Analysis and Application of an Orthogonal Nodal Basis on ......the approximation is not continuous across the element boundary. The difference is resolved by solving a local The difference

Appl. Num. Anal. Comp. Math. 2, No. 3, 326 – 345 (2005) / DOI 10.1002/anac.200510007

Analysis and Application of an Orthogonal Nodal Basis onTriangles for Discontinuous Spectral Element Methods

Shaozhong Deng∗ and Wei CaiDepartment of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223,USA

Received 10 January 2005, revised 15 May 2005, accepted 28 August 2005Published online 22 November 2005

Key words Orthogonal nodal basis, spectral methods, discontinuous Galerkin methodsSubject classification 65N30, 65N35, 78M25

In this paper, we propose and analyze an orthogonal non-polynomial nodal basis on triangles for discontinu-ous spectral element methods (DSEMs) for solving Maxwell’s equations. It is based on the standard tensorproduct of the Lagrange interpolation polynomials and a “collapsing” mapping between the standard squareand the standard triangle. The basis produces diagonal mass matrices for the DSEMs and is easy to imple-ment. Numerical results for electromagnetic scattering in heterogeneous media are provided to demonstratethe exponential convergence of the proposed basis, and its application to the simulation of optical coupling bywhispering gallery modes between two microcylinders is presented as well.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

There has been active recent research on the development of discontinuous Galerkin methods (DGMs) for han-dling material interfaces arising from electromagnetic scattering and porous media flows. The main advantagesof the DGMs are their high order accuracy and high suitability for parallel implementation when explicit schemesare utilized. In the DGMs piecewise continuous approximations are used to represent solutions of partial differen-tial equations, while the material interfaces are conformingly approximated in the underlying triangulation of thesolution domains. There are three approaches in implementing the DGMs, namely, the h-version, the p-version,and the h-p version. Similar to finite element methods [1], the h-version allows the mesh size to be decreased toachieve the convergence at a rate of the order of the employed polynomial basis in each element, resulting in afinite order method. The alternative p-version, popular in the area of computational electromagnetics due to highwave numbers possibly involved, allows the order of the polynomial basis to be increased while the elements arekept the same as in the initial triangulation. A hybrid h-p version can also be considered [2]. Since orthogonalpolynomials are often used in the p-version DGM, it is called a discontinuous spectral element method (DSEM)[3], a term which will be used in this paper.

An important issue in the implementation of a DSEM is the choice of the approximation basis functions. Fornumerical stability and accuracy concerns, the elementwise mass matrices arising from the DSEM should bemade as simple as possible. Because the order of the basis functions may have to be taken large to obtain desiredaccuracy, in the framework of spectral element methods, it is critical to have elementwise mass matrices thatare easy to invert and well-conditioned. The ideal case will be that the elementwise mass matrices are diagonal.This has actually been achieved for both quadrilateral and triangular elements. In the former case, the standardtensor product of the Lagrange interpolation polynomials over the classical Gauss points in the interval [-1,1] willyield diagonal mass matrices [3] in the DSEM. While in the later case, the ingenious construction of the Dubinerorthogonal polynomial basis on triangles [4] provides an answer.

In this paper, we propose a non-polynomial basis on triangles which will be orthogonal, like the Dubinerpolynomial basis, and at the same time will be a nodal basis over the appropriately chosen collocation points. This

∗ Corresponding author: e-mail: [email protected], Phone: +01 704 687 6657, Fax: +01 704 687 6415


Appl. Num. Anal. Comp. Math. 2, No. 3 (2005) / www.anacm.org 327

orthogonal nodal basis is again based on the standard tensor product of the Lagrange interpolation polynomialsover the classical Gauss points in the interval [-1,1]. The resulting basis functions on triangles produce diagonalmass matrices for the DSEM, and are easy to implement. Previous nodal basis functions on triangles have beenproposed in [5] by using specially chosen points inside the triangles based on static charge distribution; however,the resulting basis functions are not orthogonal on the triangles.

In the next section, we summarize the DSEM for solving Maxwell’s equations. In Section 3, after reviewingthe orthogonal polynomial nodal basis on rectangles and the Dubiner orthogonal polynomial basis on triangles,we present and analyze the orthogonal non-polynomial nodal basis on triangles. In Section 4, numerical resultsare first provided to demonstrate the exponential convergence of the orthogonal nodal basis for approximatingboth oscillatory functions and the solutions of electromagnetic scattering by single dielectric cylinder, and numer-ical simulation of optical coupling by whispering gallery modes between two microcylinders is then presented aswell.

