis�ed by the solution u, up to O�w2�. Second order approximation is needed because weare interested in large contrast problems, so that O(w) terms cannot be neglected, becausethey are multiplied by large constants.For simplicity, we restrict ourselves to piecewise constant conductivities, with values ��,5 Please note that w is di�erent from w in x3.5.

79� +� �+���DDDD w(t)Figure 3.16: The crack � lies between the regions � and +. It is described by thecurve � = ~X(t) = (X(t); Y (t)), where t is arclength and the width w(t): for t 2 [0; T ],�+ = ~X(t) + w(t)~�(t)=2 and �� = ~X(t)� w(t)~�(t)=2.�� and �+ in �, � and +, respectively. To avoid problems at the endpoints of the crack,we think of the situation where w0(0) = w0(T ) = 0 and �� = �+. The cases of a conductiveand resistive crack are of interest, i.e. the conductivity �� in � is either much larger ormuch smaller than in � and +. First we need some notation. As usual, q0 will denotederivatives of a quantity with respect to t along �. ~� = ~�(t) is the \outward normal" to �,pointing into +, and ~� = ~�(t) = (X 0(t); Y 0(T ))=k(X 0(t); Y 0(T ))k2 is the unit tangent. Inaddition to the usual u+, u� etc., which denotes limits in � and +, we also introduceu��, u�+, u��� , u�+� etc. to denote the limits in �. Jumps across �� will be denoted by[q]�(t) = lim~x2�;~x! ~X(t)�w(t)~�(t)=2q(~x)� lim~x2�;~x! ~X(t)�w(t)~�(t)=2q(~x):For example [u]� = u�� � u�, with analog de�nitions across �+, for example [u�]+ =u+� � u�+� .For our derivations, we use a linear model of u and u� (as functions of w) inside �. Tobe precise (but as usual suppressing the dependence on t), we assume that �, the normalto �, is a very good approximation to the normals to �� and �+.u�� = u� + [u]�;u�+ = u�� + wu��� +O�w2� ;u+ = u�+ + [u]+;u��� = u�� + [u�]�;u�+� = u��� + wu���� +O�w2� ;u+� = u�+� + [u�]+:

80 We copy the usual jump conditions (for piecewise constant coe�cients) across �� and�+ into our current notation:[u]� = 0;[u�]� = 0;[u�]� = ����� � 1� u�� ;[u��]� = (��)0����� � 1� u�� ;[u��]� = ����� � 1� (u�� )0;[u��]� = �f��� � (��)0����� � 1�u�� :[u]+ = 0;[u�]+ = 0;[u�]+ = � ���+ � 1� u�+� ;[u��]+ = (�+)0� ���+ � 1�u�+� ;[u��]+ = � ���+ � 1� (u�+� )0;[u��]+ = �f��+ � (�+)0� ���+ � 1� u�+� :We wish to replace the curvatures of �� and �+, (��)0 and (�+)0 by the curvature of �,�0. Observing that(X+)0(t) = (X 0(t); Y 0(t)) + w0(t)~�(t)=2 + w(t)�0(t)~�(t)=2;we see that this is legitimate as long as w0(t) is small and �0(t) is O(1), which follow fromour earlier assumption that the normals to �, �� and �+ are very close.The essential jump conditions follow by successively substituting earlier conditions:[u] = u+ � u� = u� + [u]� + wu��� + [u]+ � u� + O�w2�= w�u�� + ����� � 1�u�� �+ O�w2�= w���� u�� +O�w2� ; (3.47)

81[u�] = u+� � u�� = u�� + [u�]� + wu���� + [u�]+ � u�� +O�w2�= ����� � 1� u�� + wu���� + � ���+ � 1� u�+� + O�w2�= ����+ � 1� u�� + w ���+ u��� + �f��� � �0����� � 1� u�� !+ O�w2� :After replacing u��� = f�=�� � u��� = f�=�� � (u�)0 � �0u�� , we get[u�] = ����+ � 1� u�� + w ���+ �f+�+ � (u�� )0 � �0���� u�� �+ O�w2� : (3.48)Now, using the following general formulas (c.f. x3.5)ddt [u] = [u�];ddt [u�] = [u�� � �0u�];ddt [u�] = [u�� + �0u�];[u��] = �f��� [u��];we �nd the remaining jumps [u�] = w0���� u�� + w���� (u�� )0; (3.49)[u��] = [u�]0 + �0[u�]; (3.50)[u��] = [u�]0 � �0[u�]; (3.51)[u��] = �f��� �[u�]0 + �0[u�]� : (3.52)The last equation assumes f�� = f�+, i.e. f is constant on cross sections of the crack.The jump conditions (3.47){(3.52) can be discretized with EJIIM. We have a complete setof jumps to second order, where the only unknown quantities are u�� and u�� . By using\primed" quantities, to be computed by �tting a spline and then taking derivatives on thatspline, we avoid second and third derivatives in � and �. Those would come up when gettingrid of the arclength derivative for example in the [u�]0 term.Finally, in the important special cases where w0 and w00 can be neglected and f isuniformly zero, the following approximations may be used, now writing � = ��=�� =��=�+.

82a) �� � �+ = �� (resistive crack) [u] = wu�� =�;[u�] = w(u�� )0=�;[u�] = 0;[u��] = w(u�� )00=�;[u��] = �w(u�� )00=�;[u��] = �w�0(u�� )0=�: (3.53)b) �� � �� = �+ (conductive crack) [u] = 0;[u�] = 0;[u�] = �w(u�� )0�;[u��] = �w�0(u�� )0�;[u��] = w�0(u�� )0�;[u��] = �w(u�� )00�: (3.54)This predicted behavior agrees well with the following study of thin ellipses that are justwide enough to allow treatment as two interfaces with the jump conditions from x3.6.2.3.10.2 Example: Inclusions that are almost cracksThe examples shown here are similar to example 4.2.2, with constant Dirichlet values onthe right and left boundaries, and high contrast (� = 1000 and � = 1=1000) inclusions.However, we have Neumann conditions on the top and bottom boundaries. Figure 3.17and Figure 3.18 show contour lines for solutions similar to 4.2.1 b) (resistive object), whileFigure 3.19 and Figure 3.20 show the contour lines of solutions similar to 4.2.1 a) (conductiveobject). The di�erence in the sequences are the rotation of the ellipses. These �gures areincluded both to demonstrate the \qualitative" validity of (3.53) and (3.54) (at least for [u],[u�] and [u�]), and again to demonstrate the continuous dependence (here on the angle of theellipse) of the solutions found with EJIIM. In particular, the \ at" portion in Figure 3.19and Figure 3.20 remains almost at the exact same height.

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a) b)Figure 3.20: Contour lines for a thin, conductive ellipse, with Dirichlet conditions on thesides, and Neumann conditions on the top and bottom boundaries. a) 40 degrees fromparallel to the y-axis, b) 50 degrees from parallel to the y-axis.

