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The Pennsylvania State University
The Graduate School
Department of Mathematics
FINITE ELEMENT APPROXIMATIONS OF HIGH ORDER
PARTIAL DIFFERENTIAL EQUATIONS
A Dissertation in
Mathematics
by
Bin Zheng
c© 2008 Bin Zheng
Submitted in Partial Fulfillmentof the Requirements
for the Degree of
Doctor of Philosophy
August 2008
The dissertation of Bin Zheng was reviewed and approved∗ by the following:
Jinchao XuDistinguished Professor of MathematicsThesis AdviserChair of Committee
Chun LiuProfessor of Mathematics
Eric MockensturmAssociate Professor of Mechanical Engineering
Victor NistorProfessor of Mathematics
Ludmil ZikatanovAssociate Professor of Mathematics
John RoeProfessor of MathematicsHead of the Department of Mathematics
∗Signatures on file in the Graduate School.
iii
Abstract
Developing accurate and efficient numerical approximations of solutions of high
order partial differential equations (PDEs) is a challenging research topic. In this disser-
tation, we study finite element approximations of high order PDEs that arise in many
physics and engineering applications.
A common method of solving a high order PDE is to split it into a system of
lower order equations. By carefully studying the biharmonic equation with different
types of boundary conditions, we are able to justify the fact that the lower order system
of equations and the original problem may have different solutions. Our analysis shows
that direct discretizations are much better suited for the numerical solution of high order
problems.
We construct two finite elements to directly discretize high order equations arising
from magnetohydrodynamics (MHD) models. These elements provide nonconforming
approximations for which the number of degrees of freedom is much smaller than that of
a conforming method. The inter-element continuity is only imposed along the tangential
directions which is appropriate for the approximation of the magnetic field. A detailed
construction of basis functions for the new elements is given, and we also prove that these
finite element approximations converge for a model problem containing both second order
and fourth order terms.
iv
Another important property of high order PDEs that model physical phenomena
in material sciences, fluid mechanics and plasma physics is that they often involve dif-
ferent time and spatial scales. The solutions exhibit sharp interfaces, such as shocks,
current sheets and other singularities. Adaptive mesh refinement techniques are there-
fore crucial for reliable numerical computations of high order problems. We develop a
post-processing derivative recovery scheme and a posteriori error estimates that can be
used in local adaptive mesh refinement. A nice feature of the scheme is that it is inde-
pendent of the PDE and a single implementation can be used to solve many different
problems.
v
Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Examples of high order PDEs . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2. Biharmonic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Simply supported plates . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Primary formulation of simply supported plate model . . . . 11
2.1.2 Mixed formulation of simply supported plate model . . . . . 13
2.1.3 On the regularity of solutions . . . . . . . . . . . . . . . . . . 15
2.1.4 On the positivity of solutions . . . . . . . . . . . . . . . . . . 17
2.2 Clamped plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Primary formulation of clamped plate model . . . . . . . . . 19
2.2.2 Mixed formulation of clamped plate model . . . . . . . . . . . 20
2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 3. Finite Element Approximations of High Order MHD Equations . . . 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
vi
3.2 Descriptions of some high order MHD models . . . . . . . . . . . . . 27
3.2.1 Generalized Ohm’s law . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Electron magnetohydrodynamics . . . . . . . . . . . . . . . . 29
3.2.3 About boundary conditions . . . . . . . . . . . . . . . . . . . 33
3.3 A simplified model problem with different types of boundary conditions 34
3.4 Nedelec elements for second order problems and relevant De Rham
diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 A nonconforming finite element for problem (3.10) . . . . . . . . . . 45
3.5.1 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.2 Unisolvence of the finite element . . . . . . . . . . . . . . . . 55
3.5.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 A nonconforming finite element for problem (3.11) . . . . . . . . . . 66
3.6.1 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6.2 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter 4. Recovery Type A Posteriori Error Estimates . . . . . . . . . . . . . . 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Derivative recovery scheme . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 A posteriori error estimates . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter 5. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendix. Simply Supported Boundary Conditions . . . . . . . . . . . . . . . . 94
vii
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
viii
List of Tables
2.1 Simply supported L-shaped plate. . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Clamped L-shaped plate. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 Error estimates for uniform refinement. . . . . . . . . . . . . . . . . . . 90
4.2 Error estimates for adaptive refinement. . . . . . . . . . . . . . . . . . . 90
ix
List of Figures
2.1 Two discrete solutions for simply supported plates . . . . . . . . . . . . 22
3.1 Degrees of freedom of the first new element . . . . . . . . . . . . . . . . 46
3.2 Degrees of freedom of the second new element . . . . . . . . . . . . . . . 69
4.1 Parameters associated with the triangle τ . . . . . . . . . . . . . . . . . . 85
4.2 Graph of the exact solution . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Top left: 3×3 initial mesh. Top right: uniform refinement with nt = 128.
Bottom left: adaptive refinement with nt = 137. Bottom right: adaptive
refinement with nt = 131105. Elements are colored according to size. . . 92
x
Acknowledgments
I am most grateful and indebted to my thesis advisor, Prof. Jinchao Xu, for his
guidance, patience, and encouragement during my study at Penn State. I would like to
thank him for the numerous stimulating and fruitful discussions about my research. I
have really benefited a lot from his deep insights.
I am truly grateful to Prof. Ludmil Zikatanov and Prof. Chun Liu, for inspira-
tion and enlightening discussions on a wide variety of topics. Prof. Ludmil Zikatanov
also provided invaluable help on the preparation of this dissertation. Prof. Qiya Hu
at Chinese Academy of Sciences generously shared with me his insights for Maxwell’s
equations. His invaluable help is sincerely acknowledged. I would like to thank Prof.
Randolph E. Bank at UCSD for discussions on derivative recovery scheme and sharing
the package PLTMG for the numerical experiments.
I would like to thank Prof. Victor Nistor and Prof. Eric Mockensturm, for taking
their precious time to read my thesis, providing insightful commentary on my work, and
serving on the committee. Special thanks to Dr. Pengtao Sun and Dr. Long Chen for
their valuable help and friendship.
I would also like to thank all my friends I have made at Penn State, especially
Lei Zhang, Jiakou Wang, Tianjiang Li, Yu Qiao, Guangri Xue, and Hengguang Li, etc.
Finally, I would like to thank my parents, and my sisters for constant love and
support. My dear wife, Jian Ding, deserves the most special thanks for her company
and constant support.
1
Chapter 1
Introduction
In this dissertation, we study numerical solution of partial differential equations
which involve partial derivatives of order higher than two. These equations have been
used widely to describe different physical phenomena. Examples include: biharmonic
equations modeling thin plate bending [87]; Cahn-Hilliard equations describing phase
separation of binary alloys [10, 33]; streamfunction formulation of incompressible mag-
netohydrodynamics equations [52, 54, 53, 75, 55, 82, 59]. Additional examples are found
in material sciences [11, 23, 95], fluid mechanics, quantum mechanics [37], image process-
ing [44], plasma physics [17, 30, 16], biology, and other areas of science and engineering.
In those equations, high order terms are introduced to reveal more detailed structure
and provide insight of physical phenomena.
In contrast with the theory of second order PDEs, there are only limited number
of results concerning the existence, uniqueness and regularity of solutions of high order
PDEs [32, 39, 63, 46, 4, 12]. Numerical simulation is often the only tool to obtain
quantitative results and to study high order PDEs. However, the design of efficient and
reliable numerical methods for high order PDEs is also very challenging.
Finite difference methods provide a simple way to discretize, but they have dif-
ficulties in handling complex geometries, different types of boundary conditions. Finite
2
volume methods provide conservative discretizations but they have low accuracy. Spec-
tral methods result in exponential convergence, but have limitations when applied to
problems with arbitrary boundary conditions or non-smooth solutions. Finite element
methods have a sound mathematical foundation to provide accurate discretizations on
arbitrary domains, but their convergence analysis is often complicated.
The aforementioned discretization methods can be categorized into two basic
approaches for the discretizations of high order PDEs. One is to discretize the original
high order equation directly [68, 52, 52], and the other is to first split the high order PDEs
into a system of lower order PDEs, and then discretize the resulting system of PDEs
[10, 11, 95, 33, 59, 82, 55]. Currently the later approach is more popular. However, it is
known that for some problems, such a technique cannot be applied. For example, when
modeling the bending of simply supported plate on non-convex polygonal domains, the
original biharmonic problem is not equivalent to the lower order system of two Poisson
equations [78, 18]. In this dissertation we study this problem in a systematic way, and
provide detailed analysis. Recently we have seen two papers that address the same issue
[69, 100], but our analysis is more complete and more transparent.
Another issue related to the direct discretization of high order PDEs with finite
element method is that a conforming method would require high smoothness of the ap-
proximating functions (e.g. C1 functions for a fourth order PDE [26, 99]). This means 21
degrees of freedom per element in two dimensions and 220 degrees of freedom per element
in three dimensions, thus increasing the computational cost significantly. One possible
way to reduce the number of degrees of freedom is to use nonconforming discretizations,
3
allowing weaker inter-element smoothness constraints, but still providing convergent ap-
proximations [68]. Among the class of nonconforming finite elements for fourth order
problems, Morley element is special in the sense that it provides approximation with
polynomials of minimal degree. In [89], an elegant and systematic construction of Mor-
ley type elements is provided for solving even higher, 2m-th order partial differential
equations in Rn.
One of the main results in this dissertation is the construction of two finite element
approximations of fourth order equations in MHD. MHD models the dynamics of elec-
trically conducting fluids in magnetic field. The high order equations in MHD systems
describe the evolution of the magnetic field. We construct two types of nonconforming
finite elements to directly discretize a model problem which contains both second order
term and fourth order term. By introducing properly designed degrees of freedom, we
are able to show the approximated solution converges for the model problem. For both
theoretical and practical considerations, we have constructed nodal basis functions for
the corresponding finite element spaces.
High order problems often exhibit multiscale phenomena and solution singular-
ities. Adaptive mesh refinement techniques based on efficient and reliable a posteriori
error estimators are therefore essential for accurate numerical approximations of high
order PDEs, [88, 82, 59, 38, 84]. There are basically two types of a posteriori error
estimators: residual type estimators, and recovery type estimators. In our work we
propose and analyze error estimators of recovery type. In the recovery error estimators
approach, the finite element solution is post-processed to obtain better approximation to
the derivatives of the solution to the original PDE [7, 8]. We develop a post-processing
4
derivative recovery scheme independent of the PDE and a posteriori error estimates that
can be used in local adaptive mesh refinement algorithms. A single implementation of
our scheme can be used to solve different problems. Currently, we have implemented this
scheme to second order elliptic problems, including a nonlinear problem and a problem
whose solution has singularities. Extensions of this procedure to high order problems
are under investigation and are expected to share similar properties.
Next, we introduce several examples of high order PDEs and briefly describe the
organization of this dissertation.
1.1 Examples of high order PDEs
Example 1: The classical biharmonic equation modeling thin plate bending and
stream function formulation of Stokes equation in 2D.
∆2u = f, in Ω
u =∂u
∂n= 0, on ∂Ω.
where ∆ denotes the Laplacian operator, n is the outward normal to ∂Ω.
Example 2: The electron magnetohydrodynamics equation modeling plasma dy-
namics dominated by electron flow. This is a single vector equation describing the
evolution of the magnetic field B.
∂t(B − d2e∆B)−∇× (ve × (B − d2
e∆B)) =
ηc2
4π∆B − νed
2e∆2B,
5
where
ve = − c
4πen∇×B,
de is an intrinsic scale length.
The above two high order PDEs will be studied in this dissertation. In the
following, we list a few other examples of high order PDEs.
Example 3: Cahn-Hilliard equation modeling spinodal decomposition and coars-
ening phenomena in binary alloys.
∂u
∂t+∇ · (b(u)∇∆u) = 0, in ΩT := Ω× (0, T ),
u(x, 0) = u0(x), ∀x ∈ Ω,
∂u
∂n= b(u)
∂∆u∂n
= 0, on ∂Ω× (0, T );
where Ω ∈ R3 is a bounded domain. In this parabolic equation u represents a relative
concentration of one component in a binary mixture. The function b(u) is the degenerate
mobility, which restricts diffusion of both components to the inter-facial region. For
example, one may take a mobility of the form
b(u) = u(1− u),
which significantly lowers the long-range diffusion across bulk regions [33, 32, 10, 38].
6
Example 4: The generalized Perona-Malik equation, a fourth order diffusion equa-
tion proposed for noise reduction in images [91].
ut +∇ · [g(m)∇∆u] = 0
u(x, 0) = u0(x),
where g is an edge indicator, m is some measurement of u. A typical example is
g(s) =1
1 + ( sK )2 , m = |∇u|,
where K is a subjective parameter.
Example 5: The Kuramoto-Sivashinsky equation arising in the context of flame
propagation, viscous film flow and bifurcation solutions of Navier-Stokes equations [1]
∂φ
∂t+ |∇φ|2 + ∆φ+ ∆2φ = 0.
Example 6: A sixth order equation modeling the oxidation of silicon in supercon-
ductor devices [11, 34]:
∂u
∂t= ∇(b(u)∇∆2u), in ΩT := Ω× (0, T )
u(x, 0) = u0(x), ∀x ∈ Ω
∂u
∂n=
∂∆u∂n
= b(u)∂∆2u∂n
= 0, on ∂Ω× (0, T ),
where b(u) = |u|γ , γ ∈ (0,∞).
7
Example 7: A sixth order phase field simulation of the morphological evolution
of a strained epitaxial thin film on a compliant substrate [96]:
∂c
∂t=
1ε20∇ · (M(c)∇µ) + S(c),
µ = F (c)− ε2∆c+ κ20∆2c,
where ε0 is the gradient energy parameter; M is the surface mobility.
More high order PDEs with useful formulas for particular solutions can be found in
a book by Polyanin [77]. Some applications of high order PDEs in physics and mechanics
are presented in another book by Peletier [76].