2 Discontinuous spectral element method for Maxwell’s equations

Without losing any generality, we consider the non-dimensionalized two-dimensional TM Maxwell’s equationson a domain Ω. To approximate the Maxwell’s equations in the time domain, we write them in conservation formas

∂Q∂t

+ ∇ · F = S, (2.1)

where Q = (Ez,Hx,Hy)T with Ez and (Hx,Hy) representing the electric field and the magnetic field, respec-tively, and the flux F = (Fx,Fy) = (AxQ,AyQ) with

Ax =

⎛⎝ 0 0 −1/�0 0 0

−1/µ 0 0

⎞⎠ , Ay =

⎛⎝ 0 1/� 01/µ 0 0

0 0 0

⎞⎠ .

Here � is the material permittivity, and µ is the material permeability.To solve Eq. (2.1) in a general two-dimensional geometry, the physical domain Ω under consideration is

divided into non-overlapping quadrilateral and/or triangular physical elements. Each physical element is thenmapped onto a reference element, either the standard square R0 = [−1, 1]× [−1, 1] or the standard triangle T0 ={(x, y)| 0 ≤ x, y; x + y ≤ 1}, by an isoparametric transformation x = χ(ξ), where x = (x, y) and ξ = (ξ, η) arethe coordinates in the physical element and the reference element, respectively. In cases that curvilinear elementsare involved, the blending function method [1] can be used to construct appropriate transformations.

Under the transformation x = χ(ξ), Eq. (2.1) on each element becomes

∂Q̂∂t

+ ∇ξ · F̂ = Ŝ, (2.2)

where Q̂ = JQ, Ŝ = JS, F̂ = (F̂ξ, F̂η) with F̂ξ = (∂y/∂η,−∂x/∂η) · F, F̂η = (−∂y/∂ξ, ∂x/∂ξ) · F, and Jbeing the Jacobian of the transformation, i.e., J = |∂χ/∂ξ|.

In the DSEM, the solution Q̂ is approximated by a linear combination of the basis functions of the approxima-tion space on each reference element, and the approximation is not required to be continuous across the elementboundary. Let us generally denote the basis functions by β1(ξ, η), β2(ξ, η), · · · , βN (ξ, η), and then approximatethe solution Q̂ element-by-element in terms of the basis functions as

Q̂(ξ, η, t) ≈ Q̂N (ξ, η, t) =N∑

j=1

Q̂j(t)βj(ξ, η), (2.3)

where Q̂j(t) are time-dependent expansion coefficients. The residual of the approximation is then required to beorthogonal to the approximation space locally within each element K, yielding the following equations(

∂Q̂N∂t

, βi

)+

∫∂K

βiF̂ · nds −(F̂ · ∇ξβi

)=

(Ŝ, βi

), i = 1, 2, · · · , N, (2.4)


328 S. Deng and W. Cai: An Orthogonal Nodal Basis on Triangles

where (u, v) =∫

K

uv dξ represents the usual L2 inner product, ∂K the element boundary, and n = (nx, ny)

the outward unit normal to the element boundary.The integrals in Eq. (2.4) are calculated numerically by quadratures, depending on the element type and the

basis functions, and the discretization requires the evaluation of the fluxes along the element boundary. However,the approximation is not continuous across the element boundary. The difference is resolved by solving a localRiemann problem for the numerical normal flux, which is discussed in detail in [6].

Substituting Q̂N in Eq. (2.3) into Eq. (2.4), we obtainN∑

j=1

(mij

dQ̂j(t)dt

+ mξijF̂ξ,j(t) + mηijF̂η,j(t)

)+

∫∂K

βiF̂ · nds =(Ŝ, βi

), i = 1, · · · , N, (2.5)

where F̂j = F̂(Q̂j) = (F̂ξ,j , F̂η,j), and the local mass matrix and the local derivative matrices are

M = (mij), mij =∫

K

βiβjdξ, (2.6)

Mξ = (mξij), mξij =

∫K

∂βi∂ξ

βjdξ, (2.7)

Mη = (mηij), mηij =

∫K

∂βi∂η

βjdξ. (2.8)

Equation (2.5) is a system of ordinary differential equations for the time-dependent expansion coefficientsQ̂j(t), j = 1, 2, · · · , N , which can then be solved by explicit methods.