Chapter 4NUMERICAL METHODS FOR ELECTRICAL IMPEDANCETOMOGRAPHYElectrical Impedance Tomography (EIT) is a technique that is used in the context ofnon-destructive testing. Its purpose is to �nd the conductivity distribution inside a regionusing electrostatic measurements collected on the boundary of the region. Compared toother techniques used in medical imaging, such as X-ray or magnetic resonance imaging,EIT has poor resolution but allows real-time imaging at relatively low cost and long termmonitoring due to the small, low frequency currents involved [3]. Mathematically, EITreduces to an elliptic inverse problem as described in the introduction.A simpli�ed version of the general EIT problem was studied by A. Friedman [13] andlater by K. Bryan in his thesis [5]: suppose the medium has constant background conductiv-ity but might contain some buried object of unknown shape and di�erent, but also constant,unknown conductivity. Can we, by making electrostatic measurements on the boundary,detect whether there is a buried object or not? In the presence of such an object, we as-sume we know its conductivity and try to determine the shape of the object boundary, theinterface between di�erent conductivity values.Friedman proved that one can detect a perturbation in the medium by making onemeasurement. Bryan extended this result to the case in which the background conductivityhas two continuous derivatives, and also developed a fast algorithm to �nd the approximatelocation of an object buried in the unit disk numerically. An alternative method to �nd anapproximate solution can be based on EJIIM. In particular, in the case that we know thetwo values of a piecewise constant conductivity, the problem reduces to �nding the locationof the interface between the two values. EJIIM would work on two interfaces, the otherbeing the boundary of the unit disk which is embedded in a square. In x4.2 and x4.3, wegive examples of the infeasibility of the IIM, and of the improved performance of EJIIMas a forward solver for this problem. Once the approximate solution is found, the questionarises: can we �nd a better \approximation" to the shape of the buried object than theinitial approximation? Our answer to this question is yes, at least under certain conditions.We present a procedure to reconstruct the interface approximately, based on the as-sumption that we have already �tted a circular object with known conductivity to theboundary data, by estimating its center and radius. The use of a circular domain is neces-sary for our method, but also used in experiments; see for example [8]. The main di�erencebetween our approach and Bryan's approach in [5] is that he linearized about a constantconductivity �, while we linearize about a circular interface. This avoids the reliance onsmall contrast in the conductivity which is present in Bryan's and other methods. Dob-son and Santosa [10] also assume that the conductivity is some small perturbation abouta constant background, which is clearly not valid for large contrast problems. Their totalvariation diminishing method allows them to recover \images" that are sharper than the

86usual pseudo-inverse solutions. Their approach is similar to ours in that they assume a pri-ori knowledge, namely that the conductivity is piecewise constant. We assume even more,that we know the piecewise constant values of the conductivity, and we solve a di�erentproblem: we locate the discontinuity.We will assume that the object is centered at the origin, since one can always achievethis using linear fractional transformations from the disk to the disk. Assuming that theunknown interface is star-shaped with respect to the origin and not too di�erent from thecircle, we develop a linearized relation of integrals yielding the output voltage data thatresult from the interface under two types of �xed input currents, trigonometric functionsand dipoles. Our algorithm is based on this integral relation. The linearization of anintegral equation and inversion via Fourier coe�cients follows Kaup and Santosa [18], whodetect material loss at an inaccessible boundary of an originally rectangular object, i.e. theyestimate the (small) perturbation from a straight line segment in that portion of the objectboundary. The speci�c input patterns we select are commonly used in EIT; see [32].The quality of the perturbation recovered by our method depends on the input frequencyas well as the frequencies of the perturbation. Low frequencies work best as input, and aremost easily recovered, in that case even if the amplitude of the perturbation is not verysmall. On the other hand, several dipoles, corresponding to very close pairs of electrodesto induce the current, can also be used successfully.4.1 Experience with the IIMConsider the problem of �nding the unknown location of a circular object � of radius 0:25and conductivity � = 25, \buried" in the domain = [�3; 3]� [�2; 0], with backgroundconductivity � = 1 in n �. Suppose we have one measurement u� for �u = 5 on the\surface" fy = 0g and zero on the rest of the boundary. We attempt to �nd the object bysolving r � (�(xc; yc)ru) = 0 in with u = �u on @, where �(xc; yc) is the conductivitythat is 25 inside a circle of radius 0:25 centered at (xc; yc), and 1 elsewhere. We denote theoutward normal derivative to this solution by u�(xc; yc), and would like to solve u�(xc; yc) =u� via �nding the solution of min(xc;yc)2B ku�(xc; yc)� u�k22: (4.1)Numerically, we �nd U(xc; yc) with the IIM, then use �nite di�erences to get U�(xc; yc).Unfortunately, the IIM exhibits the following instability. Figure 4.1 a) shows the potentialsU found by the IIM when using a uniform grid with h = 0:1, with � centered at (0,-1.075)and (0,-1.076). Estimates of the normal derivative at the surface based on these (completelyincorrect) potentials would point an algorithm that is trying to �nd the location of � inopposite directions. Figure 4.2 shows the solution computed with mesh-width h = 0:05for (0,-1.075), which exhibits the correct behavior, and Figure 4.1 b) shows the di�erencebetween this potential and the one for (0,-1.076). The di�erence is very small in magnitude,as it should be. A small shift of the object should not change the potential much. Thisproblematic \discontinuous behavior" is typical for the IIM for discontinuous coe�cientsin the conductive case (the included object has larger conductivity than the background,

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a) b)Figure 4.2: a) Solution found by the IIM on a uniform grid with h = 0:05, when the interfaceis centered at (0;�1:075). b) Di�erence between the solution in a) and the one found for aninterface centered at (0;�1:076) on this �ner grid. This is the physically correct behavior.

88� = ��=�+ � 1), and can be observed also on much �ner meshes (note that we have notchosen a very complicated interface geometry, or very large contrast in �).The problems of the IIM for discontinuous coe�cients were tracked to what is nowknown as \bad stencils". For certain con�gurations of the interface{mesh geometry, the�nite di�erence coe�cients ( s) blow up in magnitude, and even worse, are far from \di-agonally dominant". Diagonal dominance is a nice feature that (with additional conditionson the signs of the coe�cients to guarantee that the scheme is \of positive type") allowsconvergence proofs for �nite di�erence methods for elliptic problems via a discrete maxi-mum principle. Recently, Adams and Calhoun [2] have reported some success on this issueby varying the expansion point, and Moskow and Santosa [31] are trying to prove that it isalways possible to �nd an expansion point that guarantees to generate a \scheme of positivetype".Excessive run-times for the IIM for discontinuous coe�cients on �ne meshes, togetherwith the observed discontinuous behavior which we were not able to predict, led us toabandon temporarily the use of the IIM for inverse problems. Instead of \�ltering theresults", we worked on stabilizing the IIM, namely the development of EJIIM. Our failureto develop an intuition for the \meaning" of the s of the IIM suggested the need to goback to 1D, and ultimately resulted in Chapter 2 and Chapter 3.Any version of the IIM can at best determine the location of an object to grid scale, soit is useful to have a method that can do the \�ne tuning". We will describe such a methodin x4.3. But now let us see what can be gained using EJIIM as a forward solver.4.2 Numerical stability of EJIIM and FIIIMBoth FIIIM and EJIIM improved the stability of the IIM signi�cantly, so that we willfocus here on hard problems with larger contrast and sometimes larger curvature than thetroublesome example for the IIM.4.2.1 Example: Composite material problemThis example is interesting because it shows how EJIIM compares with the Fast IterativeIIM [25] for problems with piecewise constant coe�cients. For large contrast in materialproperties, it can be crucial to do the one-sided interpolations on the correct side of theinterface. Let u�(r; �) = 2r cos��+ 1 + s2(�� 1) ;u+(r; �) = (r(�+ 1)� s2r�1(�� 1)) cos��+ 1 + s2(�� 1) ;u(x; y) = (u�(r; �) for (r; �) 2 � = f(x; y) 2 [0; 1]� [0; 1]jr < sg;u+(r; �) for (r; �) 2 + = f(x; y) 2 [0; 1]� [0; 1]jr > sg;where s is the radius of the circular interface. Here r = p(x� 0:5)2+ (y � 0:5)2 and� = arctan((y� 0:5)=(x� 0:5)) are polar coordinates centered at (0:5; 0:5). Then u satis�es

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xya) b)Figure 4.3: Exact solutions for Example 4.2.1. In a), we see the solution for � = 5000,in b) we see the solution for � = 1=5000. In both cases, the boundary data are foundby evaluating the analytic solutions. The solutions are scaled so that kuk1 = 1 in thecomputational domain, so that absolute errors are also relative errors.�u = 0 with [u] = 0 and u+n = �u�n at the interface r = s. This is a special case of a generalanalytic solution for circular interfaces found in [39], and also given in x4.3.3. Figure 4.3shows exact solutions on the square [0; 1]� [0; 1] for � = 5000 and � = 1=5000, with s = 0:25in both cases. The solutions are scaled so that kuk1 = 1 in the computational domain, sothat absolute errors in the tables are also relative errors.Tables 4.1 and 4.2 show the error in the in�nity norm for the Fast Iterative IIM [25],EJIIM with jump approximations based on interior, and EJIIM with jump approximationsbased on exterior points for these two problems, where the boundary values on the squareare found by evaluating the analytic solutions.The solution computed by the Fast Iterative IIM for the �rst problem (� = 5000) on the�nest mesh is good, but the behavior on the coarser meshes does not allow us to deducethis fact. For the second problem (� = 1=5000), the Fast Iterative IIM converges well withsecond order. EJIIM based on interior points performs extremely well for both examples,and in particular converges with second order. For the �rst problem, the solution found onthe coarsest mesh is better than the approximations on the �nest meshes found by theother two methods. EJIIM with jump approximations in the exterior performs well for the�rst example and poorly (in terms of error magnitude) for the second.This may be explained by looking at the truncation error. In �, u is planar in bothcases: the interior �nite di�erences for the normal derivatives are exact, the jump correctionsare exact and the truncation error results purely from the discretization of the partialdi�erential equation, which is con�rmed by comparing the truncation errors for interior