1.2 Outline of the dissertation
Chapter 2 contains a careful validation study of two different boundary value
problems of the biharmonic equation. Depending on the boundary conditions prescribed,
the biharmonic equation models the bending of either clamped plate or simply supported
plate. We study both the mixed formulation and primary formulation for each case. We
show that the two formulations of the bending problem for simply supported plate are
not equivalent, due to the lack of regularity of the solution.
In chapter 3 we study a fourth order equation in MHD models (Example 2 listed
above). The high order term appears in the magnetic induction equation which describes
the evolution of magnetic field. We construct two nonconforming finite elements for the
discretization of this equation in three dimensions on tetrahedral mesh. The first element
that we propose has twenty degrees of freedom per tetrahedron. The second new element
8
has only fourteen degrees of freedom per tetrahedron. We give a detailed construction
of the basis functions for the two elements and show their convergence analysis.
In chapter 4 we study a posteriori error estimators for adaptive mesh refinement.
We present a derivative recovery scheme for Lagrange-type finite elements of degree p on
general unstructured (shape regular) meshes. We prove that the recovered derivatives
superconverge to the derivatives of the solution to continuous problem. The recovered
derivatives can be used to provide asymptotically exact a posteriori error estimators and
local error indicators to design efficient adaptive mesh refinement algorithms. We provide
several examples to demonstrate the usefulness of our derivative recovery scheme.
In chapter 5 we draw conclusions and describe directions for future work.
9
Chapter 2
Biharmonic Equations
In this chapter we study the Kirchhoff-Love model for the bending of a thin, linear
elastic plate. The plate is subjected to an external transversal load f and one seeks the
resulting deflection u as a solution to a fourth order PDE, biharmonic equation. Vari-
ous boundary conditions can be prescribed, and they correspond to simply supported,
clamped, or free plate edges. We will consider plates with all edges simply supported or
clamped. For each of these two cases, two different variational formulations are intro-
duced: a primary formulation and a mixed formulation. We compare solutions to these
two formulations on polygonal domains. In our study, we give a detailed and transpar-
ent analysis leading to the conclusion that for the bending of simply supported plates
on non-convex polygonal domains, the original biharmonic problem is not equivalent to
the system of two Poisson equations, and illustrate the theoretic results by a numerical
example. Similar results can be found in [18, 78, 65, 64, 69, 100].
2.1 Simply supported plates
The Kirchhoff thin plate model satisfies the biharmonic equation
42u = f in Ω. (2.1)
10
For simply supported plates we have the following boundary conditions:
u = 0, nTMn = 0 on ∂Ω, (2.2)
where the bending moment M is defined by
M = (mij) = ((1− ν)∂iju+ ν∇uδij) = (1− ν)∇2u+ ν∆uI,
n = (n1, n2) is the unit outward normal along the boundary ∂Ω and ν is a constant with
0 < ν < 1.
There are several other ways to write this boundary condition, for example
u = 0,∂2u
∂n2 + ν∂2u
∂t2= 0, on ∂Ω, (2.3)
where t = (t1, t2) is the unit tangential vector along the boundary. Another equivalent
form of the boundary conditions is given by
u = ∆u− κν ∂u∂n
= 0, on ∂Ω, (2.4)
where κ is the curvature of ∂Ω.
For a polygonal domain, which is the case of interest, the curvature of the bound-
ary is zero almost everywhere. The boundary conditions then simplify to (see Appendix
A):
u = ∆u = 0, on ∂Ω. (2.5)
11
These conditions are often referred to as Navier boundary conditions [39].
2.1.1 Primary formulation of simply supported plate model
To introduce the weak formulation of the biharmonic problem, let V = H2(Ω) ∩
H10 (Ω), the variational formulation is as follows. Find u ∈ V such that
a(u, v) = (f, v), ∀ v ∈ V,
where the bilinear form is defined by
a(u, v) = (M,∇2v)0,Ω =∫
Ω
∑i,j
mij∂ijv =∫
Ω
∑i,j
(1− ν)∂iju+ ν∆uδij∂ijv
=∫
Ω(1− ν)
∑i,j
∂iju∂ijv +∫
Ων∆u∆v.
Equivalently, this bilinear form can be written in a more compact form:
a(u, v) = (1− ν)(∇2u,∇2v)0,Ω + ν(∆u,∆v)0,Ω
=∫
Ω∆u∆v − (1− ν)(∂11u∂22v + ∂22u∂11v − 2∂12u∂12v)dxdy.
If u is sufficiently smooth, then it also satisfies the corresponding strong form:
∆2u = f, in Ω
u = 0, on ∂Ω
nTMn = 0, on ∂Ω
(2.6)
12
The derivation of the form (2.6) can be done as follows. First, observe that
∆2u = div divM =∑i,j
∂ijmij ,
and use integration by parts twice (note: 〈·, ·〉 denotes boundary integral)
(M,∇2v) = −(divM,∇v) + 〈M · n,∇v〉(= −∫
Ω
∑i,j
∂jmij∂iv +∫∂Ω
∑ij
mijnj∂iv)
= (div divM,v)− 〈n · divM, v〉+ 〈M · n,∇v〉
=∫∂
∑∂j∂imijv −
∫∂Ω
∑∂jmijniv +
∫∂Ω
∑mijnj∂iv.
The first term of the right hand side is
(div divM,v) = (∆2u, v).
The second term equals 0 if v ∈ V . For the last term we have
∂iv =∂v
∂ssi +
∂v
∂nni, or ∇v =
∂v
∂ss+
∂v
∂nn,
13
hence
〈M · n,∇v〉 =∫∂Ω
∑mijnj∂iv
= 〈M · n, ∂v∂ss〉+ 〈M · n, ∂v
∂nn〉
= 〈nTMn,∂v
∂n〉+ 〈sTMn,
∂v
∂s〉
= 〈nTMn,∂v
∂n〉 − 〈∂(sTMn)
∂s, v〉,
where the second term of the right hand side equals 0 if v ∈ V .
From the above discussion, we get
a(u, v) = (∆2u, v) + 〈nTMn,∂v
∂n〉 − 〈n · divM +
∂(sTMn)∂s
, v〉.
Since v ∈ V is arbitrary, we conclude that if u is sufficiently smooth it satisfies
(2.6).
2.1.2 Mixed formulation of simply supported plate model
Consider the following boundary value problem for simply supported plates
∆2u = f, in Ω,
u = ∆u = 0, on ∂Ω.(2.7)
It is know that this problem has a unique solution u ∈ V := H2(Ω) ∩H10 (Ω). We now
introduce auxiliary variable v = −∆u, to decouple the biharmonic equation into two
Poisson equations.
14
We assume that f ∈ V ∗, and define v ∈ L2(Ω) as the unique solution to
−∫
Ωv∆φdx =
∫Ωfφdx, ∀ φ ∈ V. (2.8)
Then the solution to (2.7) satisfies the following equation:
−∆u = v, in Ω,
u = 0, on ∂Ω.(2.9)
The second boundary condition in (2.7), namely ∆u = 0 on ∂Ω, is included in
the choice of test function φ in (2.8).
Primary variables for the mixed formulation are u,−∆u. The mixed variational
formulation of (2.7) can be written in the following abstract form: Given real Banach
spaces V and W , right hand side f ∈W ∗, a(·, ·) and b(·, ·) are bilinear forms on V × V ,
and V ×W , respectively. We seek u ∈W, v ∈ V solutions of
a(v, ψ) + b(ψ, u) = 0, ∀ ψ ∈ V,
b(v, φ) = −(f, φ), ∀ φ ∈W.(2.10)
We will write the variational formulations of (2.9) and (2.8) in the form (2.10).
We set V = W ≡ H10 (Ω) and
a(v, ψ) =
∫Ω vψ dx, ∀ v, ψ ∈ H1
0 (Ω),
b(v, φ) = −∫Ω∇v · ∇φ dx, ∀ v, φ ∈ H1
0 (Ω).
15
Hence, the variational formulation corresponding to (2.8) and (2.9) can be written as :
Find u, v ∈ V × V such that
(v, ψ)− (∇ψ,∇u) + (∇v,∇φ) = (f, φ), ∀ ψ, φ ∈ V × V,
where f is as in (2.7). Equivalently, we have
(v, ψ)− (∇ψ,∇u) = 0, ∀ ψ ∈ H1
0 (Ω),
−(∇v,∇φ) = −(f, φ), ∀ φ ∈ H10 (Ω).
(2.11)
Next, we discuss the regularity to the solution of (2.11) and compare with the regularity
of the solution to the problem in primary formulation (see Section 2.1.1).
2.1.3 On the regularity of solutions
If the solution u is sufficiently smooth (say u ∈ H3(Ω)) then obviously u satisfies
(2.7) as well as the following system of two Poisson equations:
−∆w = f, in Ω,
w = 0, on ∂Ω,and
−∆u = w, in Ω,
u = 0, on ∂Ω.(2.12)
However, such regularity requirement may not be valid if the domain is polygonal, and
as we will show, in this case (2.7) and (2.12) are not equivalent.
For L-shaped domains, the weak solution of (2.7) u is not in H3(Ω), and hence,
in general ∆u is not in H1(Ω). In fact, if u /∈ H3, then according to expansion results
16
due to Kondratiev [57]
u = UR +∑k
cksk
where sk is the set of singular functions due to the reentrant corners. The precise form
of a typical singular function is S = r1+zu(θ) with u := v1u2 − v2u1 where
u1 := (z − 1)−1 sin((z − 1)θ)− (z + 1)−1 sin((z + 1)θ),
u2 := cos((z − 1)θ)− cos((z + 1)θ),
v1 := (z − 1)−1 sin((z − 1)a)− (z + 1)−1 sin((z + 1)a),
v2 := cos((z − 1)a)− cos((z + 1)a),
where a = ω = 3π/2 and z is a solution of the characteristic equation
(sin(zω))2 − z2(sin(ω))2 = 0,
satisfying Re (z) ∈ (0, 2).
The characteristic equation has a real solution z ≈ 0.6, so in general u ∈ H2.6,
but u /∈ H2.7. We also note that
∆S = c1(z)rz−1 sin((z − 1)θ) + c2(z)rz−1 cos((z − 1)θ),
where c1(z), c2(z) are nonzero when z is the real solution of the characteristic equation.
Thus ∆u is not an element of H1.
17
On the other hand for the mixed formulation (2.11), if the right hand side f /∈ L2,
then a classical regularity result tells us that w /∈ H1+2/3−ε, where ε is any arbitrarily
small positive number [45]. For the weak solution u, because right hand side w is in
H1+2/3−ε the same regularity result tells us u is in H1+2/3+(1+2/3−ε)−ε, i.e. u ∈
H3+1/3−2ε, and this implies u ∈ H3 and ∆u ∈ H1. Clearly, this is different from the
regularity of the weak solution of (2.7).
From the above analysis, we can conclude that in general the solution of (2.7) and
the solution of (2.12) are not the same because we have just shown that they may have
different regularities. The above theoretical analysis is verified through the numerical
experiments in Section 2.3.
Our considerations here are related to the famous paradox of I. Babuska, saying
that the finite element method which uses polygonal approximations of a smooth do-
main Ω for the biharmonic problems with simply supported boundary condition fails to
approximate the exact solution.
2.1.4 On the positivity of solutions
It is well known that a significant difference between high order PDEs and second
order PDEs is the lack of a general maximum principle (positivity preserving property)
for high order PDEs. For the following second order elliptic equation
−∆u = f, in Ω,
u = 0, on ∂Ω,(2.13)
18
one has:
f ≥ 0 ⇒ u ≥ 0.
Apply the above result twice, we can conclude that for problem (2.12), if f ≥ 0
then u ≥ 0. However this conclusion in general is not true for biharmonic problem (2.7).
In fact, it was shown in [69] that the solution of (2.7) may change sign for f ≥ 0 if the
domain has concave corner. Hence, one can also see that the solution of (2.7) and the
solution of (2.12) are not the same because of different positivity properties.
2.2 Clamped plates
In the following we study another biharmonic problem which provides a model
for the bending of a clamped plate. It also appears in the stream-function formulation
of a steady state planar Stokes flow. Given a convex polygonal domain Ω in R2 with
boundary ∂Ω, consider the model problem
∆2u = f in Ω ,
u = ∂u∂n = 0 on ∂Ω .
(2.14)
The following result is well known for convex polygonal domains: If f ∈ H−1,
then the solution to (2.14) is unique and satisfies
||u||3 ≤ C||f ||−1,
19
for a constant C independent of f (see, [45]). In general (see, [47]), if Ω is a plane
polygonal domain (may be non-convex) with maximum internal angle ω0 < 2π, then
for some δ ∈ [0, 1/2) the problem (2.14) has a unique solution (in the weak sense)
u ∈ H3−δ(Ω) ∩H20 (Ω) satisfying an a priori estimate of the form
‖u‖3−δ ≤ C‖f‖−1−δ.
2.2.1 Primary formulation of clamped plate model
Let V = H20 (Ω). A variational formulation of problem (2.14) is: Find u ∈ V such
that
a(u, v) = L(v), ∀ v ∈ V, (2.15)
where
a(u, v) =∫
Ω∆u ∆v dx, L(v) =
∫Ωf v dx.
Another variational formulation of the clamped plate problem used in the study
of linear elasticity corresponds to the following bilinear form:
a(u, v) =∫Ω[ ∆u∆v + (1− ν)(2∂12u∂12v − ∂11u∂22v − ∂22u∂11v)]dx
=∫Ω[ ν ∆u∆v + (1− ν)(∂11u∂11v + ∂22u∂22v + 2∂12u∂12v)]dx.
If Poisson ratio ν satisfies 0 < ν < 1/2; then the bilinear form is H20 (Ω)-elliptic, since
a(v, v) = ν|∆v|20,Ω + (1− ν)|v|22,Ω, ∀ v ∈ H2(Ω).