3 Orthogonal bases on rectangles and triangles

As mentioned earlier, for numerical stability and accuracy concerns when we solve Eq. (2.5), the local massmatrix M should be made as simple as possible. The ideal case will be that the local mass matrix becomesa diagonal matrix, and this can actually be achieved for both quadrilateral and triangular elements by usingappropriate orthogonal bases.

3.1 Orthogonal nodal basis on rectangles

For quadrilateral elements, the approximation space on the standard reference square R0 is normally chosenas PM,M = PM × PM , where PM represents the space of polynomials of degree M or less. Let τi, ωi, i =0, 1, · · · ,M be the classical Gauss points and weights in the interval [-1, 1]. Then an orthogonal basis for PM,Mis the set of standard tensor products of the Lagrange interpolation polynomials on the interval [-1, 1], i.e.,

qmn(ξ, η) = φm(ξ) φn(η), 0 ≤ m,n ≤ M, (3.1)where

φi(x) =M∏

j=0,j �=i

x − τjτi − τj , i = 0, 1, · · · ,M.

Note that qmn(τi, τj) = δmiδnj , 0 ≤ i, j, m, n ≤ M. Therefore, the basis is also a nodal basis over thecollocation points (τi, τj) ∈ R0, 0 ≤ i, j ≤ M, i.e., for any function f(ξ, η), it can be approximated by

f(ξ, η) ≈M∑

m,n=0

fmnqmn(ξ, η),

where fmn=f(τm, τn) are the point values of f(ξ, η) at the collocation points. And for the same reason, we have

(qmn, qij) =∫∫

R0

qmn(ξ, η)qij(ξ, η)dξdη = ωmωnδmiδnj . (3.2)

Also note that the above choice of the classical Gauss collocation points requires interpolation of the expansionto the boundary to evaluate the boundary flux terms in the DSEM.



3.2 Orthogonal polynomial basis on triangles

For triangular elements, the approximation space on the standard reference triangle T0 is frequently chosen as

PM (T0) = span {ξmηn, 0 ≤ m,n, m + n ≤ M} . (3.3)

A natural basis for this space is {ξmηn, 0 ≤ m, n,m + n ≤ M} . However, this basis is not orthogonal andworks fine only for small expansion order M , about 2 or 3. When M is taken large, say M ≥ 7, the basis isnearly dependent and leads to ill-conditioned approximations.

Investigations of orthogonal polynomials on triangles have been done for long time. In particular, spectralmethods on triangles have been studied in [4][7]-[13], where two different approaches have been developed.One approach is to use transformations between triangles and squares and warped tensor product grids withintriangles, designed for the accurate approximation of integrals [4][7]-[9], while the other approach is to usecritically sampled points in triangles designed for accurate approximation rather than for accurate integration,such as the Fekete points [10]-[13]. However, for triangular spectral element methods, probably the most popularbasis is the Dubiner orthogonal polynomial basis discussed in [4]. This basis is briefly summarized in this section,and for more details the readers may consult [4] and [7].

The Dubiner basis on triangles is obtained by transforming the Jacobi polynomials defined on intervals toform polynomials on triangles. The n-th order Jacobi polynomials Pα,βn (x) on the interval [−1, 1] are orthogonalpolynomials under the Jacobi weight w(x) = (1 − x)α (1 + x)β , i.e.,

∫ 1−1

(1 − x)α (1 + x)β Pα,βl (x)Pα,βm (x) dx = δlm.

To construct an orthogonal polynomial basis on the triangle T0, we follow the same idea as in [7] and considerthe transformations in Fig. 1 between the reference square R0 and the reference triangle T0 defined by

⎧⎪⎨⎪⎩

ξ =(1 + a) (1 − b)

4,

η =1 + b

2,

or

⎧⎪⎨⎪⎩

a =2ξ

1 − η − 1,

b = 2η − 1.(3.4)

0T

1

1

η

ξ

A

B

C

b

−1

1

−1 1

a

Fig. 1 Illustration of the mapping between the square R0 and the triangle T0.