90 Table 4.1: Results for Example 4.2.1 with � = 5000.Fast Iterative IIM Interior EJIIM Exterior EJIIMn kEnk1 � kEnk1 � kTnk1 � kEnk1 � kTnk1 �25 5.7e�3 6.9e�4 5.1e�1 4.4e�2 6.6e+050 4.5e�2 0.1 1.7e�4 4.0 7.2e�1 0.7 1.2e�2 3.7 5.2e+0 1.3100 2.9e�2 1.5 4.4e�5 3.9 4.7e�1 1.5 3.3e�3 3.7 2.2e+0 2.4200 3.7e�3 7.7 1.1e�5 4.0 2.6e�1 1.8 9.5e�4 3.5 1.8e+0 1.2Table 4.2: Results for Example 4.2.1 with � = 1=5000.Fast Iterative IIM Interior EJIIM Exterior EJIIMn kEnk1 � kEnk1 � kTnk1 � kEnk1 � kTnk1 �25 1.6e�3 5.7e�4 3.6e�1 4.0e+0 2.4e+450 4.7e�4 3.3 1.7e�4 3.4 5.1e�1 0.7 1.7e+0 2.4 1.8e+4 1.4100 7.0e�5 6.8 3.9e�5 4.3 3.3e�1 1.5 1.9e�1 8.7 7.8e+3 2.3200 1.5e�5 4.6 9.8e�6 4.0 1.9e�1 1.8 2.4e�2 7.8 6.7e+3 1.2di�erences in column 6 in Tables 4.1 and 4.2. In +, u varies mildly but the third derivativesdo not vanish. In the �rst problem, the exterior di�erences are multiplied with a factor1=5000� 1 while in the second problem the factor is 5000� 1. Note that the � used in thede�nition of u is exactly the factor � = ��=�+ entering the interior di�erences in (3.22),while for exterior di�erences the the meaning of � is reciprocal.For the use as a forward solver, we also have to compare the solutions for di�erentinterfaces. Again, the analytic solutions can be used. For �xed �, we shift the center ofthe circular interface to random points in the interior of the unit square, and read o� thevalues of the solution on the boundary of the unit square. Then we scale the solution, sothat its maximum value on the grid is one. With this setup we can study the dependenceof the error on the interface location. By repeating the experiment for the same locations,but a di�erent value of �, we can also study the in uence of the contrast. We look at theworst case behavior and the mean behavior of the FIIIM and EJIIM with both exterior andinterior di�erences.4.2.2 Example: Stability of EJIIM as a forward solver for inverse problems.Here we study the use of EJIIM and the FIIIM as forward solvers. The goal is to establishnumerically the \continuous" dependence of the solution on the location of the interface,for grids that are as coarse as possible. We consider the following interface �, shown inFigure 4.4 a) for xc = 1:115 and yc = 1:795, marked by a \+" in Figure 4.4 a).X(�) = 0:6(cos� � 0:5 cos3�) + xc;Y (�) = 0:7(sin � � 0:1 sin 3� + 0:4 sin 7�) + yc:

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a) b)Figure 4.5: Response surfaces found with a) EJIIM and b) FIIIM for interfaces centered onthe dotted triangle in Figure 4.4 a). For each location, we plot uy(x; 3; �l) � uy(x; 3; ��).For locations 13 and 22 the responses are closest to zero, which is in good correlation withthe fact that those center points lie closest to the original center.

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a) b)Figure 4.6: Cross sections through the response surfaces, i.e. values of the normal derivativeat a point on the boundary as the interface is varied over the triangle. Results found withEJIIM (curve) and FIIIM (circles) at a) x = 0:26 and b) x = 0:66.This particular shape was used by Adams [1] and Li [25] to test the IIM. The geometryis interesting because the characteristics of both conductive and resistive included objectswill occur, since two portions of the exterior domain are almost surrounded by the interiordomain.We compute u satisfying r � (�ru) = 0 in (0; 2)� (0; 3), were � is discontinuous across�, with � = 1 in the exterior and � = � in the interior domain. On the boundary, u = y.Figure 4.4 b) shows the solution u computed with EJIIM for this interface, where � = 5000.It is interesting to vary �, rotate or otherwise deform the interface, and to shift it to seehow the solutions depend on changes in these parameters. We chose � = 5000, and neededa 152� 216 mesh in order to be able to do exterior di�erences1 for all interface locations(xc; yc) that are indicated by the dots in Figure 4.4 a). Let �� be the conductivity withinterface centered at (xc; yc), and �i (for i = 1; 2; : : : ; 30) the conductivity when the object iscentered at a dot on the triangle, indexed in counterclockwise order, starting in the bottomleft corner. For each \trial" �i, we compute an approximation of the potential u(x; y; �i),for the boundary condition fu = yg. From u we estimate the outward normal derivativeat fy = 3g by second order �nite di�erences, and subtract the estimate for the derivativefor the original location, i.e. we estimate uy(x; 3; �i)� uy(x; 3; ��). An inversion algorithmcould work with the resulting collection of responses, shown in Figure 4.5. Figure 4.6 hascross sections through these response surfaces; it shows how the response changes at a singlepoint (a): (0:26; 3) and b): (0:66; 3)) as we vary the interface. In Figure 4.5 a) and the1 Recall that this has better worst case behavior than interior di�erences, but far worse average behavior.

93s� + @DFigure 4.7: The interface �0 (inner circle) and the perturbed interface � (dashed interface).A current g is applied on @D, and the voltage response G is measured on @D. We �ndapproximations for the Fourier coe�cients of ��(�), the di�erence in the radial directionbetween � and �0.curves in Figure 4.6, we see that the responses for EJIIM vary smoothly with (xc; yc) (up tokinks occurring in the \location direction" corresponding to corners of the triangle), whilethe responses for FIIIM (Figure 4.5 b) and circles in Figure 4.6) are erratic especially forthe \oblique" steps in the diagonal portion of the triangle. Our mesh width and steps thereare chosen so that the relative position of interface and grid is di�erent for all points on thisdiagonal, which seems to cause the problems for the FIIIM. The behavior is best capturedin animations; see http://www.math.washington.edu/~awiegman/Math/index.html.The main observation is the following. \In a grid sense" the solutions found with EJIIMdepend continuously on the location of the interface (with error norm roughly independentof the interface location, error distribution following the interface, and errors independentof the magnitude of �). This is what one could hope for by analogy with what we can provefor EJIIM in 1D, Corollary 2.35. The error of the solution found with the FIIIM dependsmuch more on the relative locations of mesh and interface. These �ndings agree with allour experiments for large contrast (� � 1 or � � 1) and complicated geometries, andjustify our pursuit of one-sided di�erences in order to keep the constants in the truncationerror small (see results in x2.5 and x3.6.3). In inverse problems, where many solutions fordi�erent interface locations have to be compared with each other, this continuity (on rathercoarse grids) is a very attractive feature. Similarly, for the extension to 3D, this \coarsegrid" behavior is crucial, in both cases in order to keep the computational cost bearable.4.3 Recovery of perturbations of an interface in EIT via linearizationIn this section we will assume that a good circular approximation (centered at the origin)to the interface has been found with EJIIM.