20
Thus, there exists a unique function u ∈ H20 (Ω) that solves the following variational
equations
∫Ω
[∆u∆v + (1− ν)(2∂12u∂12v − ∂11u∂22v − ∂22u∂11v)]dx =∫
Ωf v dx, ∀ v ∈ H2
0 (Ω).
For a smooth u, by integration by parts, one can show that the contribution of
∫Ω
(2∂12u∂12v − ∂11u∂22v − ∂22u∂11v)dx =∫
Γ(−∂ττu∂nv + ∂nτu∂τ v) dx,
is zero. Indeed, by integration by parts, we have
∫ω
∆u∆v dx =∫
Ω∆2u v dx−
∫Γ∂n∆u v dγ +
∫Γ
∆u ∂nv dγ.
Thus, we find that u satisfies the biharmonic equation (2.14), independent of the choice
of ν.
2.2.2 Mixed formulation of clamped plate model
Consider the following variational problem: Find u ∈ H10 (Ω), v ∈ H1(Ω) such
that (v, ψ)− (∇ψ,∇u) = 0, ∀ ψ ∈ H1(Ω),
−(∇v,∇φ) = −(f, φ), ∀ φ ∈ H10 (Ω).
(2.16)
The existence of a solution to (2.16) is not straightforward as the bilinear form
a(·, ·) is not coercive on H1(Ω). Since the solution of mixed problem (2.16) should be a
solution of (2.1) in H20 (Ω), one can show the existence of a solution to (2.16) indirectly
21
as the following. From a regularity result on the biharmonic problem, we know (for
instance if Ω is a convex polygon ([62], [45]) that for f ∈ H−1(Ω)), the solution of (2.1)
belongs to H3(Ω), so that v = −∆u belongs to H1(Ω), and hence u is also a solution of
(2.16).
Hanisch [47] established a similar result for non-convex polygons. In fact, he
studied the following variational problem: Find v, u ∈ H1−δ(Ω)×H1+δ0 (Ω), such that
for f ∈ H−1−δ(Ω),
(v, ψ)−DΩ(ψ, u) = 0, ∀ ψ ∈ H1−δ(Ω),
−DΩ(v, φ) = −(f, φ), ∀ φ ∈ H1+δ0 (Ω),
(2.17)
where the form
DΩ(ψ, u) ≡2∑i=1
(Diψ,Diu).
He showed that the weak formulation (2.16) has, for each f ∈ H−1−δ(Ω), a unique
solution v, u ∈ H1−δ(Ω)×H1+δ0 (Ω). This same u solves (2.14) with u ∈ H3−δ(Ω) ∩
H20 (Ω) and v = −∆u.
From the above analysis, we can see that the regularity issue related to the mixed
formulation of simply supported plates does not cause problem for the clamped plates.
This conclusion is verified through the numerical experiments given below.
2.3 Numerical results
In this section we solve the biharmonic problem with right hand side equals 1 on
L-shaped domain. Denote by UAdini the solution of (2.7) by Adini nonconforming finite
22
element [26] and Ulinear solution of (2.12) by linear finite element. The numerical results
are summarized in Table 2.1, and they show no convergence rate which means that the
discrete solution corresponding to (2.7) and (2.12) are indeed different. This can also
be seen by looking at the level sets of the solutions plotted in Fig. 2.1. Next set of
Fig. 2.1. Two discrete solutions for simply supported plates
numerical examples is for the L-shaped clamped plate. In this case the resulting lower
order system is not decoupled. The numerical tests confirm the conclusions we made
in the previous sections; that the mixed formulation for simply supported plate is not
equivalent to the primary formulation, while for clamped plate the mixed formulation can
be used to construct lower order finite element discretizations for biharmonic equation.
As seen in Table 2.2, the convergence for the clamped plate is first order.
23
Table 2.1. Simply supported L-shaped plate.
h ‖UAdini − Ulinear‖0 |UAdini − Ulinear|∞2.0000E-001 3.0874E-003 8.7865E-0031.0000E-001 3.3904E-003 9.3017E-0035.0000E-002 3.5258E-003 9.4164E-0032.5000E-002 3.5805E-003 9.4388E-0031.2500E-002 3.6016E-003 9.4222E-003
order N/A N/A
Table 2.2. Clamped L-shaped plate.
h ‖UAdini − Umixed‖0 |UAdini − Umixed|∞2.0000E-001 5.1507E-005 2.0012E-0041.0000E-001 2.3881E-005 9.0024E-0055.0000E-002 1.1178E-005 4.2885E-0052.5000E-002 5.2285E-006 2.0248E-0051.2500E-002 2.4503E-006 9.5324E-006
order 1.0934 1.0869
24
Chapter 3
Finite Element Approximations of High Order
MHD Equations
3.1 Introduction
Magnetohydrodynamics (MHD) describes the macroscopic dynamics of electri-
cally neutral fluid that moves in a magnetic field. It is a single-fluid model of a fully
ionized plasma. The single hydrodynamic fluid is made up of moving charged particles,
electrons and ions, that are acted upon by electric and magnetic forces. The governing
PDEs are obtained by coupling Navier-Stokes equations with Maxwell equations through
Ohm’s law and the Lorentz force. Several MHD models have been proposed to explain
physical phenomena under various assumptions [14, 15]. As an example, a resistive MHD
system is described by the following equations:
ρ(ut + u · ∇u) +∇p =1µ0
(∇×B)×B + µ∆u,
∇ · u = 0,
Bt −∇× (u×B) = − η
µ0∇× (∇×B)− di
µ0∇× ((∇×B)×B)− η2
µ0(∇×)4B,
∇ ·B = 0,
where ρ is the mass density, u is the velocity, B is the magnetic induction field, η is
the resistivity, η2 is the hyper-resistivity, µ0 is the magnetic permeability of free space,
25
and µ is the viscosity. The primary variables in MHD equations are fluid velocity u and
magnetic field B.
MHD models have widespread applications in thermonuclear fusion, magneto-
spheric and solar physics, plasma physics, geophysics, and astrophysics. Mathematical
modeling and numerical simulations of MHD have attracted much research effort in the
past few decades. The numerical simulations of MHD are challenging because of nonlin-
earities in the equations, different time scales involved, coupling of fluid mechanics with
electromagnetism, and divergence-free constraints.
Various numerical algorithms have been used in MHD simulations; examples in-
clude finite difference methods, finite volume methods, finite element methods, and
Fourier-based spectral and pseudo-spectral methods [85]. In [28], a comparison of a
finite element simulation of a turbulent MHD system with a pseudo-spectral simulation
of the same system shows that the results agree. In this dissertation we focus on fi-
nite element discretizations of the MHD equations since they have the advantages of
handling realistic geometries and boundary conditions, as well as the capability of ap-
plying adaptive mesh refinement. One of the major difficulties in MHD simulations is
the constraint ∇ ·B = 0. In [86], seven schemes designed to numerically maintain this
divergence-free constraint are compared. In [52, 53, 55, 75, 58], two-dimensional, incom-
pressible MHD problems are studied in terms of finite element approximations of the
stream function-vorticity advection formulation. The stream-function approach has the
advantage that the divergence-free constraints of the velocity and magnetic fields are
satisfied exactly. However, this approach increased the order of derivatives that appear
in the original equations, i.e., one need to solve a fourth-order equation. To discretize
26
this fourth-order PDE, either mixed finite element methods [48, 55, 75, 80], or con-
forming C1 finite elements [52, 53] have been used. Since MHD flow tends to develop
sharp interfaces, adaptive h-refinement techniques have been applied in MHD simula-
tions [59, 82, 102]. Finite element computations in three-dimensions have been reported
in [27, 40, 60, 79, 80, 94].
In MHD models, the second-order term ∇ × ∇ × B is usually replaced by ∆B
because ∇ ·B = 0. The resulting Helmholtz formulation is widely used in the literature
[79]. There are basically two approaches to the solution of MHD equations: one approach
is to use a stable element for velocity and a Lagrange nodal element for the magnetic field
(based on the Helmholtz formulation [40, 94]); the other approach is to use a standard
stable or stabilized finite element to discretize the fluid equation and use an edge element
to discretize the magnetic variable [80]. It is well known that standard Lagrange nodal
elements may produce spurious solutions when used to discretize the magnetic field. This
is because the magnetic field is only required to have a continuous tangential component,
while Lagrange elements impose a continuous normal component as well. The advantage
of using an edge element is that it provides a consistent approximation of the magnetic
field with only tangential continuity, and it avoids spurious solutions [19]. In fact, the
lowest-order edge element and its generalizations to higher-order elements by Nedelec
have been used extensively in computational electromagnetics. In our study, we have
used properties of Nedelec elements to construct and analyze the new types of finite
elements.
27
We are mainly interested in investigating those MHD equations that contain
fourth-order terms. In the literature, the major tool used for performing MHD sim-
ulations involving high order equations[17] has been the pseudo-spectral method. By
choosing appropriate formulations, we are able to construct two new finite element ap-
proximations for the solutions of these high order equations in MHD systems.
3.2 Descriptions of some high order MHD models
3.2.1 Generalized Ohm’s law
Ohm’s law describes the balance of current. It shows how the current density
is related to the electromagnetic fields and other quantities. Thus, Ohm’s law plays a
crucial role in the derivation of MHD equations. Various versions of Ohm’s law corre-
spond to several MHD models: ideal MHD (including no dissipation), resistive MHD
(including dissipation due to plasma resistivity η), Hall MHD (allowing relative drifts
between ions and electrons), and extended MHD (allowing additional electron dynamics
and/or non-Maxwellian species effects), etc. The resulting MHD systems differ signifi-
cantly especially on the order of the spatial derivative of B in the magnetic induction
equation. For example, if a resistive term is included, a second-order equation for the
magnetic field accrues.
A generalized Ohm’s law is written in the following form [14, 15, 35]:
E + u×B = ηj + dij×B− di∇pe − η2∇2j + d2
e
∂j∂t,
28
where σ = η−1 is the electrical conductivity; j is the current density; E is the electric
field; di = c/ωpi is the collisionless ion skin depth; and de = c/ωpe is the electron inertial
skin depth. The u×B term is the convective electric field. On the right-hand side of this
equation, the first term is the field associated with Ohmic dissipation caused by electron-
ion collisions; the second term j×B corresponds to the Hall effect; the third term is the
electron pressure; and the fourth term corresponds to hyper-resistivity, η2 = (c/ωpe)2µe
(µe is electron viscosity), which describes the effect of electron viscosity.
Usually under certain assumptions, it is necessary to keep only a few dominant
terms in the generalized Ohm’s law. For example, in the ideal MHD model the effects
of resistivity and electron inertia are neglected, resulting in an “ideal” Ohm’s law:
E + u×B = 0.
Another frequently used version is the “resistive” Ohm’s law:
E + u×B = ηj.
In our study we are concerned with case in which the hyper-resistivity term re-
mains in the equations, for example,
E + u×B = ηj + dij×B− di∇pe − η2∇2j.
This version of Ohm’s law is often used in the numerical studies of turbulence as the
high order diffusion term. It is useful to separate dissipative and non-dissipative scales
29
more clearly [17]. Substituting E into Faraday’s law:
∇×E = −∂B∂t.
Notice the relationship between the magnetic field and the current density given by
Ampere’s law:
∇×B = µ0j,
where µ is the magnetic permeability of free space, we obtain
∂B∂t
= ∇× (u×B)− η
µ0(∇×)2B− di
µ0∇× ((∇×B)×B)− η2
µ0(∇×)4B.
When the velocity u is known, the above induction equation could be used to
determine the evolution of the magnetic field B and ∇ ·B = 0 should be imposed as an
initial condition.
Usually this induction equation is written as
∂B∂t
= ∇× (u×B) +η
µ0∆B− di
µ0∇× ((∇×B)×B)− η2
µ0∆2B.
3.2.2 Electron magnetohydrodynamics
In this section we study the electron magnetohydrodynamics equation (Electron
MHD) in which electron flow dominates plasma dynamics. The Electron MHD model
describes the behavior of plasmas in which ions can be assumed to be immobile and
the motion of electrons keeps the plasmas quasi-neutral. This model applies to a wide
30
range of plasma phenomena such as plasma switches, Z pinches, and quasi-collisionless
magnetic reconnection [16, 17, 22, 36, 56].
There are two conditions in the Electron MHD model. First, phase velocities
are small compared to the speed of light, so that lωpi << c. It follows that l << c/ωpi
where c/ωpi is the ion inertial skin depth; c is the speed of light; and ωpi =√
4πnie2/mi
is the ion plasma frequency (ni is the ion number density, e is the magnitude of the
electron charge, and mi is the ion mass). Second, the time scales of the electromagnetic
phenomena are shorter than the ion cyclotron period: t << ω−1ci/2π, ωci = ZieB/mic
is the ion cyclotron frequency (Zi is the ion charge number).
Under the above two conditions, vi << ve; therefore, ions can be assumed to be
immobile neutralizing background and the electron flow can be assumed to dominate the
plasma dynamics. Because the phase velocities of the electromagnetic waves are much
smaller than the speed of light, the displacement current is negligible in comparison to
the conduction current, and a direct relationship between the magnetic field B and the
fluid (electron) velocity: ve exists,
ve = − jne
= −α∇×B (3.1)
where α = c/(4πne) is the Hall constant. This relationship is indeed the main difference
between the Electron MHD and the ordinary MHD as in the latter case no such equation
holds.
31
The Electron MHD equation is a single vector equation given by
∂t(B− d2e∆B)−∇× (ve × (B− d2
e∆B)) =
ηc2
4π∆B− νed
2e∆2B, (3.2)
where ve satisfies (3.1), and de = c/ωpe is an intrinsic scale-length. This equation
governs the evolution of the magnetic field in plasmas that have a short time scale and
a small length scale.