The transformations in Eq. (3.4) basically collapse the top edge of the square R0 into the top vertex (0,1) ofthe triangle T0. The Jacobians of the transformations are

J (ξ, η) =∂ (a, b)∂ (ξ, η)

=4

1 − η , (3.5)

J−1 (a, b) =∂ (ξ, η)∂ (a, b)

=1 − b

8=

1 − η4

. (3.6)

The Dubiner basis is then constructed by a generalized tensor product (warped product [7]) of the Jacobipolynomials on the interval [-1, 1] to form a basis on the square R0, which is then transformed by the above



“collapsing” mapping to a basis on the triangle T0, i.e., the Dubiner basis on the triangle T0 is defined as

gmn (ξ, η) = P 0,0m (a) (1 − b)m P 2m+1,0n (b) (3.7)

= 2mP 0,0m

(2ξ

1 − η − 1)

(1 − η)m P 2m+1,0n (2η − 1) , (3.8)0 ≤ m, n,m + n ≤ M.

Note that the Dubiner basis functions are polynomials in (a, b) space as well as in (ξ, η) space. For example, thefirst six un-normalized Dubiner basis functions on the triangle T0 are

g00(ξ, η) = 1,

g10(ξ, η) = 4ξ + 2η − 2,g01(ξ, η) = 3η − 1,g20(ξ, η) = 24ξ2 + 24ξη + 4η2 − 24ξ − 8η + 4,g11(ξ, η) = 20ξη + 10η2 − 4ξ − 12η + 2,g02(ξ, η) = 10η2 − 8η + 1.

Moreover, the Dubiner basis is orthogonal in the Legendre inner product defined by

(gmn, gij) =∫∫

T0

gmn(ξ, η)gij(ξ, η)dξdη =18δmiδnj ,

and is complete in the polynomial space PM (T0) defined in Eq. (3.3) [7].However, the Dubiner orthogonal polynomial basis on the triangle T0 is not a nodal basis, which makes its

implementation much more inconvenient than that of a nodal basis. Previous nodal basis functions on triangleshave been proposed by using specially chosen points inside the triangles based on static charge distribution [5],but the resulting polynomial basis functions are not orthogonal on the triangles. Additionally, for the Dubinerbasis a warped tensor product grid has to be employed for the accurate approximation of integrals, but that gridis over-sampled since it requires twice as many grid points as there are degrees of freedom in the polynomialexpansion (i.e., the number of polynomial basis functions).

3.3 Orthogonal nodal basis on triangles

Similar to the Dubiner orthogonal polynomial basis on the triangle T0, the orthogonal non-polynomial nodal basisis constructed by the standard tensor product of the Lagrange interpolation polynomials on the interval [-1, 1] toform a basis on the square R0, which is then transformed by the same “collapsing” mapping (3.4) to a basis onthe triangle T0, i.e.,

ψmn (ξ, η) = qmn(a, b) = φm(a)φn(b) (3.9)

= φm

(2ξ

1 − η − 1)

φn(2η − 1), 0 ≤ m, n ≤ M. (3.10)

Clearly, this basis is a nodal basis over the collocation points on the triangle T0

(ξi, ηj) =(

(1 + τi) (1 − τj)4

,1 + τj

2

), 0 ≤ i, j ≤ M, (3.11)

namely, ψmn(ξi, ηj) = φm(τi)φn(τj) = δmiδnj , 0 ≤ i, j ≤ M, and thereby for any function f(ξ, η) defined onthe triangle T0, it can be approximated by

f(ξ, η) ≈M∑

m,n=0

fmnψmn(ξ, η), (3.12)

where fmn=f(ξm, ηn) are the point values of f(ξ, η) at the collocation points given by Eq. (3.11). And again,the interpolation of the expansion is required to evaluate the boundary flux terms since all collocation points areinterior.



Remark 3.1 Unlike the Dubiner basis defined by Eqs. (3.7) and (3.8), the nodal basis defined by Eqs. (3.9)and (3.10) is no longer a polynomial basis in (ξ, η) space even though the basis functions are still polynomialsin (a, b) space. Moreover, it has roughly twice as many basis functions as the Dubiner basis does for the sameexpansion order M . Although it is clear that the computational cost to evaluate a double integral in Eqs. (2.5)-(2.8) by Gaussian quadrature under the non-polynomial nodal basis is only O(1), one or two orders less thanthe O(M) or O(M2) computational cost of the polynomial basis, most integrals if not all can be computed inthe preprocessing phase. And for these reasons, we are tempted to conclude that in terms of computational cost(CPU time) per time step, the Dubiner polynomial basis should be more efficient than the non-polynomial nodalbasis because the latter requires almost twice the number of degrees of freedom as compared to the polynomialbasis. Our numerical experiments in Section 4, however, indicate that for some applications the non-polynomialnodal basis is more efficient than the polynomial basis in terms of CPU time needed per time step.