944.3.1 The mathematical formulation of the problemConsider the Neumann BVP: �nd u satisfyingr � (�ru) = 0 in D; (4.2)@u@n ����@D = g on @D; (4.3)where D is the unit disk. D is divided into two open regions � and + by an interface� = @� = f(r; �)jr(�) = s + ��(�); � 2 [0; 2�], where 0 < � � 1, �(0) = �(2�) and� = O(1)g, an unknown simple closed curve. Here � = f(r; �)jr < s+ ��(�)g is assumedto be star-shaped with respect to the origin to allow this representation. The conductivity� is given as �(r; �) = (�� for (r; �) 2 �;�+ for (r; �) 2 +:Here and throughout this section, �� and �+ are known constants. The quantity that issigni�cant is their quotient, � = ��=�+. The special case � � 0 gives rise to the circularinterface �0 and conductivity �0. In this case, �0 is the open disk of radius s and +0 isthe open annulus of radii s and 1. Figure 4.7 shows the geometry. We impose the usualnormalization R@D u = 0 and take equation (4.2) to mean that both u and �@u=@n (here nis the unit outward normal to �, @u=@n = n �ru = run, since gradients are row vectors)are continuous across � (an assumption valid also in applications other than electrostaticsgoverned by the same equations) to make the solution u to the BVP unique, for g 2 H 12 .We are interested in the inverse problem to (4.2) and (4.3), namely, �nding � frommeasurements on the boundary. Our goal is to recover an approximation of the perturbedinterface by linearization about the interface �0 = f(r; �)jr(�) = sg from correspondingpairs of Dirichlet (G) and Neumann (g) boundary values (voltages and current uxes) on@D. More precisely, we give approximations for the Fourier coe�cients of ��.4.3.2 The linearized integral relation formulaWe need to consider the two interfaces � and �0 at the same time. Since they may intersectmany times, c.f. Figure 4.7, we need to treatD as a �nite number of open domains, separatedby � and �0. The weak form of (4.2) that we need isZD �rurvT dx dy = �+ Z@D gv ds: (4.4)Here u solves the perturbed BVP (4.2-4.3) with the interface �. We assume there are �nitelymany regions i with piecewise smooth boundary, so that �D = [i �i, @D\([i@i) = ;, andv is continuous on the closure �D of D, v is smooth in i for all i, and v extends smoothlyto �i for all i.Remark 4.1 To see that (4.4) holds, observe that D is cut into �nitely many open domains~i by � and the boundaries @i. On each of these domains, the analog of (4.4) holds by

95Green's �rst identity, and the interior boundary contributions eliminate each other by thecontinuity of both v and �@u=@n throughout . Note that this uses the divergence theoremfor regions with piecewise smooth boundaries.Example 4.1 Consider the situation shown in Figure 4.8. Both u and v are continuousin �. The function v is smooth in n 0 and smooth in 0, with smooth extensions tothe closures, while u is smooth in n (�1 [ �3) with r � (�ru) = 0. Here � is piecewiseconstant (values �� in 1 and �+ in n �1) with jump across @1 = �1 [ �3. By Green's�rst identity in the four sub domains in the following integrals, �u = 0 implies thatZ1n0 �rurvT dx dy = �� Z�1[��2 v @u@n ds; (4.5)Zn(1[0) �rurvT dx dy = �+ Z�[��1[��4 v @u@n ds; (4.6)Z0n1 �rurvT dx dy = �+ Z�4[��3 v @u@n ds; (4.7)Z0\1 �rurvT dx dy = �� Z�2[�3 v @u@n ds: (4.8)By continuity of v and �@u=@n on the boundaries, the contributions on �1, �2, �3, and �4cancel out, and Z �rurvT dx dy = �+ Z� v @u@n ds: (4.9)� @0 @1�4 �1�3 �2r r? 6� �Figure 4.8: Both u and v are continuous in �. The function v is smooth in n 0 andsmooth in 0, with smooth extensions to the closures, while u is smooth in n (�1 [ �3)with r � (�ru) = 0 where � is piecewise constant (values �� in 1 and �+ in n �1) withjump across @1 = �1 [ �3. Then R �rurvT dx dy = �+ R� v@u=@n ds:

96 Let u0 be the solution of the BVP (4.2-4.3) with the interface �0. We express � as aperturbation of �0, and also u as a perturbation of u0, with the notation u = u0+ �u1. Forany function q on D for which the limits exist, the jump [q] in q across � is given by thedi�erence of limits in + and �:[q]�(�) � lim(r;#)!(r(�);�)+ q(r; #)� lim(r;#)!(r(�);�)� q(r; #):As before, by O(�) we denote a quantity q(�) so that q(�)=� is uniformly bounded in �, forall � with 0 < � < �0 for some �0.Theorem 4.2 With this notation, the following equality holds to O(�):Z 2�0 �(�)��s ��0@u0@r @v@r�s + 1s ��0@u0@� @v@��s� d� = �+ Z@D u1(1; �)@v@� d�; (4.10)where v is any function that satis�es (4.2) with � replaced by �0.Proof. We change coordinates on the left hand side of (4.4) from Cartesian to polar co-ordinates (Jacobian J1, coordinate transformation G), and then to interface �tted polarcoordinates (Jacobian J2, coordinate transformation H), sending � to �0 and �xing 0 andthe boundary @D, via (r; �) = �r ln sln(s+��(�)) ; �� :The result is (to O��2�)ZD �rurvT dx dy = Z 10 �0 Z 2�0 J1J2rr;�uHGGTHTrr;�vT d� dr;whereJ1 = r�1 + ��(�) ln rs ln s � ; G = cos � sin �� sin �r �1� ��(�) ln rs ln s � cos �r �1� ��(�) ln rs ln s �! ;J2 = 1 + ��(�)(ln r + 1)s ln s ; H = 1� ��(�)(ln r+1)s ln s � ��0(�)r ln rs ln s0 1 ! ;are exact to O(�2). We write J1J2HGGTHT = M + �m whereM = �r 00 1r� and m = 1s ln s � �r�(�) ��0(�) ln r��0(�) ln r �(�)r � :With this notation (4.4) becomes (to O(�2))Z 10 Z 2�0 �0r (u0 + �u1) (M + �m)rvT d� dr = �+ Z@D gvds: (4.11)

97Dropping the remaining O(�2) terms, we getZ 10 Z 2�0 �0ru0MrvT| {z }1 +��0 �ru1MrvT +ru0mrvT� d� dr = Z@D �+gv| {z }2 ds: (4.12)The integrals over terms 1 and 2 in (4.12) cancel, because u0 satis�es (4.2, 4.3) for �0, andM is just the matrix for the change from Cartesian to polar coordinates, i.e. for �0. So weget � Z 10 Z 2�0 �0ru0mrvT d� dr = Z 10 Z 2�0 �0ru1MrvT d� dr: (4.13)We write out the left hand side of (4.13):� 1s ln s Z 10 Z 2�0 �0�(�)�@u0@� @v@� 1r � r @u0@r @v@r�+ �0�0(�) ln r��@u0@r @v@� � @u0@� @v@r� d� dr:Integrating the �0 term by parts gives� 1s ln s Z 10 Z 2�0 �0�(�)�@u0@� @v@� 1r � r @u0@r @v@r� d� dr� 1s ln s Z 10 �0 ln r ��(�)��@u0@r @v@� � @u0@� @v@r���=2��=0 dr| {z }1+ 1s ln s Z 10 Z 2�0 �0�(�) ln r @@� ��@u0@r @v@� � @u0@� @v@r�| {z }2 d� dr: (4.14)Term 1 drops out by periodicity of �, u0 and v in the � variable. Using Laplace's equationin polar coordinates, we rewrite term 2 as@@r �@u0@� @v@�� � @@r �r2@u0@r @v@r� :Integrating by parts in r; (4.14) becomes1s ln s 0BB@� Z 10 Z 2�0 �0�(�)�@u0@� @v@� 1r � r @u0@r @v@r� d� dr| {z }3+ Z 2�0 ��(�)�� ln r r2@u0@r @v@r�r=sr=0 + ��(�)�+ ln r r2@u0@r @v@r�r=1r=s d�� Z 2�0 Z 10 �(�)�0r@u0@r @v@r dr d�| {z }4 + Z 2�0 Z 10 �(�)�0r @u0@� @v@� dr d�| {z }5� Z 2�0 ��(�)�� ln r @u0@� @v@��r=sr=0 + ��(�)�+ ln r @u0@� @v@��r=1r=s d�! : (4.15)