To derive the equation (3.2), we start from the electron momentum equation [25]
mene(∂
∂t+ ve · ∇)ve = −ene(E +
1cve ×B)−∇pe + j +meneνe∆ve, (3.3)
which is often written in the form of the generalized Ohm’s law [17], i.e.,
E = −1cve ×B− 1
ene∇pe −
mee
(∂tve +∇ · (veve)) + ηj− νemee
∆ve. (3.4)
Remark 1. vv is a tensor defined by
vv =
v1v1 v1v2 v1v3
v2v1 v2v2 v2v3
v3v1 v3v2 v3v3
;
32
the divergence of vv is obtained by applying the divergence operator row-wise; hence,
∇ · (vv) =
v1(∇ · v) + v · (∇v1)
v2(∇ · v) + v · (∇v2)
v3(∇ · v) + v · (∇v3)
=
v · (∇v1)
v · (∇v2)
v · (∇v3)
= (v · ∇)v.
Remark 2. The following identity is necessary to derive Equation (3.2):
∇× (v · ∇v) = ∇× (v× (∇× v)).
This identity can be verified by
v× (∇× v) =
12∂∂x(|v|2) + v · (∇ · v1)
12∂∂y (|v|2) + v · (∇ · v2)
12∂∂z (|v|2) + v · (∇ · v3)
=12∇(|v|2)− (v · ∇)v.
The Electron MHD equation (3.2) can be derived by taking the curl of the gen-
eralized Ohm’s law (3.4) and using Faraday’s law of induction:
∂tB = −c∇×E. (3.5)
Using identity
∇× (∇×B) = −∆B +∇(∇ ·B)
33
in Equation (3.2) and the fact that ∇ ·B = 0 leads to the following formulation
∂tB−∇× (ve × B) = −ηc2
4π(∇×)2B− νed
2e(∇×)4B, (3.6)
where
B = B− d2e∆B.
It can be seen that with the above formulation, the divergence-free condition of the
magnetic field B is built in, i.e.,
∂t(∇ ·B) = 0.
In numerical simulations of turbulence, one can introduce even higher order dif-
fusion terms, see e.g. [17],
∂tB−∇× (ve × B) = −ην(−∆)νB,
where ν = 1 corresponds to resistivity; ν = 2 corresponds to electron viscosity; and
ν > 2 is introduced to separate nondissipative and dissipative scales more clearly.
3.2.3 About boundary conditions
The MHD equations are usually supplemented by boundary conditions of different
types. In the following, we describe two typical boundary conditions often used in second-
order MHD models, e.g., [41, 27].
34
The simplest essential condition on ∂Ω is
B× n = k,
where k satisfies the compatibility condition
k · n = 0.
Another possible set of boundary conditions is given by
B · n = q,
and
∇×B× n = k,
where q and k satisfy the compatibility conditions∫∂Ω q = 0 and k · n = 0 respectively.
We will discuss boundary conditions for fourth-order models in the next section.
3.3 A simplified model problem with different types of boundary con-
ditions
In the following, we introduce model problems for the fourth-order magnetic in-
duction equations described in the previous section. Assume that Ω ⊂ R3 is a bounded
polyhedron. By consider the semi-discretization in time of these equations and then
35
ignoring the nonlinear terms, we obtain the following system of equations:
α(∇×)4u+ β(∇×)2u+ γu = f, in Ω,
div u = 0, in Ω,
(3.7)
where div f = 0. It is associated with two types of boundary conditions,
u× n = g1, ∇× u = g2, on ∂Ω, (3.8)
or
u× n = g1, (∇× u)× n = g3, on ∂Ω. (3.9)
The above choices of boundary conditions arise naturally in the variational formu-
lation (see below). On the other hand, in the numerical simulations of these problems
using the pseudo-spectral method, one often uses periodic boundary conditions, e.g.,
[16, 42].
It is worth pointing out that the parameter α is usually much smaller than either
β or γ. This fact imposes some difficulties in designing robust numerical methods, as
have been studied in the context of biharmonic problems, e.g., in [72] and [90]. In this
study, we focus on finite element methods that are robust with respect to the parameters.
Indeed, this is one of the key features of our new elements.
Remark: The above fourth-order curl equations also arise from the interior trans-
mission problem in the study of inverse scattering problems for inhomogeneous medium,
e.g., [24].
36
In order to provide an appropriate framework for analysis, we define the following
function spaces:
H(curl; Ω) = u ∈ (L2(Ω))3 | ∇ × u ∈ (L2(Ω))3,
H0(curl; Ω) = u ∈ H(curl; Ω) | u× n = 0, on ∂Ω,
V = v ∈ H0(curl; Ω) | ∇ × v ∈ H10 (Ω),
W = v ∈ H0(curl; Ω) | ∇ × v ∈ H0(curl; Ω).
V and W are Hilbert spaces with scalar products and norms given by
(u, v)V , (∇(∇× u),∇(∇× v)) + (∇× u,∇× v) + (u, v),
(u, v)W , (∇×)2u, (∇×)2v) + (∇× u,∇× v) + (u, v),
||u||V ,√
(u, u)V , ||u||W ,√
(u, u)W .
Lemma 3.1. If v is piecewise smooth, and v×n and ∇×v are continuous across element
interfaces, then v ∈ V .
We introduce two bilinear forms a(·, ·) and a(·, ·) defined on V × V and W ×W ,
respectively:
a(u, v) = α(∇(∇× u),∇(∇× v)) + β(∇× u,∇× v) + γ(u, v),
37
a(u, v) = α((∇×)2u, (∇×)2v) + β(∇× u,∇× v) + γ(u, v).
The corresponding variational formulations are
Find u ∈ V such that a(u, v) = (f, v), ∀ v ∈ V, (3.10)
Find u ∈W such that a(u, v) = (f, v), ∀ v ∈W. (3.11)
The well-posedness for the above variational problems follows from the Lax-
Milgram lemma.
The next lemma indicates that the weak solution satisfies the divergence-free
constraint.
Lemma 3.2. Assume ∇ · f = 0, and let u be the solution of problem (3.10) or (3.11).
Then ∇ · u = 0.
Proof. Choose test function v = ∇ϕ where ϕ ∈ C∞0 (Ω), then
(u,∇ϕ) = (f,∇ϕ);
hence, ∇ · u = ∇ · f = 0.
3.4 Nedelec elements for second order problems and relevant De Rham
diagrams
In the following, we study Nedelec elements for second order problems. We also
introduce a powerful tool for the study of error estimates of vector finite elements. This
38
is the so-called De Rham diagram which relates standard H1-conforming elements with
H(curl)-conforming elements of Nedelec and H(div)-conforming elements of Raviart-
Thomas [21]. These technical tools have been used in the error analysis of our new finite
elements.
We first introduce some notations:
• Pk - space of multivariate polynomials of degree (less than or equal to) k
• Pk - space of homogeneous multivariate polynomials of degree k
• Pk - space of vector-valued multivariate polynomials of degree (less than or equal
to) k
Denote by FK : K → K the affine map such that FK(K) = K and
FK x = BK x+ bK ,
where
|BK | ≤ ChK , |B−1K| ≤ Cρ−1
K, C1ρK ≤ |det BK | ≤ C2h
3K
.
Consider sequences involving incomplete polynomial spaces that correspond to
Nedelec elements of the first family for H(curl) and the Raviart-Thomas elements for
H(div). The use of incomplete polynomials is motivated by the fact that, due to the
mixed formulation, approximability of both electric field E and its curl E, will affect the
ultimate convergence rates. One can add those polynomials of order k from Pk that are
39
not gradients, and, therefore, do not contribute to the kernel of curl,
Rk = Pk−1 ⊕ p ∈ (Pk)3 | p · x = 0.
Another incomplete polynomial space is given by
Dk = Pk−1 ⊕ (Pk−1)x,
where the homogeneous polynomials from the second set do not contribute to the kernel
of the operator div. In other words, the divergence-free vectors of Dk belong to Pk−1;
hence, the spaces Dk and Pk−1 contain the same divergence-free vectors. Moreover,
div Dk(K) = Pk−1(K).
An exact sequence is given by (k ≥ 1):
Pk∇−→ Pk−1 ⊕ p ∈ (Pk)3 | p · x = 0 ∇×−−−→ Pk−1 ⊕ p(x)x | p ∈ Pk−1
∇·−−→ Pk−1.
Or it can be written in a more compact form (k ≥ 1),
R → Pk∇−→ Rk
∇×−−−→ Dk∇·−−→ Pk−1 → 0.
At the end, the order of polynomials in the new sequence drops from k to only k − 1.
Next, we give two examples of Nedelec elements. For a general definition, we refer
to [70, 71, 66].
40
Definition 3.1. The second order Nedelec element of the first family is defined by
• K is a tetrahedron.
• PK
= R2(K).
• ΣK
is the set of degrees of freedom given by
– edge degrees of freedom:
Me(u) =
∫eu · τ q ds | ∀ q ∈ P1(e), ∀ e ⊂ K
,
– face degrees of freedom:
Mf
(u) =
1|f |
∫fu× n · q dA | ∀ q ∈ (P0(f))2
, ∀ f ⊂ K.
Then, ΣK
= Me(u) ∪M
f(u).
Definition 3.2. The first order Nedelec element of the second family is defined by
• K is a tetrahedron.
• PK
= P1(K).
• ΣK
= Me(u) is the set of degrees of freedom given by
Me(u) =
∫eu · τ q ds | ∀ q ∈ P1(e), ∀ e ⊂ K
.
The interpolation properties of these two Nedelec elements are given by the fol-
lowing results.
41
Theorem 3.1. [70, 66] If u ∈ (H2(Ω))3, ∇ × u ∈ (H2(Ω))3, and rhu is the standard
nodal interpolant of u in the second order Nedelec space of the first family, then
‖u− rhu‖0,Ω + ‖∇ × (u− r
hu)‖0,Ω . h
2(‖u‖2,Ω + ‖∇ × u‖2,Ω).
Theorem 3.2. [71, 66] If u ∈ (H2(Ω))3, and rhu is the standard nodal interpolant of
u in the first order Nedelec space of the second family, then
‖u− rhu‖0,Ω + h‖∇ × (u− r
hu)‖0,Ω . h
2|u|2,Ω.
For the purpose of error analysis, it is also necessary to introduce Raviart-Thomas
elements for H(div) problems.
Definition 3.3. The second order Raviart-Thomas is defined by
• K is a tetrahedron.
• PK
= D2(K).
• ΣK
is the set of degrees of freedom given by
– face degrees of freedom:
Mf
(u) =∫
fu · n q dA | ∀ q ∈ P1(f), ∀ f ⊂ K
,
– element degrees of freedom:
MK
(u) =∫
Ku · q dx | ∀ q ∈ R3
.
42
Then ΣK
= Mf
(u) ∪MK
(u).
Lemma 3.3. If v and v are related by the Piola transformation
u · FK
=1
det (BK
)BKu,
then
|v|s,K≤ C‖B−1
K‖s‖B
K‖|det (B
K)|−1/2|v|
s,K.
Proof. By the Piola transformation, we have for |α| = s,
∂αv
∂xα=
BK
det BK
∂αv
∂xα.
By the chain rule and the fact that FK
is an affine transformation, we obtain
‖∂αv
∂xα‖0,K = (
∫K|BK
det BK
∂αv
∂xα|2 |det B
K| dV )1/2
≤ C|BK| |det B
K|−1/2|B−1
K|s‖∂
αv
∂xα‖0,K .
To describe the De Rham diagrams, we consider a bounded and convex polyhe-
dron Ω ⊂ R3. Let Th
be a triangulation of Ω consisting of tetrahedra with diameters
bounded by h. We first define some conforming finite elements (Lagrange, edge, face, and
discontinuous finite elements, denoted by Hgrad
h, H
curl
h, H
div
h, L
2
h, respectively). Then it
can be shown that the following diagram commutes and has exact rows (assume the
43
functions are regular enough to ensure the existence of the corresponding canonical in-
terpolations). The following diagram is also referred to as discrete De Rham complex:
R id−−−−→ C∞(Ω) ∇−−−−→ C
∞(Ω) ∇×−−−−→ C∞(Ω) ∇·−−−−→ C
∞(Ω) 0−−−−→ 0y yΠ1h
yΠch
yΠdh
yΠ0h
yR id−−−−→ H
1(Ω) ∇−−−−→ H(curl; Ω) ∇×−−−−→ H(div; Ω) ∇·−−−−→ L2(Ω) 0−−−−→ 0
.
Next, we prove a maximum norm estimate for the interpolation operator rh
cor-
responding to the second order Nedelec element of the first family, which is needed in
our convergence analysis later.
Lemma 3.4. Let rKu be the local interpolant of u in the second order Nedelec space of
the first family, then
‖∇ × (rKu− u)‖∞,K . h
1/2‖∇ × u‖2,K .
Proof. Lemma 3.3 implies for s = 2 that
|v|2,K ≤ Ch5/2|v|2,K .
Let πh
be the interpolation operator corresponding to the second order Raviart-Thomas
element space. By Bramble-Hilbert lemma, we obtain
‖∇ × u− πh∇ × u‖∞,K . |∇ × u|2,K .
44
Hence, the commuting diagram property and standard scaling argument gives,
‖∇ × (r2,Ihu− u)‖∞,K = ‖π
h(∇× u)−∇× u‖∞,K
= ‖ 1det B
K
BK
(∇ × u− πh∇ × u)‖∞,K
. | 1det B
K
BK| |∇ × u|2,K
. h−2|∇ × u|2,K
. h1/2|∇ × u|2,K .
We can also show the following boundary estimate.
Lemma 3.5. Let rh
be the interpolation operator corresponding to the second order
Nedelec space of the first family and f be a face of the tetrahedron K, then
‖∇ × (rhu− u) · n‖0,f . h
3/2‖∇ × u‖2,K .