Remark 3.2 The advantage of a polynomial basis is that the basis functions are easy to differentiate andthereby the derivative matrices (2.7) and (2.8) are easy to compute. The nodal basis is not polynomial in (ξ, η)space, but by simply applying the chain rule, we can differentiate the nodal basis functions by

∂

∂ξψmn(ξ, η) =

21 − ηφ

′m

(2ξ

1 − η − 1)

φn (2η − 1) ,∂

∂ηψmn(ξ, η) =

2ξ(1 − η)2 φ

′m

(2ξ

1 − η − 1)

φn (2η − 1) + 2φm(

2ξ1 − η − 1

)φ′n (2η − 1) ,

where φ′m and φ′n represent the derivatives of the Lagrange interpolation polynomials φi(x), i = 0, 1, · · · ,M

with respect to x, which can be evaluated in many different ways.

Remark 3.3 The nodal basis functions defined by Eq. (3.10), though unconventional compared with tradi-tional polynomial basis functions, appear to but actually do not have singularity at the corner (ξ, η) = (0, 1) ∈ T0since on the triangle T0 we always have ξ ≤ 1 − η. Furthermore, when applied in a DSEM where all grid pointsare interior [3], the basis (3.10) need not be defined at the corner point at all. On the other hand, the nodal basisis not smooth at the corner (ξ, η) = (0, 1), but that does not prevent it from accurately approximating smoothfunctions. And as pointed out in [4], any spectral element approximation is globally unsmooth. Figure 2 showsthe contour shapes of all triangular nodal basis functions and the corresponding collocation points used in theright triangle T0 for the expansion order M = 2. And the plot clearly suggests that the non-polynomial basisconcentrates a lot of resolution in a single corner of the triangle.

As observed earlier, the nodal basis on the triangle T0 is not a polynomial basis in (ξ, η) space, but it stillmaintains orthogonality. In addition, for the same expansion order M the approximation space characterized bythe nodal basis can be proved without great difficulty to contain the polynomial space PM (T0). In fact, we havethe following theorems.

Theorem 3.4 The nodal basis on the triangle T0 is orthogonal in the Legendre inner product defined by

(ψmn, ψij) =∫∫

T0

ψmn(ξ, η)ψij(ξ, η)dξdη =ωmωn(1 − τn)

8δmiδnj . (3.13)

P r o o f. Note that for any function f(ξ, η) ∈ C(T0), by performing the coordinate transformation (3.4) from(ξ, η) to (a, b), we have∫∫

T0

f(ξ, η)dξdη =18

∫∫R0

f(a, b)(1 − b)dadb.

Therefore, the inner product in Eq. (3.13) becomes

(ψmn, ψij) =∫∫

T0

ψmn(ξ, η)ψij(ξ, η)dξdη

=18

∫∫R0

φm(a)φn(b)φi(a)φj(b)(1 − b)dadb

=18

(∫ 1−1

φm(a)φi(a)da)(∫ 1

−1φn(b)φj(b)(1 − b)db

). (3.14)



0

0.5

1

1.5

2ψ

11

o

o

o

o

o

o

o

o

o

AB

C

−0.5

0

0.5

1ψ12

o

o

o

o

o

o

o

o

o

AB

C

0

0.5

1

1.5

2ψ

13

o

o

o

o

o

o

o

o

o

AB

C

−0.5

0

0.5

1ψ21

o

o

o

o

o

o

o

o

o

AB

C

−0.5

0

0.5ψ

22

o

o

o

o

o

o

o

o

o

AB

C

−0.5

0

0.5

1ψ23

o

o

o

o

o

o

o

o

o

AB

C

0

0.5

1

1.5

2ψ

31

o

o

o

o

o

o

o

o

o

AB

C

−0.5

0

0.5

1ψ32

o

o

o

o

o

o

o

o

o

AB

C

0

0.5

1

1.5

2ψ

33

o

o

o

o

o

o

o

o

o

AB

C

Fig. 2 Contour shapes of all orthogonal nodal basis functions on the triangle T0 for the expansion orderM = 2, where ‘o’ indicates a collocation point defined in Eq. (3.11). Note that each basis function has a unitat one nodal point and varies to zero at all other nodal points.