98Terms 4 and 5 cancel out against term 3. For the remaining two terms, the contributionsat 0 and 1 vanish (near zero, ln r gets canceled by the r2 and (@u0=@�)(@v=@�) terms, whileln 1 = 0), and (4.15) reduces toZ 2�0 �(�)��s ��0@u0@r @v@r�s + 1s ��0@u0@� @v@��s� d�: (4.16)By Lemma 4.3, taking v to satisfy the equation (4.2) for �0, the right hand side of (4.13)is equal to Z 2�0 �+u1(1; �)@v(1; �)@� d�; (4.17)which, together with (4.13) and (4.16), proves the Theorem.Lemma 4.3 With the notation from Theorem 4.2 and its proof we haveZ 10 Z 2�0 �0ru1MrvT d� dr = Z 2�0 �+u1(1; �)@v(1; �)@� d�; (4.18)where v satis�es (4.2) for �0.Proof. Simply reverse the change of variables from the beginning of the proof of Theo-rem 4.2 observing that in this situation � = 0, and observe that now v plays the role of uin equation (4.4), while u1 plays the role of v there.4.3.3 Analytic solutions for circular interfacesThe simple nature of the \background" interface �0 allows analytic solutions to the BVPin this case, which we use for the inversion of the integral equation derived in Theorem 4.2.These solutions are found as follows. Given Neumann data g on @D, we make an ansatz ofunknown but �xed Dirichlet data f on �0. Let ue0 solve Laplace's equation in the annuluss < r < 1 with R@D ue0 = 0. Continuity of the conductive current �un(= �ur) and potentialu in all of D determine f and and the solution ui0 of Laplace's equation in the inner diskr < s. The form of the solutions becomes particularly simple when written in terms ofFourier coe�cients. On the annulus s < r < 1,ue0 = a0 + b0 ln r+ Xn2Znf0gein� �anrjnj + bnr�jnj� ; (4.19)which gives u0 on the boundary when setting r = 1. On the inner disk,ui0 =Xn2Z ein� fnsjnj rjnj (4.20)where g = Pn2Z gnein� and f = Pn2Z fnein� : For n = 0, R@D u = 0 implies a0 = 0,continuity of u in D implies b0 = 0, and hence we can only solve for input satisfying g0 = 0,and we always have f0 = 0.

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a) b)Figure 4.9: a) The harmonic u resulting from Neumann input cos(3�). b) The potentialu2=3;5 resulting from � = 5, with s = 2=3 for the same input.For n 2 Z n f0g, the coe�cients an, bn and fn have to satisfy the following linear systemof equations 0@ jnj �jnj 0sjnj s�jnj �1�+sjnj ��+s�jnj ���1A0@anbnfn1A = 0@gn001A : (4.21)The matrix on the left has determinant jnj(1��)sjnj�jnj(1+�)s�jnj, which is zero only for� = (s2jnj � 1)=(s2jnj + 1) < 0, so that we have analytic solutions for all problems (4.2{4.3)with circular interfaces where � > 0 and g0 = 0. Evaluating these solutions on the boundaryof an arbitrary domain containing � provides useful test problems for numerical methodsand are used in x4.2.1 and x3.8.3. In those examples we use real pure frequency solutions,i.e. an and bn are zero for all n 62 f1;�1g (x4.2.1) and for all n 62 f2;�2g (x3.8.3), and theremaining coe�cients satisfy an = �a�n and bn = �b�n.Figure 4.9 a) shows the harmonic function u with @u=@n = cos(3�) on @D, and R@D u =0. Figure 4.9 b) and Figure 4.10 a) show analytic solutions u2=3;5 and u3=4;5 for � =��=�+ = 5, where s = 2=3 and s = 3=4 respectively. Figure 4.10 b) shows the responses(us;5�u2=3;5, on @D) for s = 1=2; 17=24; 3=4. The amplitude of the response increases withjs� 2=3j, while the phase depends on the sign of s� 2=3. For example, s = 17=24 results inthe curve with the smallest amplitude, while s = 1=2 produces the curve with the oppositephase, since 1=2 < 2=3 < 17=24 < 3=4.It should be clear how these analytic solutions extend to a non-concentric circular in-terfaces, and to multiple concentric interfaces. In the electrical engineering literature thistechnique is called the series expansion method [14] and is praised both for its e�ciency,

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thetaa) b)Figure 4.10: a) The potential u3=4;5 resulting from from Neumann input cos(3�) for � = 5and s = 3=4. b) The responses (voltage di�erences on the boundary) us;5 � u2=3;5 fors = 1=2; 17=24; 3=4. The amplitude of the response increases with js� 2=3j, while the phasedepends on the sign of s� 2=3.and because the geometry is easy to set up accurately in a laboratory environment [3].4.3.4 Inversion FormulaSince we assume that u = u0 + �u1, we can �nd u1 on @D as the di�erence between themeasured response and ue0 on @D. At �rst, consider Neumann data of the form g(�) =e�ik� ; k 2 Z n f0g: By x4.3.3, v that satis�es (4.2) with � replaced by �0 can be foundanalytically. We use speci�c vm that satisfy the boundary condition gm(�) = e�im� ; m 2Z n f0g: This choice is made to make the right hand side of the integral equation (4.10)simple: it is �+u1(m), i.e. a multiple of the mth Fourier coe�cient of u1. The jump termson the left hand side of (4.10) can also be computed explicitly from the analytic solutions(we used the program Maple for all derivations):�s ��0@u0@� @v@��s = 4���+(�+ � ��)e�i(k+m)�s �(�+ + ��)s�jmj + (�� � �+)sjmj� �(�+ + ��)s�jkj + (�� � �+)sjkj�and1s ��0@u0@� @v@��s = 4 sgn(m) sgn(k)�+�+(�� � �+)e�i(k+m)�s �(�+ + ��)s�jmj + (�� � �+)sjmj� �(�+ + ��)s�jkj + (�� � �+)sjkj� :

101So we �nd the following formula for the Fourier coe�cients of �:�(k +m) = �+ck;m u1(m); (4.22)whereck;m = � 4(�� � �+)(�� � sgn(m) sgn(k)�+)s �(�+ + ��)s�jmj + (�� � �+)sjmj� �(�+ + ��)s�jkj + (�� � �+)sjkj� :Remark 4.4 With formula (4.22), one can recover all but one frequency. Sincem 6= 0, theinput frequency k can not be recovered. With two measurements, one gets two independentestimates for all but the two input frequencies, and one estimate each for the two inputfrequencies.Example 4.2 Fix � > 0, �+ � 1, �� � � and �(�) = 1. Use f(�) = e�ik� as input.We can construct the analytic solution u0 to the unperturbed problem, and also u for theperturbed problem, the latter by replacing s in the formula for u0 by s+�. Since u = u0+�u1,we check that u1 has Fourier coe�cientsu1(m) = 8><>:� 4 (�� 1) (�+ 1)s�(�+ 1) s�jkj + (�� 1) sjkj�2 + O(�) m = �k;0 otherwise. (4.23)Plugging this into equation (4.22) we get �(j) = 0 for all j 6= 0 and �(0) = 1+ O(�). Nowwe derive (4.23): looking at (4.21) it is clear that gm = 0 implies am = bm = fm = 0, whichgives u1(m) = 0 for m 6= �k. On the other hand, g�k = 1 impliesa�k = 1+ �jkj((1� �)(s+ �)2jkj � (1 + �)) :Also b�k = a�k+1=jkj. So u(�k) = 2a�k+1=jkj, and u0(�k) is given by the same formula,substituting � = 0. Linearization gives2a�k = 2(1 + �)jkj 1(1� �)s2jkj � (1 + �) + 2jkjs2jkj�1�+ O(�2)= 2(1 + �)jkj 1(1� �)s2jkj � (1 + �) � �2jkjs2jkj�1(1� �)((1� �)s2jkj � (1 + �))2 + O(�2)! :By de�nition, u1(�k) = � (u(�k)� u0(�k)), which impliesu1(�k) = �2(1 + �)jkj �2jkjs2jkj�1(1� �)((1� �)s2jkj � (1 + �))2 +O(�)= � 4(�� 1)(�+ 1)s((�� 1)sjkj + (1� �)s�jkj)2 + O(�):Remark 4.5 The inversion (4.22) generalizes for arbitrary input g toXk2Z �(k +m)gkck;m = �+u1(m); (4.24)where the ck;m are de�ned above and the gk are the Fourier coe�cients of the input g.