Proof. Let πh
be the interpolation operator corresponding to the second order Raviart-
Thomas space, and ph
be the L2 projection operator onto the linear nodal finite element
space on face f . Then,
‖∇ × (rhu− u) · n‖2
0,f= ‖[π
h(∇× u)−∇× u] · n‖2
0,f
= ([(πh− I)∇× u] · n,
[(πh− I)∇× u] · n− (π∇× u) · n+ p
h(∇× u · n))
f.
45
Hence,
‖∇ × (rhu− u) · n‖0,f ≤ ‖p
h(∇× u · n)− (∇× u · n)‖0,f
. h3/2‖(∇× u) · n‖3/2,f
. h3/2‖(∇× u‖2,K .
3.5 A nonconforming finite element for problem (3.10)
In this section, we will construct a nonconforming finite element to solve the
fourth-order equations arising from the MHD models. The use of a nonconforming
element has the advantage that the number of degrees of freedom is small compared to
that for conforming elements. The following construction is based on Nedelec elements
of the first family that consist of incomplete polynomials [70]. One advantage of using
incomplete polynomial space is that it provides the same order of convergence in terms of
energy norms as the one given by the corresponding complete polynomial space. In the
following, we define the degrees of freedom in a special way to ensure that the consistency
error estimate holds.
Definition 3.4. The new finite element on an element K is defined by the following
finite element triple (K,PK,ΣK
).
• K is a tetrahedron.
• PK
= R2(K).
• ΣK
is the set of degrees of freedom given by (see Figure 3.1)
46
– edge degrees of freedom:
Me(u) =
∫eu · τ q ds | ∀ q ∈ P1(e), ∀ e ⊂ K
,
– face degrees of freedom:
Mf
(u) =
1
|f |2
∫f
(∇× u)× n · q dA | ∀ q ∈ (P0(f))2, ∀ f ⊂ K
.
Then, ΣK
= Me(u) ∪M
f(u).
Fig. 3.1. Degrees of freedom of the first new element
The above finite element triple can be considered as a modification of a Nedelec
element of the first family for the second order H(curl) problem. The only difference
is the definition of the second set of degrees of freedom which is designed to ensure
47
consistency for the fourth-order problems. The total number of the degrees of freedom
for this new element is 20, which is the same as the dimension of the polynomial space
R2(K).
It should be pointed out that the scaling factor 1/|f |2 in the definition of the
second set of degrees of freedom is associated with the definition of the nodal basis
functions to be constructed later.
The next lemma provides a relationship between the edge integral and face inte-
gral; this relationship is useful in error analysis.
Lemma 3.6. [70] If u ∈ R2 is such that the edge degrees of freedom vanish, then
∫f
(∇× u) · n dA = 0.
As a direct consequence of the previous lemma, if both the edge degrees of freedom
and face degrees of freedom vanish, then
∫f
(∇× u) · q dA = 0, ∀ q ∈ R3.
3.5.1 Basis functions
In order to show the unisolvence of the finite element, we construct nodal basis
functions corresponding to the above defined degrees of freedom. The explicit form of
these basis functions also provides a tool for the interpolation error estimate.
48
Let K be an arbitrary tetrahedron with four vertices a1, a2, a3 and a4. The
corresponding barycentric coordinates are given by λ1, λ2, λ3, and λ4 respectively.
Denote by eij
the edge connecting vertices ai
and aj.
On each of the four faces, we choose two tangential direction vectors as below.
Face 1 (with vertices a2, a3, a4):
q1
1= −−→a2a4 = 6|K|(∇λ1 ×∇λ3),
q2
1= −−→a4a3 = 6|K|(∇λ1 ×∇λ2).
Face 2 (with vertices a1, a3, a4):
q1
2= −−→a3a1 = 6|K|(∇λ2 ×∇λ4),
q2
2= −−→a1a4 = 6|K|(∇λ2 ×∇λ3).
Face 3 (with vertices a1, a2, a4):
q1
3= −−→a1a2 = 6|K|(∇λ3 ×∇λ4),
q2
3= −−→a2a4 = 6|K|(∇λ3 ×∇λ1).
Face 4 (with vertices a1, a2, a3):
q1
4= −−→a1a3 = 6|K|(∇λ4 ×∇λ2),
49
q2
4= −−→a3a2 = 6|K|(∇λ4 ×∇λ1).
The edge degrees of freedom are defined explicitly in the following:
M1
e(u) =
∫eu · τ ds, M
2
e(u) =
∫eu · τ(3− 6
|e|s) ds,
where τ is the unit direction vector of edge e.
The basis functions of the second order Nedelec element of the first family in
barycentric coordinates are given below:
(1) Two basis functions on each edge ij (1 ≤ i < j ≤ 4):
Leij ,1
= λi∇λ
j− λ
j∇λ
i,
Leij ,2
= λi∇λ
j+ λ
j∇λ
i.
(2) Two basis functions on each face f with vertices ai, aj
and ak:
Lf,1 = λ
i(λj∇λ
k− λ
k∇λ
j),
Lf,2 = λ
j(λi∇λ
k− λ
k∇λ
i).
Some useful facts are listed as follows:
(1) The normal vector of face fl
(the face opposite vertex al) is given by
∇λl
‖∇λl‖.
50
(2) The two tangential vectors of face fl
are given by
∇λl×∇λ
i‖∇λ
l×∇λ
i‖,∇λ
l×∇λ
j
‖∇λl×∇λ
j‖.
(3) Let hi
be the height of the tetrahedron corresponding to the face fi, then
∇λi
=1
6|K|(aj− a
l)× (a
k− a
l),
|∇λi| = 1
hi
.
(4) Let |K| be the volume of the tetrahedron K, then
6|K| = |(ai− a
l) · [(a
j− a
l)× (a
k− a
l)]| = 1
(∇λi×∇λ
j) · ∇λ
k
.
Some vector identities:
A× (B×C) = B(A ·C)−C(A ·B),
∇× (A×B) = (B · ∇)A− (A · ∇)B + A(∇ ·B)−B(∇ ·A),
∇× (fA) = f(∇×A) + A×∇f.
We construct nodal basis functions in barycentric coordinates and are dual basis
with respect to the prescribed degrees of freedom by the following two steps.
51
Step 1. Construct eight nodal basis functions φi8i=1
corresponding to the face
degrees of freedom such that
Me(ϕi) = 0, i = 1, · · · , 8, (3.12)
and
Mf,j
(ϕi) = δ
j,i, i, j = 1, · · · , 8. (3.13)
We use the basis functions of the second order Nedelec element as building blocks as
they automatically satisfy the first condition (3.12). Using the facts listed above, we can
justify that the basis functions corresponding to the facial degrees of freedom are given
by the following:
Face 1:
φ1 = 3|K|[λ1(λ4∇λ2 − λ2∇λ4)− λ1(λ2∇λ3 − λ3∇λ2)],
φ2 = 3|K|[λ1(λ3∇λ4 − λ4∇λ3)− λ1(λ2∇λ3 − λ3∇λ2)].
Face 2:
φ3 = 3|K|[λ2(λ1∇λ3 − λ3∇λ1)− λ2(λ3∇λ4 − λ4∇λ3)],
φ4 = 3|K|[λ2(λ4∇λ1 − λ1∇λ4)− λ2(λ3∇λ4 − λ4∇λ3)].
52
Face 3:
φ5 = 3|K|[λ3(λ1∇λ2 − λ2∇λ1)− λ3(λ4∇λ1 − λ1∇λ4)],
φ6 = 3|K|[λ3(λ2∇λ4 − λ4∇λ2)− λ3(λ4∇λ1 − λ1∇λ4)].
Face 4:
φ7 = 3|K|[λ4(λ3∇λ1 − λ1∇λ3)− λ4(λ1∇λ2 − λ2∇λ1)],
φ8 = 3|K|[λ4(λ2∇λ3 − λ3∇λ2)− λ4(λ1∇λ2 − λ2∇λ1)].
As an example, we check that φ1 satisfies the second condition (3.13) by the
following calculations.
Let ϕ1 = 3|K|λ1(λ4∇λ2 − λ2∇λ4), ϕ2 = 3|K|λ1(λ2∇λ3 − λ3∇λ2), then φ1 =
ϕ1 − ϕ2.
53
M1
f1
(ϕ1) =1
|f1|2
∫f1
(∇× ϕ1)× n · q1 dA
=1
|f1|2
∫f1
(∇× ϕ1)×∇λ1‖∇λ1‖
· (6|K|∇λ1 ×∇λ3) dA
= 3|K|6|K||f1|
2h1
∫f1
[(∇λ1 ×∇λ2) · ∇λ3](−λ4(∇λ1 · ∇λ1))
+[(∇λ1 ×∇λ4) · ∇λ3](−λ2(∇λ1 · ∇λ1)) dA
= 3|K|6|K||f1|
2h123
|f1|
6|K|h21
=23.
Similarly,
M2
f1
(ϕ1) = −13, M
1
f2
(ϕ1) =13, M
2
f2
(ϕ1) =13,
M1
f3
(ϕ1) = −13, M
2
f3
(ϕ1) =23, M
1
f4
(ϕ1) =13, M
2
f4
(ϕ1) = −23,
M1
f1
(ϕ2) = −13, M
2
f1
(ϕ2) = −13, M
1
f2
(ϕ2) =13, M
2
f2
(ϕ2) =13,
M1
f3
(ϕ2) = −13, M
2
f3
(ϕ2) =23, M
1
f4
(ϕ2) =13, M
2
f4
(ϕ2) = −23.
Hence, Mfj
(φ1) = Mfj
(ϕ1)−Mfj
(ϕ2) = δj,1, j = 1, · · · , 8.
54
Step 2. Construct twelve nodal basis functions ψ12
j=1corresponding to the
edge degrees of freedom such that
Me,i
(ψj) = δ
ij, i, j = 1, · · · , 12, (3.14)
Mf
(ψj) = 0, j = 1, · · · , 12. (3.15)
Here, we use the edge basis functions of the second order Nedelec element as
building blocks as they automatically satisfy the first condition (3.14). Notice that
∇ × (λi∇λ
j+ λ
j∇λ
i) = 0, such that the following functions automatically satisfy the
second condition:
λi∇λ
j+ λ
j∇λ
i.
For functions of the form λi∇λ
j−λ
j∇λ
i, we need to subtract from them a linear
combination of face basis functions such that (3.14) and (3.15) hold. This can be done
because by construction, our face basis functions have no edge moments. This strategy
for constructing nodal basis functions can also be found in [43, 83].
Finally, we can write the nodal basis functions of the new element as the following:
(1) Two basis functions on each face k (1 ≤ k ≤ 4):
ψm
k= 3|K|λ
k((−1)mL
k+m,k+2 − Lk+1,k+2), (m = 1, 2), mod (4),
where Lij
= λi∇λ
j− λ
j∇λ
i.
55
(2) Two basis functions on each edge ij (1 ≤ i < j ≤ 4):
ψ1
ij= λ
i∇λ
j+ λ
j∇λ
i,
ψ2
ij= L
ij+
4∑l=1
2∑m=1
Mm
l(Lij
)ψml,
where Mm
l(u) = 1
|f |2∫fl
(∇× u)× n · qmdA.
3.5.2 Unisolvence of the finite element
Let us recall the definition of unisolvent for a finite element [26].
Definition 3.5. The finite element (K,PK,ΣK
) is said to be unisolvent if a function
in PK
can be uniquely determined by specifying values for degrees of freedom in ΣK
.
In order to prove the unisolvence property of our element, we consider any function
v in PK
whose degrees of freedom are all zero. It suffices to show that v is identically
equal to zero. This can be done by simply observing that v can be written as
v =4∑
k=1
2∑m=1
Mk,m
f(v)ϕm
k+
∑1≤i<j≤4
2∑m=1
Mij,m
e(v)ψm
ij.
3.5.3 Convergence analysis
Denote by Th
= KiNh
i=1the triangulation of the domain Ω into tetrahedra.
Denote by Vh
the finite element space associated with Th
. The degrees of freedom of
functions in Vh
vanish on ∂Ω.
56
Define the discrete norm ‖ · ‖h
by
‖v‖h
=
∑K∈Th
(‖v‖2
0,K+ ‖∇ × v‖2
0,K+ ‖∇(∇× v)‖2
0,K
)1/2
.
In the following, we first consider interpolation error estimates.
Let wK
= rKu be the second order Nedelec interpolant. K
fand K
′f
are tetra-
hedra sharing a common face f . Define wK
such that
Me(wK
) = Me(wK
),
Mf
(wK
) = [Mf
(wKf
) +Mf
(wK′
f)]/2.
If f ⊂ ∂Ω, we set Mf
(wK
) = Mf
(wKf
).
Define a special interpolant uI
such that uI|K
= wK
, then uI∈ V
h.
By triangle inequality,
‖u− uI‖h≤ ‖u− w
h‖h
+ ‖wh− w
h‖h.
Notice that
wK− w
K=∑f⊂K
2∑m=1
Mm
f(wK− w
K)ϕmf,
where ϕmf
are nodal basis functions on face f , and Mm
f(·) are degrees of freedom corre-
sponding to face f .
57
2|Mf
(wK− w
K)| = |M
f(wK′
f)−M
f(wKf
)|
=1
|f |2
∣∣∣∣ ∫f
(∇× wK′
f−∇× w
Kf)× n · q dA
∣∣∣∣≤ 1
|f |2
∣∣∣∣ ∫f
(∇× (rK′
fu)−∇× u)× n · q dA
∣∣∣∣+
1
|f |2
∣∣∣∣ ∫f
(∇× (rKf
u)−∇× u)× n · q dA∣∣∣∣
.1
|f |2(‖(∇× (r
K′fu)−∇× u)× n‖∞,f
+‖(∇× (rKf
u)−∇× u)× n‖∞,f )∫f|q| dA
. h−1/2‖∇ × u‖2,K .
We want to show that
‖u− uI‖h
. h(‖u‖2 + ‖∇ × u‖2).
By the interpolation error estimates of the Nedelec element, we have
‖u− wh‖h
. h(‖~u‖2 + ‖∇ × ~u‖2);
hence, it suffices to show that
‖wh− w
h‖h
. h(‖u‖2 + ‖∇ × u‖2).
58
Since ∇λi
= O(h−1), ‖ϕf‖20,K
= O(h7), by the Cauchy-Schwarz inequality, we
obtain
‖wK− w
K‖0,K ≤
∑f⊂K
2∑m=1
|Mm
f(wK− w
K)|21/2∑
f⊂K
2∑m=1
‖ϕmf‖20,K
1/2
. h3‖∇ × u‖2,K .
Hence,
‖wh− w
h‖0,Ω = (
∑K
‖wK− w
K‖20,K
)1/2. h
3‖∇ × u‖2,Ω.
By inverse inequality, we have
‖∇ × (wh− w
h)‖0,Ω . h
2‖∇ × u‖2,Ω,
‖∇(∇× (wh− w
h))‖0,Ω . h‖∇ × u‖2,Ω.
In the end, we reach the following estimate
‖wh− w
h‖h
. h‖∇ × u‖2,Ω,
which implies that
‖u− uI‖h
. h(‖u‖2,Ω + ‖∇ × u‖2,Ω).
Next, we show the consistency error estimate.
59
Consider the following discrete bilinear form:
ah
(uh, vh
) =∑K∈Th
α(∇(∇×uh
),∇(∇×vh
))L2(K)+β(∇×u
h,∇×v
h)L2(K)+γ(u
h, vh
)L2(K).
The nonconforming finite element discretization of problem (3.10) is
Find uh∈ V
hsuch that for all v
h∈ V
h,
ah
(uh, vh
) = (f, vh
). (3.16)
Given a face f , we define an average operator P 0
fby
P0
fv =
1|f |
∫fv dA.
The following two lemmas are standard results.
Lemma 3.7. ∫∂K|w|2 dA . h
−1
K||w||2
0,K+ h
K|w|2
1,K.
Proof. Using the facts that
|B−1
K| . h
−1
K, |B
K| . h
K, |detB
K| . h
3
K,
and
|w|1,K . |BK|2|detB
K|−1/2|w|1,K ,
60
we obtain,
∫∂K|w|2dA =
∫∂K|B−TK
w|2 |A||A|
dA
.∫∂K|w|2dA
. ‖w‖21,K
= ‖w‖20,K
+ |w|21,K
=∫K|BTKw|2 |K||K|
dx+ |w|21,K
. h−1
K‖w‖2
0,K+ |B
K|4(det B
K)−1|w|2
1,K
. h−1
K‖w‖2
0,K+ h
K|w|2
1,K.
Lemma 3.8. Given any face f ⊂ K and w ∈ H1(K),
∫f|w − P 0
fw|2 dA ≤ Ch
K|w|2
1,K.
Proof. Notice that P 0
fis an orthogonal projection, from Lemma 3.7 and the local
error estimate; therefore, we obtain,
∫f|w − P 0
fw|2 dA .
∫f|w − P 0
Kw|2 dA
. h−1
K||w − P 0
Kw||2
0,K+ h
K|w − P 0
Kw|2
1,K
= h−1
K‖w − P 0
Kw‖2
0,K+ h
K|w|2
1,K
≤ ChK|w|2
1,K.
61
A few more lemmas are needed for the consistency error estimates.
Let K be a tetrahedron, rK
be the local interpolation operator for the second
order Nedelec space of the first family, and rK
be the interpolation operator for the first
order Nedelec space of the second family.
Consider the two tetrahedra Kf
and K′f
that share a common face f . Given a
function vh
in the new finite element space Vh
, denote vK
= vh|K
. By definition,
rKf
vKf
= rK′
fvK′
f, on face f.
Hence,
vKf× n− v
K′f× n = (v
Kf− r
KfvKf
)× n− (vK′
f− r
K′fvK′
f)× n,
where n is the unit normal vector of face f . As a direct consequence, we have
∑K
∫∂K
ϕ · [(rKvK
)× n] dA = 0. (3.17)
Lemma 3.9 can be found in [50]:
Lemma 3.9.
‖rKwK− w
K‖0,K . h‖∇ × v
K‖0,K .
Lemma 3.10.
‖rKvK− v
K‖0,K . h‖∇ × v
K‖0,K .
62
Proof. Consider the Helmholtz decomposition
vK
= ∇pK
+ wK,
where wK∈ L2(K)3, div w
K= 0, w
K· n|
∂K= 0, and p
K∈ P2(K).
Using the interpolation operators defined above, we obtain
rKvK
= rK∇p
K+ r
KwK
= ∇pK
+ rKwK.
Hence,
rKvK− v
K= r
KwK− w
K.
By Lemma 3.9, we obtain
‖rKvK− v
K‖0,K . h‖∇ × v
K‖0,K .
Now, we can show the following lemma, which is critical for the consistency error
estimate.
Lemma 3.11. For ϕ ∈ H(curl;K),
|∑K
∫∂K
ϕ · (vh× n) dA . h(‖ϕ‖0,Ω + ‖∇ × ϕ‖0,Ω)(
∑K
‖∇ × vh‖1,K).
63
Proof. By the interpolation error estimates of the Nedelec elements,
‖∇ × (rKvK− v
K)‖0,K . h‖∇ × v
K‖1,K ,
Lemma 3.10, and Equation (3.17), we have
∣∣∣∣∑K
∫∂K
ϕ · (vK× n) dA
∣∣∣∣ =∣∣∣∣∑K
∫∂K
ϕ · [(rKvK− v
K)× n] dA
∣∣∣∣=
∣∣∣∣∑K
∫K
(∇× ϕ) · (rKvK− v
K) dx+ ϕ · [∇× (r
KvK− v
K)] dx
∣∣∣∣≤
∑K
(‖∇ × ϕ‖0,K‖rKvK − vK‖0,K + ‖ϕ‖0,K‖∇ × (rKvK− v
K)‖0,K)
. h(‖ϕ‖0,Ω + ‖∇ × ϕ‖0,Ω)
∑K
‖∇ × vh‖1,K
.
Next, we show the consistency error estimate for the nonconforming finite element
approximation defined above.
Theorem 3.3. Assume that u ∈ V is sufficiently smooth and vh∈ V
h, then
|ah
(u, vh
)− (f, vh
)| . h(‖∇ ×∆(∇× u)‖+ ‖∇2(∇× u)‖
+ ‖∇ ×∇× u‖+ ‖∇ × u‖)∑K
‖∇ × vh‖1,K .
64
Proof. Applying integration by parts we get
(∇(∇× u),∇(∇× v
h))K
= −(∆(∇× u),∇× vh
)K
+ (∇(∇× u) · n,∇× vh
)∂K
= −(∇×∆(∇× u), vh
)K
+ (∆(∇× u), vh× n)
∂K+ (∇(∇× u) · n,∇× v
h)∂K
,
and
(∇× u,∇× vh
)K
= ((∇×)2u, v
h)K− (∇× u, v
h× n)
∂K.
Hence,
ah
(u, vh
)− (f, vh
)
=∑K∈Th
α(∆(∇× u), vh× n)
∂K+ α(∇(∇× u) · n,∇× v
h)∂K− β(∇× u, v
h× n)
∂K
=∑K∈Th
[(α∆(∇× u)− β∇× u, v
h× n)
∂K
]+∑K∈Th
[α(∇(∇× u) · n,∇× v
h)∂K
].
By Lemma 3.11, we can estimate the first term on the right-hand side of the last
equation:
∑K∈Th
[(α∆(∇× u)− β∇× u, v
h× n)
∂K
]
. h(‖∆(∇× u)‖0,Ω + ‖∇ ×∆(∇× u)‖0,Ω + ‖∇ × u‖0,Ω + ‖∇ ×∇× u‖0,Ω)∑K
‖∇ × vh‖1,K .
65
For the second term, we apply Lemma 3.8. We also use the inter-element con-
tinuity of ∇ × vh
given by Lemma 3.3 to add a constant term Pf
(∇(∇ × u) · n) to
∇(∇× u) · n:
∑K∈Th
[α(∇(∇× u) · n,∇× v
h)∂K
]
≤ α
∣∣∣∣∣∣∑K∈Th
∑f⊂∂K
(∇(∇× u) · n− Pf
(∇(∇× u) · n),∇× vh− P
f(∇× v
h))f
∣∣∣∣∣∣. h|∇(∇× u)|1,Ω|
∑K∈Th
|∇ × vh|1,K .
Finally, we have the following convergence result.
Theorem 3.4. Let u and uh
be the solutions of the problems (3.10) and (3.16) respec-
tively, then
||u− uh||0,h + ||∇ × (u− u
h)||0,h + ||∇(∇× (u− u
h))||0,h . h||u||4,Ω
when u ∈ (H4(Ω))3.
Proof. Apply the second Strang’s lemma, i.e.,
||u− uh||0,h + ||∇ × (u− u
h)||0,h + ||∇(∇× (u− u
h))||0,h
. infwh∈Vh
(||u− wh||0,h + ||∇ × (u− w
h)||0,h + ||∇(∇× (u− w
h))||0,h)
+ supwh∈Vh,wh 6=0
ah
(u,wh
)− (f, wh
)
||∇ × wh||1,h
,
and previous lemmas, and the desired inequality follows.
66
3.6 A nonconforming finite element for problem (3.11)
We first introduce three special vectors:
p1 =
0
−x3
x2
, p2 =
x3
0
−x1
, p3 =
−x2
x1
0
.
In fact, pi
= ei× x. By direct calculations we get
∇ · pi
= 0, ∇× pi
= 2ei.
Define
ϕi
= xi+1pi, i = 1, 2, 3,
we have
∇× ϕi
= 3xi+1e
i, ∇×∇× ϕ
i= −3e
i+2, i = 1, 2, 3.
We also define
ψi
= x2
ipi, i = 1, 2, 3, ψ4 = x1x3p1, ψ5 = x1x3p3,
which satisfies
∇×∇× ψi
= −2pi, i = 1, 2, 3,
67
and
∇×∇× ψ4 =
x2
3x1
0
, ∇×∇× ψ5 =
0
−3x3
−x2
.
We choose the local polynomial space as
P = Spanei, pi, ϕi, ψj, i = 1, 2, 3, j = 1, 2, 3, 4, 5.
One important property of the above defined vector polynomial space is that the
set ∇ × pi,∇ × ϕ
i,∇ × ψ
j and the set ∇ × ∇ × ϕ
i,∇ × ∇ × ψ
j are both linear
independent. As a consequence, the curl operator acting on P is similar to the gradient
operator acting on scalar polynomials. For example, if v = x×a, then ∇×v = 0 implies
v = 0.
Lemma 3.12.
‖∇u‖ . ‖∇ × u‖, ∀ u ∈ P.
Proof. It suffices to show this inequality for any u ∈ P\R3. Notice that dim
P\R3 = 11 is a finite number; by norm equivalence it suffices to show that ‖∇u‖ and
‖∇ × u‖ are two norms in P\R3.
It is obvious that ‖∇u‖ is a norm in space P\R3. To show that ‖∇ × u‖ is also
a norm in the same space, we only need to prove that u ≡ 0 if ‖∇ × u‖ = 0.
68
Since u =∑3
i=1(aipi
+ biϕi) +
∑5
j=1cjψj, we have
‖∇ × u‖ = 0⇒3∑i=1
ai∇× p
i+ b
i∇× ϕ
i+
5∑j=1
cj∇× ψ
j= 0.
Using the fact that ∇ × pi,∇× ϕ
i,∇× ψ
j is linear independent, we conclude
that ai
= bi
= cj
= 0, ∀ i, j, i.e. u ≡ 0.
Definition 3.6. The finite element triple (K,PK,ΣK
).
• K is a tetrahedron.
• PK
is the incomplete cubic vector polynomial space defined by the span of the
functions given above.
• ΣK
is the set of degrees of freedom given by (see Figure 3.2)
– edge degrees of freedom:
Me(u) =
∫eu · τ q ds | ∀ q ∈ P0(e), ∀ e ⊂ K
,
– face degrees of freedom:
Mf
(u) =∫
f(∇× u)× n · q dA | ∀ q ∈ (P0(f))2
, ∀ f ⊂ K.
Then, ΣK
= Me(u) ∪M
f(u).
The first set of degrees of freedom is the same as the one for the lowest order edge
element. The second set of degrees of freedom is designed solely for the consistency error
69
Fig. 3.2. Degrees of freedom of the second new element
estimate. The total number of the degrees of freedom of this new element is 14, which
is the same as the dimension of the local polynomial space.
Denote by ΠK
the local interpolation operator associated with the given set of
degrees of freedom; therefore, we have the following local error estimate.
Lemma 3.13. There exists a constant C independent of h such that
|v −ΠKv| ≤ Ch|v|1,K ,
|(∇×)2(v −ΠKv)| ≤ Ch|v|3,K , ∀ K ∈ Th.
The global interpolation operator Πh
is defined by
(Πhv)|K
= ΠK
(v|K
), ∀ K ∈ Th.
70
3.6.1 Basis functions
In the following, we construct an orthogonal set of vector basis functions. Given
any tetrahedron K, denote by ei
the i-th edge, and denote by fj
the j-th face. We begin
with some notations:
Mei
=∫ei
u · τ ds, i = 1, · · · , 6,
M(k)
fj
=∫fj
(∇× u)× n · qkdA, j = 1, · · · , 4, k = 1, 2,
where τ is the unit direction vector of edge ei; q1 and q2 are two tangential
direction vectors of face fj; and n is the outer unit normal vector to face f
j.
Consider ϕ1, ϕ2, ϕ3 defined as before, and denote
ϕ3+j = ψj, j = 1, 2, · · · , 5.
Denote the lowest order edge basis functions as
Leij
= λi∇λ
j− λ
j∇λ
i.
We first modify ϕi’s such that they have vanishing edge degrees of freedom:
ϕi
= ϕi−∑e⊂K
Me(ϕi)Le.