1024.3.5 Remarks on numerical resultsClearly, many parameters could be studied in numerical experiments. We present just afew examples. We �x the radius of the unperturbed interface s = 1=3, i.e. the disturbanceoccurs fairly far inside the object, the case in which EIT methods are reported to havedi�culties. We will also �x the inner conductivity � = 2 for simplicity. Please note that weare not trying to recover the value of � here, and that larger values would result in greaterdi�erences and hence make the shape recovery easier.We present results for two types of input, one which is theoretically convenient and onethat may be more relevant for practical purposes (i.e. using electrodes for actual measure-ments). In the examples in this section we will use �+ = 1 and denote � = ��.The �rst type of input is cos(k�) and sin(k�) for k 2 IN. The second type is a dipole at(1; 0). In the weak sense, on the torus, the Dirac � and its derivative (the dipole �0) at thepoint �0 are given by�(� � �0) � 12�Xk2Z e�ik�0eik� ; �0(� � �0) � i2�Xk2Zke�ik�0eik� :4.3.6 Solving the forward problem with a Boundary Integral MethodFor numerical tests, we need an independent method to create data, i.e. to solve the forwardproblem for arbitrary interfaces. Bryan based his work on (4.25) and (4.26), proved in theappendix of [5]: for x 2 @D,u(x) = u0(x) + (1� �) Z� u(�)@��N(x; �)dS�; (4.25)where u0 is the harmonic that solves equations (4.2{4.3) with � � 1 in D. This formula isused to �nd the response, based on knowledge of the solution u on the interface �, and N ,the Neumann function on D: N solves��N(x; �) = ��x;@n�N(x; �)���2@D = �12� ;Z@DN(x; �)dS� = 0;for each x 2 D, where �x denotes the Dirac delta at x and �� is the Laplacian appliedin the �-variable. N , @n�N and in particular the limit @n�N(�; �) for � on � are knownanalytically for D.The solution u also satis�es another identity on the interface. For x 2 �,u(x) = 21� �1 + � Z� u(�)@n�N(x; �)dS�+ 21+ �u0(x): (4.26)Equation (4.26) can be solved numerically as follows. Parameterize � as x(t) for t 2 [0; 2�]and de�ne xj = x(2�j=M) for j = 0; 1; : : :M �1. The Nystr�om method replaces (4.26) with

103the discrete approximationuj = 2(1� �)2�(1 + �)M M�1Xi=0 @n�N(xj; xi)ui + 21 + �u0;j for j = 0; 1; : : : ;M � 1; (4.27)where uj is an approximation to u(xj) and u0;j = u0(xj). Using the points xj and valuesuj , we can approximate u on @D by using any quadrature method on equation (4.25).We worked with Bryan's program from [5], which uses dipole input, and extended it toallow input of the form cos(k�) and sin(k�) for k 2 IN. Table 4.3 shows the quality of theboundary integral method. We compare the analytic solution for s = 1=3 and � = 2 forthe input function g = cos �. Both the interface and boundary are approximated using nequally spaced points. It is clear from the table that for n = 100 we can expect very goodsolutions to the forward problem.Table 4.3: Results comparing the analytic solution for s = 1=3, � = 2 and g = cos � withthe approximation computed with Bryan's Boundary Integral code, discretizing both �0 and@D with n equally spaced points. n error25 3.32e-0550 1.32e-09100 6.10e-14200 6.12e-144.3.7 Examples: Pure frequency inputFrom a theoretical point of view, it is most convenient to work with Neumann data of theform eik� , because of the particularly easy inversion via (4.22). We �rst consider the caseof a shifted circular interface of the form� = f(x0; y0) + s(cos(�); sin(�)) j � 2 [0; 2�)g;i.e. ��(�) = x0 cos �+y0 sin �+ps2 � x2 sin2 � + 2xy sin � cos � � y2 cos2 ��s. We show thetrue and recovered perturbation of the interface shifts of di�erent magnitudes in Figure 4.11.In Figure 4.12, we see the same results, now showing the true �� and recovered ��� asfunctions of �. We compute the response for ei� with Bryan's program on 80 equallyspaced boundary points and 100 equally spaced (in �) interface points, taking advantageof ei� = cos � + i sin �. To �nd the Fourier coe�cients of �, we use only the lowest 21frequencies of the response (i.e. m = 0;�1;�2; : : : ;�20 in (4.22)), and �ll in the missingFourier coe�cient (m = 0 corresponds to k = 1) from e�i� . The number of frequenciesshould be adjusted to the error in the measurements. 21 worked worked well for ouraccurate forward solver. The various locations of the shifted circle are given in Table 4.4.By the norms in the table we mean the discrete max norm, taken over 41 uniformly spaced

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alpha for circle shifted by (0.128,−0.128)a) b)Figure 4.11: True shifted circle (curve) and recovered shifted circle (circles). a) Small shift,� � 0:02. b) Large shift, � � 0:2. The dashed circle is the interface that we linearize about.0 1 2 3 4 5 6

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a) b)Figure 4.12: True perturbation (curve) and recovered perturbation (circles) �� of the inter-face for a shifted circle, as a function of �. Same results as in Figure 4.11. a) Small shift,� � 0:02. b) Large shift, � � 0:2.

105Table 4.4: Results for circles of radius 1=3 centered at various points near the origin. � isthe perturbed interface and �� the recovered perturbation (corresponds to curve and circles inFigure 4.12). Input frequency was k = 1. By � we mean the relative error k�� ��k1=k�k1.x0 y0 � k�� ��k1 �2.0e�3 2.0e�3 2.8e�3 1.15e�5 4.1e�30.0e+0 -5.7e�3 5.7e�3 4.63e�5 8.2e�3-8.0e�3 8.0e�3 1.1e�2 1.88e�4 1.7e�20.0e+0 2.3e�2 2.3e�2 7.80e�4 3.5e�24.5e�2 0.0e+0 4.5e�2 3.36e�3 7.4e�26.4e�2 6.4e�2 9.1e�2 1.58e�2 1.7e�11.3e�1 -1.3e�1 1.8e�1 9.86e�2 5.4e�10 1 2 3 4 5 6

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Pure Frequency input, k =5

a) b)Figure 4.13: True perturbation (curve) and recovered perturbation (circles) of the interfacefor a rosetta interface with l = 5, as a function of �. a) Input frequency k = 1. b) Inputfrequency k = 5. Here � = 0:02.points (vertical distance from circles to curve in Figure 4.12). From the table we see thatthe relative error grows linearly for small shifts, then grows slightly faster as we increasethe shift. Even for a fairly large perturbation, the recovered interface is a \visually" goodapproximation to the true interface, Figure 4.11 b). Our results are so good because werecover a low frequency perturbation, using low frequency input.Next we consider the case of a rosetta interface of the form� = f(r(�); �)jr(�) = s+ � sin(l�)g :Now we compare responses for the same input but di�erent interfaces, with various values