71
Then the face basis functions can be written as a linear combination of ϕi’s, i.e.,
L(k)
fj
=8∑i=1
aiϕi,
where the coefficients ai’s are obtained by solving a linear system of equations:
M(m)
fl
(L(k)
fj
) = δjlδkm
, l = 1, · · · , 4, m = 1, 2.
The last step in our construction is to modify Le’s such that they have vanishing
face degree of freedoms:
Le
= Le−∑j,k
M(k)
fj
(Le)L(k)
fj
.
The orthogonal vector basis functions are given by L(k)
fj
for j = 1, · · · , 4, k = 1, 2
and Le
for any edge e ⊂ K.
3.6.2 Convergence analysis
Define the discrete bilinear form as
ah
(uh, vh
) =Nh∑i=1
α(∇×∇×uh,∇×∇×v
h)L2(Ki)
+β(∇×uh,∇×v
h)L2(Ki)
+γ(uh, vh
)L2(Ki)
.
The nonconforming finite element discretization of problem (3.7) is
Find uh∈ V
hsuch that for all v
h∈ V
h,
ah
(uh, vh
) = (f, vh
). (3.18)
72
Lemma 3.14. If v is piecewise smooth, v × n and (∇ × v) × n are continuous across
element interfaces, then v ∈W .
Let K and K′ be two tetrahedra sharing a common face f , and let rK
be the local
interpolation operator for the first order Nedelec element of the second family. Then by
the definition of the first set of degrees of freedom, we have
(rK
vK
)× n = (rK′vK′)× n on f,
where vK
= vh|K
.
Lemma 3.15.
‖vK− r
KvK‖ . h‖∇ × v
K‖0,K .
‖∇ × (vK− r
KvK
)‖ . h‖∇ ×∇× vK‖0,K .
Proof. For any element K ∈ Th
, let x = FKx = B
Kx+ b
Kbe the affine mapping
between K and the reference element K.
In K, since the interpolation operator rK
preserves constants, and ‖(∇×) · ‖ is a
norm in space VK\(P0(K))3, we have
‖(I − rK
)u‖0,K . ‖∇ × u‖0,K , ∀u ∈ Vh(K).
73
Consider the transformation
u FK
= (BTK
)−1u;
we, therefore, have
‖(I − rK
)u‖2 . h‖(I − rK
)u‖2 . h2‖∇ × u‖2
0,K.
Hence, the first estimate holds. Similarly, the second estimate follows from the fact that
‖(∇×)2 · ‖ is a norm in space VK\R1(K).
Next, we show the consistency error estimate for the above defined nonconforming
element.
Theorem 3.5. Assume that u ∈ V is sufficiently smooth and vh∈ V
h, then the following
estimate holds:
|ah
(u, vh
)− (α(∇×)4u+ β(∇×)2
u+ γu, vh
)| . h(‖(∇×)4u‖+ ‖(∇×)2
u‖1 + ‖∇× u‖)
(‖∇ × vh‖0,h + ‖(∇×)2
vh‖0,h), ∀ v
h∈ V
h.
74
Proof. For sufficiently smooth u and vh∈ V
h, applying integration by parts we
get
(∇×∇× u,∇×∇× v
h
)K
=(
(∇×)3u,∇× v
h
)K
+(∇×∇× u, n× (∇× v
h))∂K
=(
(∇×)4u, v
h
)K
+(
(∇×)3u, n× v
h
)∂K
+(∇×∇× u, n× (∇× v
h))∂K
,
and
(∇× u,∇× vh
)K
=(
(∇×∇× u, vh
)K
+ (∇× u, n× vh
).
By Lemma (3.8) we have
|ah
(u, vh
)− (f, vh
)|
=
∣∣∣∣∣∣∑K∈Th
(α(∇×)3
u+ β∇× u, vh× n
)∂K
+ α(
(∇×)2u, (∇× v
h)× n
)∂K
∣∣∣∣∣∣For the first term on the right-hand side of the last equation, we have
∣∣∣∣∣∣∑K∈Th
(α(∇×)3
u+ β∇× u, vh× n
)∂K
∣∣∣∣∣∣≤
∣∣∣∣∣∣∑K∈Th
(α(∇×)3
u+ β∇× u, (vK− r
KvK
)× n)∂K
∣∣∣∣∣∣≤
∣∣∣∣∣∣∑K∈Th
(α(∇×)4
u+ β(∇×)2u, v
K− r
KvK
)K−(α(∇×)3
u+ β∇× u,∇× (vK− r
KvK
))K
∣∣∣∣∣∣. h
(‖(∇×)4
u‖+ ‖(∇×)2u‖)‖∇ × v
h‖0,h + h
(‖(∇×)3
u‖+ ‖∇ × u‖)‖(∇×)2
vh‖0,h.
75
For the second term, we add a constant term Pf
((∇×)2u) to (∇×)2
u using the
inter-element continuity imposed by the second set of degrees of freedom to obtain
∣∣∣∣∣∣∑K∈Th
((∇×)2
u, (∇× vh
)× n)∂K
∣∣∣∣∣∣=
∣∣∣∣∣∣∑K∈Th
∑f⊂∂K
((∇×)2
u− Pf
(∇×)2u, (∇× v
h)× n− P
f(∇× v
h)× n
)f
∣∣∣∣∣∣. h|(∇×)2
u|1|∇ × vh|1,h.
The desired estimate then follows from the above estimation of the two boundary
terms.
We have the following convergence result.
Theorem 3.6. Let u and uh
be the solutions of the problem (3.11) and (3.18) respec-
tively, then
||u− uh||0,h + ||∇ × (u− u
h)||0,h + ||(∇×)2(u− u
h)||0,h . h||u||4,Ω
where u ∈ (H4(Ω))3.
Proof. Apply the second Strang’s lemma, i.e.,
||u− uh||0,h + ||∇ × (u− u
h)||0,h + ||(∇×)2(u− u
h)||0,h
. infwh∈Vh
(||u− wh||0,h + ||∇ × (u− w
h)||0,h + ||(∇×)2(u− w
h)||0,h)
+ supwh∈Vh,wh 6=0
ah
(u,wh
)− (f, wh
)
||∇ × wh||h
+ ||(∇×)2wh||h
,
76
and previous lemmas; hence, the desired inequality follows.
77
Chapter 4
Recovery Type A Posteriori Error Estimates
This chapter is devoted to developing a derivative recovery scheme and recovery
type a posteriori error estimators. The main results presented here are joint work with
Professor Randolph E. Bank and Professor Jinchao Xu [9].
4.1 Introduction
Adaptive finite element methods have been used with great success in the nu-
merical approximations of PDEs. Adaptive methods are particularly well suited for the
numerical solution of problems with singularities. Recently, several convergence results
regarding adaptive finite element methods have been proposed by Dofler [29], Morin,
Nochetto and Siebert [67], Binev, Dahmen and DeVore [13], and Stevenson [81]. In
these works, a typical adaptive finite element algorithm has the following form:
SOLVE→ ESTIMATE→ MARK→ REFINE.
Here, “SOLVE” refers to a fast iterative solver for the algebraic systems, e.g., multi-
grid methods; the step “ESTIMATE” requires certain a posteriori error estimators, e.g.,
residual type error estimators or recovery type error estimators; the “MARK” step in-
dicates the strategies for selecting appropriate elements for local refinement, e.g., the
78
error equi-distribution principle; and the last step, “REFINE”, represents algorithms,
e.g., regular refinement or newest vertex bisection refinement.
In an adaptive finite element computation, an efficient and reliable a posteriori
error estimator plays an important role. Beginning with the pioneering works of Babuska
and Rheinboldt [2, 3], and Bank and Weiser [6], adaptive local mesh refinement based
on a posteriori error estimators has attracted considerable interest.
In this study, we introduce a derivative recovery scheme for Lagrange triangular
elements of degree p. It is an extension of the gradient recovery scheme for linear elements
proposed by Bank and Xu [8]. We show that the recovered p-th derivatives superconverge
to the derivatives of the exact solution of the continuous problem on general unstructured
meshes. Based on the superconvergent properties, we design new and efficient a posteriori
error estimators and local error indicators to be used in local mesh refinement algorithms.
The recovery techniques in finite element analysis have been studied extensively in
the literature [49, 51, 73, 92, 93, 101, 103]. Finite element recovery techniques construct
better numerical approximations based on certain post-processing procedures. A typical
example is the local or global averaging of numerical approximations by applying local or
global L2 projection. The reconstructed numerical approximations often superconverge
to the exact solutions. Hence, recovery techniques are often used to construct asymp-
totically exact a posteriori error estimators (e.g., [31, 74, 97, 98, 104]). For a literature
review regarding the superconvergence properties of finite element recovery techniques,
see [8] and the references therein.
79
It should be pointed out that most recovery schemes are usually concerned with
the recovery of the finite element solution itself, its gradient, or its second order deriva-
tives. There are also studies on the recovery of higher order derivatives on uniform
grids (e.g., [20]). In our work, we provide an algorithm for the recovery of high order
derivatives based on high order Lagrange elements on unstructured meshes.
We first develop a post-processing derivative recovery scheme for the finite element
solution uh
on general unstructured but shape regular triangulation. In particular, we
compute SmhQh∂puh
. Here, Sh
is an appropriate smoothing operator; m ∈ 1, 2, . . .
is the number of smoothing steps; and Qh
is the L2 projection operator. We will show
that the recovered p-th derivatives superconverge to the exact derivatives of the solution
to the continuous problem. In case the number of smoothing steps is small (the most
interesting case), Theorem 4.1 proved in Section 4 indicates that the following estimate
holds:
||∂pu− SmhQh∂puh||0,Ω . h
(mh
1/2 +[κ− 1κ
]m)(||u||
p+2,Ω + |u|p+1,∞,Ω
),
with a constant κ > 1, independent of h and u.
Next, we develop a posteriori error estimators based on the above derivative
recovery scheme. As an example, we discuss quadratic finite elements in detail. In
fact, we define our local error indicator as
ετ
=112
3∏k=1
(`k+1∂k+1 − `k−1∂k−1
)u3φ0 +
112
3∑k=1
`3
k∂
3
ku3φk, (4.1)
80
where u3 is any cubic polynomial with third derivatives equal to ∂SmhQh∂
2uh
, `k
are the
edge lengths of the triangular element; and φk’s are hierarchical basis functions for the
4-dimensional space of cubic polynomials that are zero at the vertices and midpoints of
the element. The important feature of this local error indicator (4.1) is that it depends
only on the geometry of the element and the recovered third order derivatives.
4.2 Derivative recovery scheme
In this section, we introduce our derivative recovery scheme. For simplicity, we
consider a bounded polygonal domain, Ω in R2. Denote by V(p)
hthe finite element
space consisting of C0 piecewise polynomials of degree p associated with a shape regular
triangulation Th
. Let uh∈ V(p)
hbe the finite element approximation to a (possibly
nonlinear) second order elliptic boundary value problem.
We analyze a superconvergent approximation to the p-th order derivatives of u.
This approximation is generated by applying the global L2 projection operator Qh
and
a multigrid smoothing operator Sh
to the discrete p-th order derivatives of the finite
element solution uh
. It can be represented as SmhQh∂p
huh
.
Definition 4.1. Given a function u ∈ L2(Ω), the L2 projection Qhu ∈ V(1)
his defined
by solving the following variational problem:
(Qhu, v
h) = (u, v
h), ∀ v
h∈ V(1)
h, (4.2)
where (·, ·) denotes the inner product on L2(Ω).
81
By the Riesz representation theorem, the bilinear form,
a(u, v) = (∇u,∇v) + (u, v), (4.3)
induces a bounded linear operator Ah
: V(1)
h→ V(1)
h. It is uniquely determined by
(Ahuh, vh
) = a(uh, vh
), ∀ uh, vh∈ V(1)
h.
The discrete operator Ah
is symmetric with respect to the L2-inner product.
Indeed, Ah
is symmetric positive definite on the finite dimensional space V(1)
hand
λ ≡ ρ(Ah
) ' h−2.
We define a smoothing operator Sh
by
Sh
= I − λ−1Ah.
In the rest of this chapter, ∂pu denotes a certain p-th order derivative of u and
∂puh
denotes some discrete p-th order derivative of uh
. We also introduce the notation
|| · ||′Ω
to indicate the usual broken norm∑K∈Th
|| · ||K
defined piecewisely. For the sake
of completeness, we now state several preliminary lemmas whose proofs can be found in
[9]. These technical lemmas lead to the main Theorem 4.1 in this section.
82
Lemma 4.1. For any z ∈ V(1)
h, u ∈ Hp+2(Ω),
||(I − Smh
)z||0,Ω . mh(||z − ∂pu||1,Ω + h||u||p+2,Ω + h
1/2|u|p+1,∂Ω).
Lemma 4.2. [8] Suppose that for v ∈ V(1)
hand some 0 < α ≤ 1 we have
||v|| ≤ ω(h, v),
||v||−α ≡ ||A−α/2h
v|| ≤ (Ch)αω(h, v).
Then
||Smhv|| ≤ ε
mω(h, v),
where
εm
=
κα/2
f(m,α/2) . m−α/2 for m > (κ− 1)α/2,
[(κ− 1)/κ]m for m ≤ (κ− 1)α/2,
and κ = (Ch)2λ.
Lemma 4.3. Let w|K∈ Hp(K)∩W p−1,∞(K), for all K ∈ T
h. Then, for 1/2 < α ≤ 1,
||SmhQh∂pw||0,Ω . ε
m(h−1||w||′
p−1,Ω+ ||w||′
p,Ω+ h−α||w||′
p−1,∞,Ω),
with εm
defined as in Lemma 4.2.
83
Lemma 4.4. Let u ∈ Hp+2(Ω)∩W p+1,∞(Ω). Then for any vh∈ V(p)
hand 1/2 < α ≤ 1
we have
||∂pu− SmhQh∂pvh||0,Ω . mh
3/2(h1/2||u||p+2,Ω + |u|
p+1,∂Ω)
+ εm
(h−1||u− vh||′p−1,Ω
+ h−α||u− v
h||′p−1,∞,Ω
),
with εm
defined as in Lemma 4.2.
When considering problems with the homogeneous Dirichlet boundary condition,
the boundary terms vanish and
||∂pu− SmhQh∂puh||0,Ω . h(mh+ ε
m)||u||
p+2,Ω.
In more general cases, we have the following theorem based only on the previous lemmas.
Theorem 4.1. Let u ∈ Hp+2(Ω) ∩W p+1,∞(Ω) and uh∈ V(p)
hbe an approximation of
u satisfying
||u− uh||′p−1,Ω
. h2|u|
p+1,Ω,
||u− uh||′p−1,∞,Ω
. h2| log h||u|
p+1,∞,Ω.
Then
||∂pu− SmhQh∂puh||0,Ω . h(mh1/2 + ε
m)(||u||
p+2,Ω + |u|p+1,∞,Ω),
where εm
is defined as in Lemma 4.2 and 1/2 < α < 1.
84
The next theorem provides an estimate for (p+ 1)-st order derivatives which is a
direct consequence of Theorem 4.1.
Theorem 4.2. Assume the hypotheses of Theorem 4.1. Then
||∂(∂pu− SmhQh∂puh
)||0,Ω . (mh1/2 + εm
)(||u||p+2,Ω + |u|
p+1,∞,Ω),
where εm
is defined as in Lemma 4.2 and 1/2 < α < 1.
4.3 A posteriori error estimates
Based on the derivative recovery scheme developed in the previous section, we
can introduce the following recovery type a posteriori error estimates. Here, we follow
the approach given in [8] for the case of piecewise linear finite elements. We consider the
case of Lagrange elements of degree p and provide an expression to estimate the error,
which involves only (approximate) derivatives of order p + 1 of u and some parameters
describing the geometry of a given element τ .
First, we introduce some notations to describe the geometry of a canonical element
τ ∈ Th
. Let ptk
= (xk, yk), 1 ≤ k ≤ 3 be the three vertices oriented counterclockwise,
and ψk3k=1
be the corresponding barycentric coordinates. Let ek3k=1
denote the
edges of element τ , nk3k=1
the unit outward normal vectors, tk3k=1
the unit tangent
vectors with counterclockwise orientation, and `k3k=1
the edge lengths (see Fig. 4.1).
To fix the idea, we consider a special case p = 2, i.e., quadratic finite elements.
We first give an explicit formula for u3 − u2 on τ , where u2 is the quadratic Lagrange
85
interpolant, and u3 is the cubic hierarchical extension. Thus, u3 − u2 is a cubic polyno-
mial zero at vertices and edge midpoints of τ . A hierarchical basis for this 4-dimensional
space is given by
φ0 = ψ1ψ2ψ3,
φk
= ψk−1ψk+1(ψ
k+1 − ψk−1),
for 1 ≤ k ≤ 3, and (k − 1, k, k + 1) is a cyclic permutation of (1, 2, 3). Let ∂ku denote
the directional derivative in the direction tk. Then
u3 − u2 =112
3∏k=1
(`k+1∂k+1 − `k−1∂k−1
)u3φ0 +
112
3∑k=1
`3
k∂
3
ku3φk. (4.4)
@@
@@
@@
@@
@@
@
τ
p1 p2
p3
n1`2t2
e3
Fig. 4.1. Parameters associated with the triangle τ .
86
In order to define the local error indicator, we approximate the third derivatives
which are needed to compute the directional derivatives appearing in (4.4) by
∂xxx
u3 ≈ ατ∂xSm
hQh∂xxuh,
∂xxy
u3 ≈ατ
2(∂ySm
hQh∂xxuh
+ ∂xSm
hQh∂xyuh
), (4.5)
∂xyy
u3 ≈ατ
2(∂ySm
hQh∂xyuh
+ ∂xSm
hQh∂yyuh
),
∂yyy
u3 ≈ ατ∂ySm
hQh∂yyuh,
where ατ> 0 is a constant as described below. Let u3 be any cubic polynomial with
third derivatives given by the right-hand sides of (4.5). Then we define our local error
indicator as a cubic polynomial on each element depending only on the geometry of τ
and the approximate third derivatives derived from our superconvergent approximations
by
ετ
=112
3∏k=1
(`k+1∂k+1 − `k−1∂k−1
)u3φ0 +
112
3∑k=1
`3
k∂
3
ku3φk. (4.6)
The normalization constant ατ
is chosen such that
|ετ|22,τ
= ||(I − SmhQh
)∂2
xxuh||20,τ
+ 2||(I − SmhQh
)∂2
xyuh||20,τ
+ ||(I − SmhQh
)∂2
yyuh||20,τ≡ |u
h−R(u
h)|2
2,τ.
Normally, we would expect that ατ≈ 1, which is likely to be the case in regions where the
third derivatives of the exact solution are well defined. Near singularities, u is not smooth
87
and we anticipate difficulties in estimating the third derivatives. For elements near such
singularities, ατ
provides a heuristic for partly compensating for poor approximation.
For a discussion of more general cases where p > 2, see [9].
4.4 Numerical experiments
In this section, we provide numerical experiments to illustrate the effectiveness
of our recovery scheme and a posteriori error estimates in the cases of uniform and
adaptively refined (nonuniform) meshes. The derivative recovery scheme and a posteriori
error estimate described above for the case of continuous piecewise quadratic elements
were implemented in the PLTMG package [5], which was then used for our numerical
experiments.
We consider the solution of the problem
−∆u = f in Ω = (0, 1)× (0, 1),
u = g on ∂Ω,
where f and g are chosen such that the exact solution is given by u = ex+y (see Fig.
4.2). This is a very smooth solution that satisfies all the assumptions of our theory. In
our experiments, we choose m = 2. We begin with a uniform 3 × 3 mesh consisting of
eight right triangles as shown in Figure 4.3. Elements in Figure 4.3 are colored according
to size.
In Tables 4.1–4.2, we record the results of the numerical experiments. The error
is given as a function of the number of elements. For the purpose of comparison, we
88
Fig. 4.2. Graph of the exact solution
choose targets for the adaptive refinement procedure to produce adaptive meshes with
similar numbers of elements to the uniform refinement case. Note that the dimension
of the quadratic finite element space is approximately 2nt, where nt is the number of
89
elements. Other values are defined as follows:
L2 = ||u− uh||0,Ω,
L2 = ||εh||0,Ω,
EF0 =||εh||0,Ω
||u− uh||0,Ω
,
H1 = |u− uh|1,Ω,
H1 = |εh|1,Ω,
EF1 =|εh|1,Ω
|u− uh|1,Ω
,
H2 = |u− uh|2,Ω,
H2 = |u−R(uh
)|2,Ω,
EF2 =|R(u
h)− u
h|2,Ω
|u− uh|2,Ω
.
For each type of norm, we made a least squares fit of the data to a function of the form
F (N) = CN−p/2 in order to estimate the order of convergence p. All integrals were
approximated using a 12-point order 7 quadrature formula applied to each triangle.
From the above result, we can observe the superconvergence of the second deriva-
tives. The effectivity ratios are close to one. Despite lack of a complete theory, error
estimates L2 and H1 are also quite accurate, and the orders of convergence are optimal
in all three norms (and superconvergent for the recovered second derivatives).
90
Table 4.1. Error estimates for uniform refinement.
nt L2 L2 EF0 H1 H1 EF1 H2 H2 EF28 8.8e-3 1.0e-2 1.1 0.1 0.2 1.6 1.3 2.1 1.7
32 1.0e-3 1.8e-3 1.8 3.0e-2 0.1 1.8 0.7 1.2 2.0128 1.2e-4 2.0e-4 1.6 7.5e-3 1.2e-2 1.6 0.3 0.5 1.7512 1.6e-5 2.4e-5 1.6 1.9e-3 2.9e-3 1.5 0.2 0.2 1.5
2048 1.9e-6 2.7e-6 1.4 4.7e-4 6.5e-4 1.4 0.1 0.1 1.48192 2.4e-7 3.1e-7 1.3 1.2e-4 1.5e-4 1.3 4.2e-2 3.4e-2 1.3
32768 3.0e-8 3.5e-8 1.2 3.0e-5 3.4e-5 1.2 2.1e-2 1.3e-2 1.2131072 3.8e-9 4.1e-9 1.1 7.4e-6 8.0e-6 1.1 1.0e-2 4.7e-3 1.1
order 3.04 3.15 2.02 2.13 1.01 1.43
Table 4.2. Error estimates for adaptive refinement.
nt L2 L2 EF0 H1 H1 EF1 H2 H2 EF28 6.9e-4 3.7e-4 0.5 1.0e-2 5.6e-3 0.6 0.2 0.2 0.5
33 2.5e-4 1.8e-4 0.7 5.1e-3 4.8e-3 0.9 0.1 0.2 1.0137 1.6e-5 2.2e-5 1.4 8.9e-4 1.5e-3 1.6 0.1 0.1 1.7523 1.8e-6 2.2e-6 1.2 1.8e-4 2.6e-4 1.4 2.2e-2 3.1e-2 1.6
2063 2.0e-7 2.0e-7 1.0 3.7e-5 4.4e-5 1.2 1.0e-2 1.0e-2 1.38207 1.8e-8 1.6e-8 0.9 7.9e-6 7.9e-6 1.0 4.7e-3 2.6e-3 1.1
32775 2.2e-9 1.7e-9 0.8 1.9e-6 1.7e-6 0.9 2.3e-3 7.3e-4 1.0131105 2.6e-10 2.0e-10 0.8 4.5e-7 4.1e-7 0.9 1.1e-3 2.1e-4 1.0
order 3.15 3.24 2.12 2.20 1.06 1.83
91
For more numerical experiments on nonlinear problems as well as singular prob-
lems we refer to [9]. For the nonlinear problem in [9], the exact solution is smooth but has
large derivatives. Nonetheless, we observe similar behavior in regard to the derivative
recovery scheme and effectiveness of the a posteriori error estimates. In a problem with
singularities presented in [9], the local error indicator ετ
still provided useful information
and formed a reliable basis for adaptive refinement.
92
Fig. 4.3. Top left: 3 × 3 initial mesh. Top right: uniform refinement with nt = 128.Bottom left: adaptive refinement with nt = 137. Bottom right: adaptive refinementwith nt = 131105. Elements are colored according to size.
93
Chapter 5
Future Work
Our results suggest a number of further research directions. First, we plan to
implement the new finite elements to solve some benchmark problems for three dimen-
sional MHD models. The second future research problem is to develop adaptive methods
for MHD equations. We are interested in generalizing our derivative recovery scheme to
the finite element approximations of MHD equations. Then we can use the recovered
derivatives to define an a posteriori error estimator and combine it with adaptive mesh
refinement techniques. We are also interested in designing robust iterative solvers for
the algebraic systems resulting from finite element discretizations of high order PDEs.
Another difficulty in the numerical approximation of MHD equations is the ap-
pearance of nonlinear terms in the magnetic induction equation. Based on the ideas
presented by Lee and Xu in regard to non-Newtonian fluids [61], we are able to write the
nonlinear term together with the time derivative into a single term as a covariant deriva-
tive. Then the method of characteristics can be applied to design numerical schemes
for solving MHD problems. This is an ongoing research project with Prof. Jinchao Xu,
Prof. Chun Liu, Prof. Ludmil Zikatanov, and Yao Chen.
94
Appendix
Simply Supported Boundary Conditions
The Kirchhoff thin plate model satisfies the biharmonic equation
42u = f in Ω.
For simply supported plates the following boundary conditions also holds
u = 0, nTMn = 0 on ∂Ω, (A.1)
where the bending moment M is defined by
M = (mij
) = ((1− ν)∂iju+ ν∇uδ
ij) = (1− ν)∇2
u+ ν∆uI,
and n = (n1, n2) is the unit outward normal along the boundary ∂Ω.
There is another way to write this boundary condition
u = 0,∂
2u
∂n2 + ν∂
2u
∂t2= 0 ∂Ω, (A.2)
where t = (t1, t2) is the unit tangential vector along the boundary. The equivalence
between (A.1) and (A.2) is verified in the following.
95
By a direct calculation, one could see
nTMn = ν4u+ (1− ν)(∂11un
2
1+ 2∂12un1n2 + ∂22un
2
2),
while the second term of the right hand side is exactly (1− ν)∂2u∂n2 . Hence
nTMn = ν4u+ (1− ν)
∂2u
∂n2 . (A.3)
Next observe
4u =∂
2u
∂n2 +∂
2u
∂t2,
together with (A.3) one immediately get
nTMn = ∆u− (1− ν)
∂2u
∂t2, or, n
TMn =
∂2u
∂n2 + ν∂
2u
∂t2.
Recall the following geometric identities,
∂u
∂t=∂u
∂s
∂2u
∂t2=∂
2u
∂s2+∂ψ
∂s
∂u
∂n
where s is measured along the boundary and ∂ψ/∂s is the curvature of the boundary.
So (A.2) can be written in a slightly different form
u = 0,∂
2u
∂n2 + ν∂ψ
∂s
∂u
∂n= 0 on ∂Ω. (A.4)
96
For polygonal domain, the boundary is straight, the natural boundary condition
in(A.4) reduces to
∂2u
∂n2 = 0,
combined with (A.3) we finally get 4u = 0 on ∂Ω.
97
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Vita
Bin Zheng was born in Huainan, Anhui Province in China on January 19, 1979. In
2000 he received the B.S. degree in computational mathematics, from Peking University.
In 2003 he received the M.S degree in computational mathematics, again from Peking
University. In 2003 he enrolled in the Ph. D. program in applied mathematics at the
Pennsylvania State University. Since 2003 he has been employed in the Mathematics
Department of the Pennsylvania State University as a teaching assistant.