106Table 4.5: Results for various input (k) and rosetta (l) frequencies. � is the perturbedinterface (amplitude � = 0:02) and �� the recovered perturbation. The cases (l; k) = (5; 1)and (5; 5) correspond to curve and circles in Figure 4.13 a) and Figure 4.13 b), respectively.k�� ��k1 �k = 1 k = 2 k = 3 k = 4 k = 5 k = 1 k = 2 k = 3 k = 4 k = 5l=1 6.2e�4 2.5e�3 3.4e�3 4.8e�3 6.4e�3 3.1e�2 1.2e�1 1.7e�1 2.4e�1 3.2e�1l=2 1.3e�3 2.1e�3 3.4e�3 6.1e�3 6.6e�3 6.5e�2 1.1e�1 1.7e�1 3.1e�1 3.3e�1l=3 2.1e�3 2.8e�3 3.9e�3 5.2e�3 6.9e�3 1.1e�1 1.4e�1 1.9e�1 2.6e�1 3.4e�1l=4 3.0e�3 3.7e�3 4.5e�3 6.0e�3 7.1e�3 1.5e�1 1.9e�1 2.3e�1 3.0e�1 3.6e�1l=5 4.0e�3 4.6e�3 5.4e�3 6.4e�3 8.4e�3 2.0e�1 2.3e�1 2.7e�1 3.2e�1 4.2e�1Table 4.6: Results for various input and rosetta frequencies. � is the perturbed interface(amplitude � = 0:05) and �� the recovered perturbation.k�� ��k1 �k = 1 k = 2 k = 3 k = 4 k = 5 k = 1 k = 2 k = 3 k = 4 k = 5l=1 4.3e�3 1.7e�2 2.5e�2 3.7e�2 5.3e�2 8.5e�2 3.4e�1 5.0e�1 7.5e�1 1.1e+0l=2 9.0e�3 1.7e�2 2.7e�2 5.1e�2 5.9e�2 1.8e�1 3.4e�1 5.4e�1 1.0e+0 1.2e+0l=3 1.7e�2 2.3e�2 3.7e�2 5.0e�2 7.0e�2 3.3e�1 4.7e�1 7.5e�1 1.0e+0 1.4e+0l=4 2.7e�2 3.4e�2 4.5e�2 6.8e�2 7.3e�2 5.4e�1 6.8e�1 8.9e�1 1.4e+0 1.5e+0l=5 3.6e�2 4.3e�2 5.3e�2 6.8e�2 1.0e�1 7.3e�1 8.6e�1 1.1e+0 1.4e+0 2.0e+0of the rosetta frequency l, but keeping the amplitude � �xed at 0:02 or at 0:05. The�rst choice of � is made because we saw good results for perturbations around and belowthat magnitude in the previous example. Table 4.5 shows results for a range of input androsetta frequencies. Small frequencies work best as input, and low frequency perturbationsare recovered more easily. However, Figure 4.13 b) demonstrates that even higher inputfrequencies recover the behavior of the perturbation \in principle". In Table 4.6, where� = 0:05, we see that this quality (in terms of the supremum norm) deteriorates rapidly forincreased amplitudes of the rosetta, meaning that � is getting too large for the linearizationto be valid.4.3.8 Examples: Dipole inputWe now turn to the practically more relevant case of \discrete" input, the case of one ormore dipoles.In the case of one single dipole input, one needs to invert a system of the form (4.24),where gk = ik=2�. Given that the ck;m decrease and that we have measurement errors andlinearization errors, the question of derivation of a �nite system (in general) and the stabilityof its inversion goes far beyond the scope of this paper. We will, however, demonstrate thefeasibility (in principle) of our approach, including the improvements possible when usingmultiple measurements.

107Table 4.7: Results for circles of radius 1=3 centered at various points near the origin. �is the perturbed interface and �� the recovered perturbation. Solutions are obtained usingone and eight single dipole measurements, the latter with least square solution for Fouriercoe�cients. Single dipole 8 dipolesx0 y0 � k�� ��k1 � k� � ��k1 �2.0e�3 2.0e�3 2.8e�3 4.67e�5 1.7e�2 1.38e�5 4.9e�30.0e+0 -5.7e�3 5.7e�3 1.23e�4 2.2e�2 5.41e�5 9.7e�3-8.0e�3 8.0e�3 1.1e�2 7.25e�4 6.4e�2 2.16e�4 1.9e�20.0e+0 2.3e�2 2.3e�2 2.04e�3 9.1e�2 8.83e�4 4.0e�24.5e�2 0.0e+0 4.5e�2 1.88e�2 4.2e�1 3.74e�3 8.3e�26.4e�2 6.4e�2 9.1e�2 5.59e�2 6.2e�1 1.51e�2 1.7e�11.3e�1 -1.3e�1 1.8e�1 2.91e�1 1.6e+0 5.57e�2 3.1e�10 1 2 3 4 5 6

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

alpha for circle shifted by (0,0.02263)

Single Dipole input

0 1 2 3 4 5 6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

alpha for circle shifted by (0.128,−0.128)

Single Dipole input

a) b)Figure 4.14: True perturbation (curve) and recovered perturbation (circles) of the interfacefor a shifted circle, single dipole (at (1; 0)) input. a) Small shift, � � 0:02. b) Large shift,� � 0:2.

1080 1 2 3 4 5 6

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

alpha for circle shifted by (0,0.02263)

8 Dipole inputs, least square Fourier coefficients

0 1 2 3 4 5 6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

alpha for circle shifted by (0.128,−0.128)

8 Dipole inputs, least square Fourier coefficients

a) b)Figure 4.15: True perturbation (curve) and recovered perturbation (circles) of the interfacefor a shifted circle, using measurements for a dipole at 8 uniformly spaced points on @Dwith least square solution for Fourier coe�cients. a) Small shift, � � 0:02. b) Large shift,� � 0:2.First, we consider the case of a shifted circular interface. For a single dipole we set up asystem of 9 equations, recovering the Fourier coe�cients 0;�1;�2;�3;�4. The equationsare all of the form (4.24), with m 6= k. Table 4.7 shows results for the same problems asTable 4.4. Comparing columns 4 and 5 of Table 5 with Table 4.4, we see that one dipolecannot resolve the interface as well as the low frequency input. Figure 4.14 a) shows that fora small shift we recover the interface well, but for large shifts (see Figure 4.14 b), comparealso with Figure 4.12 b)), we have problems. Trying to recover more coe�cients is notpossible with our simple procedure. In our experience, larger systems produce larger errorsin the most negative Fourier coe�cients, creating fast oscillations even for very small shifts.Figure 4.15 shows the perturbations we recover for the same examples as Figure 4.14when using measurments for dipoles at 8 uniformly spaced locations on @D, computing theFourier coe�cients as the least squares solution to the 72� 9 system obtained by stackingthe matrices and right hand measurements. See Table 4.7 (columns 6 and 7) for the resultson the shifted circles, and compare with Table 4.4 to see that 8 dipoles resolve the shiftedcircles just as well as pure frequency input.Finally, we go back to the case of a rosetta interface for dipole input. Table 4.8 showsthe results for rosettas with frequencies 1, 2, 3, 4 and amplitudes 0.01 and 0.05 for 1measurement and 8 measurements with uniformly spaced dipoles on @D. Comparing withtables 4.5 and 4.6 we see that for 8 dipoles the results are even better than for pure frequencyinput, whereas 1 dipole is not enough even for the smallest amplitude. The quality ofthe results also depends on the locations of the dipoles. Figure 4.16 shows the recovered

109Table 4.8: Results using measurments for one and eight dipoles for various rosetta fre-quencies. � is the perturbed interface (amplitudes 0.02 and 0.05) and �� the recoveredperturbation. k�� ��k1 �dipoles 1 8 1 8 1 8 1 8� .002 0.02 0.05 0.05 .002 0.02 0.05 0.05l=1 1.6e�3 5.4e�3 9.9e�3 3.5e�2 8.0e�2 2.7e�1 2.0e�1 7.0e�1l=2 2.3e�3 1.5e�3 1.7e�2 1.6e�2 1.1e�1 7.5e�2 3.5e�1 3.2e�1l=3 2.9e�3 1.0e�3 2.5e�2 1.1e�2 1.5e�1 5.2e�2 5.1e�1 2.3e�1l=4 2.1e�3 6.2e�4 1.8e�2 4.3e�3 1.1e�1 3.1e�2 3.6e�1 8.5e�20 1 2 3 4 5 6

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

alpha for rosetta frequency 4 amplitude 0.05

Single Dipole input

0 1 2 3 4 5 6

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

alpha for rosetta frequency 4 amplitude 0.05

8 Dipole inputs, least square Fourier coefficients

a) b)Figure 4.16: True perturbation (curve) and recovered perturbation (circles) of rosetta in-terface. a) Single dipole input. b) Using measurments for a dipole at 8 uniformly spacedpoints on @D with least square solution for Fourier coe�cients.interfaces for 1 and 8 dipoles.Remark 4.6 Given the nature of the Fredholm integral equation of the �rst kind (4.10),our di�culties �nding the higher Fourier coe�cients come as no surprise. Using multiplemeasurements to regularize the solution also for pure frequency input could be a way toimprove on our numerical results.

Chapter 5CONCLUSION AND OUTLOOK5.1 ConclusionIn this dissertation, we developed a new version of the Immersed Interface Method, whichwe call the Explicit Jump Immersed Interface Method, and we developed a method thatcan resolve the details of an interface that is to be found in an inverse problem, after acircular initial guess is found with some other method, for example EJIIM.EJIIM keeps the ability of the IIM to solve elliptic interface problems with arbitraryinterfaces, that are usually not aligned with the grid. As for the IIM, numerical resultsindicate second-order accuracy for a wide variety of problems, many of them not treatableby the original IIM. Especially for large contrast problems, we observed improved continuousdependence on the interface location, and better absolute errors than for the IIM. We wereable to prove second-order convergence for singular sources in 2D (using the jumps up tothird order), and have found bounds on the coe�cients for all three basic elliptic problemsin Chapter 3. In 1D, we were able to prove second-order convergence for single pointinterfaces using only the jumps up to second order, and established error bounds that areindependent of the geometry and contrast. We found jump conditions for a variety ofnew problems (including our new extension by zero for irregular domain problems, cracks,and grid re�nement) and established interesting connections of EJIIM with certain integro-di�erential equations of potential theory.According to our insights, for discontinuous coe�cients, the problem has shifted from�nding new coe�cients of a �nite di�erence scheme (IIM) to �nding the jumps inthe solution and its derivatives. Even though these jumps may not always be thephysically meaningful quantities, the best approach for elliptic problems with piecewisesmooth solutions is to �nd the jumps �rst, then correct the scheme for the smooth problemwith these jumps. This was done for singular sources by the IIM, and is now also done fordiscontinuous coe�cients by EJIIM. This is following the spirit of the historical predecessorsof the IIM. Peskin for example derived the jumps from the physics, and Mayo found themby solving an integral equation. EJIIM (and probably also Li's FIIIM) �nd them viadiscretization of some integro-di�erential equation. Our original motivation to separate thejumps was the possibility to use fast solvers, but our experience shows that it also helps interms of generality and stability of the method.By generality, we mean that the formulas in x2.2 allow the discretization of virtuallyany di�erential operator in the presence of discontinuities in the solution. For example,u(@=@x)v is a nonlinear term appearing in the Navier-Stokes equations, which could bediscretized in the smooth case by diag U(DxV +Bv) (where Bv has boundary contributions),and in the non-smooth case by diag U(DxV + x;vCv + Bv), where Cv has the jumps of vand derivatives, and x;v distributes them in the usual fashion, here only in the x-direction.

111Another reason why we claim generality is the easy extension of our framework to generalboundary conditions and applicability to the local re�nement problem that we demonstratein x3.8.3.By stability, we mean the improvement of both the FIIIM and EJIIM over the IIM forlarge contrast problems, and the improvement of EJIIM over the FIIIM on these problems.Depending on which equation we use to determine the jumps with EJIIM, either (� �1) or (1=� � 1) enters the truncation error, which makes a di�erence for the worst caseerror behavior, especially on coarse grids, where � � 1=h or 1=� � 1=h, and a�ects theconvergence of an iterative method.By modularizing our computer code, and using explicit operators to accomplish thevarious needed approximations (discrete deltas, discrete dipoles, derivatives in Cartesianand local coordinates, etc.), we were able to solve a vast variety of problems for generalinterfaces with essentially the same mechanism: singular source problems, exterior andinterior irregular domain problems with general boundary conditions, piecewise constantand discontinuous variable coe�cient problems, and the interface problem introduced bya local re�nement of the mesh. In the continuous formulation, only the jump conditionchanges, and in the discrete setting, simply di�erent (linear) combinations of the operatorsare needed. EJIIM also naturally accommodates multiple interfaces, giving an importanttool for the di�cult \multiply connected domain" problems. Another nice feature is thatthe operators can also be used to interpolate di�erential properties of any grid function atthe interface, i.e. function and derivative values there.For the inverse problem of �nding the perturbation of a circular interface, our explicitanalytic solutions for circular interfaces already have proved very useful in creating testproblems, and our numerical results established the feasibility of this approach, which isespecially useful when combined with a method to estimate the location of the circularguess for the unknown interface.Combining our knowledge of EJIIM with our experience with nonlinear and inverseproblems gained in the course of this thesis work, a multitude of future projects becomefeasible.5.2 OutlookThe most important contribution from this thesis might be the future use of the formulasin x2.2.2 in all kinds of numerical solvers for di�erential equations. The formulas are by nomeans restricted to elliptic problems. If one knows the jump (and \location" of a jump)occurring in a time-like direction, the formulas can also be used for the discretization ofparabolic or hyperbolic equations. An interesting problem could be the extension of EJIIMto anisotropic problems with these di�erences.In some chemistry and in some electrostatics problems, the di�usion coe�cient dependson the solution, resulting in nonlinear equations. It would be interesting to see if the nicestability properties (and fast solvers) for the linear problem could be brought to use in thenonlinear problem via our way of discretizing it by the Liouville transformation.Crack problems are very important in porous media ow, for example. Numerical ex-periments with the crack interface conditions derived in this thesis, and used in the EJIIM

112or some other numerical scheme, could potentially be very helpful in forward solvers for thevery large scale inverse problems arising in oil exploration.Our grid re�nement idea (as well as codes for the IIM in general) could be extended toboundary intersecting interfaces, because one certainly would only want to re�ne in a smallneighborhood of the interface. Of course, completely di�erent ways to re�ne the grid arealso conceivable.The implementation of an IIM in 3D, and more sophisticated use of the IIM as a forwardproblem solver in inverse problems should probably be done using the EJIIM setup, due toits good coarse-grid stability and the availability of fast solution methods. It should also bevery helpful for solving systems of di�erential equations, where there may be no fast solversavailable.From a theoretical point of view, a complete convergence theory of EJIIM, or the IIMfor that matter, in several space dimensions, would be very satisfying.EJIIM might also be useful in treating stability problems in moving interface problems.In general, all the problems that have been treated with the IIM can be solved with EJIIMas well, and a comparison of results could be very interesting.It is conceivable that the discrete eigenvalues and eigenvectors obtained through EJIIMcould give good approximations of the eigenvalues and eigenvectors of the Sturm-Liouvilleproblems: �nd u, � that satisfy (�ux)x+�u = 0 where � is given and discontinuous at one ormore points. The smaller discrete eigenvalues, all of which are (2n)2 sin2(k�=(2n)), for k =1; 2; : : : ; n, give excellent approximations for the eigenvalues � = k2, k = 1; 2; 3; : : : of theproblem uxx+�u = 0. By the same token, in 2D, the eigenfrequencies of a membrane withirregular boundary could be estimated by the eigenvalues of our discretization. Forsytheand Wasow [12] contains a note on Weinberger's method, which might be an interestingapplication for EJIIM.Finally, for practical purposes, it would be useful to replace GMRES by another iter-ative solver, that does not need as much storage, i.e. does not reorthogonalize against allprevious vectors. The method we have in mind for this is BICGSTAB, after a suggestionby Greenbaum [16]. Also, a more e�cient implementation in C++ or FORTRAN ratherthan MATLAB, with carefully designed user interfaces for input and output, would be theright way to make the results of this research more useful for applications.

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VITAAndreas Wiegmann was born on April 15, 1967 in M�unchen, Germany. He attendedthe Josef Albers Gymnasium in Bottrop, served in the German air force and attendedthe Universit�at Fridericiana zu Karlsruhe (TH), where he received the degree of DiplomMathematiker in July 1992. He entered the University of Washington in 1992 and completedthe requirements for the Doctor of Philosophy degree in 1998.