Now from the property of the Gaussian quadrature that the (M + 1)-point Gaussian quadrature is exact forpolynomials of degree up to 2M + 1, we see that the first integral in Eq. (3.14) becomes

∫ 1−1

φm(a)φi(a)da =M∑

s=0

ωsφm(τs)φi(τs) = ωmδmi,

and similarly the second integral in Eq. (3.14) becomes

∫ 1−1

φn(b)φj(b)(1 − b)db =M∑

s=0

ωsφn(τs)φj(τs)(1 − τs) = ωn(1 − τn)δnj .

Therefore, we finally have

(ψmn, ψij) =ωmωn(1 − τn)

8δmiδnj .

An additional point that can be appreciated from the above proof is that integrals involving the inner productof ψmn(ξ, η) with a function f(ξ, η) can be evaluated in O(1) operations by using the Gaussian quadrature, i.e.,

(ψmn, f) ≈ ωmωn(1 − τn)8 fmn, (3.15)

where fmn are again the point values of f(ξ, η) at the collocation points defined by Eq. (3.11).



We define GM (T0) as the finite element space on the triangle T0 spanned by the nodal basis, i.e.,GM (T0) = span{ψmn(ξ, η), 0 ≤ m,n ≤ M}. (3.16)

Then clearly dim(GM (T0)) = (M + 1)2 as the basis is truly independent. Moreover, this space contains thepolynomial space PM (T0).

Theorem 3.5 The finite element space GM (T0) contains the polynomial space of degree M on the triangle T0,namely, PM (T0) ⊂ GM (T0).

P r o o f. To demonstrate this result we consider the following functions in (a, b) space

f(a, b) =(

(1 + a) (1 − b)4

)i (1 + b2

)j, 0 ≤ i, j, i + j ≤ M.

Note that every function f(a, b) is a polynomial in (a, b) with degrees i in a and i + j in b. Since both i ≤ Mand i + j ≤ M , the Lagrange interpolation polynomial for f(a, b) is thus exact, i.e., we have the followingpolynomial identity in (a, b) space

f(a, b) =M∑

m,n=0

f(τm, τn)φm(a)φn(b), (a, b) ∈ R0. (3.17)

Using the definitions (3.4) and (3.9), we then have the corresponding identity in (ξ, η) space

ξiηj =M∑

m,n=0

ξimηjnψmn(ξ, η), (ξ, η) ∈ T0,

which implies that PM (T0) ⊂ GM (T0).As a matter of fact, it can be shown that the first three finite element spaces GM (T0) are

G1(T0) = span{1},

G2(T0) = span{

1, ξ, η,ξ

1 − η}

,

G3(T0) = span{

1, ξ, η, ξ2, ξη, η2,ξ

1 − η ,ξ2

1 − η ,ξ2

(1 − η)2}

.

And in general, we have the following theorem.

Theorem 3.6 The finite element space GM (T0) equals span{SM}, where the set SM is defined as

SM ={

ξiηj , 0 ≤ i, j, i + j ≤ M ; ξt

(1 − η)s ,1 ≤ s ≤ M , s ≤ t ≤ M}

.

P r o o f. We have already shown above that ξiηj ∈ GM (T0) for 0 ≤ i, j, i + j ≤ M . For ξt/(1 − η)s, where1 ≤ s ≤ M, s ≤ t ≤ M , we see

ξt

(1 − η)s =12s

(1 + a)s(

(1 + a)(1 − b)4

)t−s=

122t−s

(1 + a)t(1 − b)t−s.

Note that (1 + a)t(1 − b)t−s/22t−s is a polynomial in (a, b) with degrees t in a and t − s in b. Again since botht ≤ M and t − s ≤ M , in (a, b) space the polynomial can be exactly represented by its Lagrange interpolatingapproximation as Eq. (3.17). After transforming coordinate variables from (a, b) to (ξ, η), we obtain

ξt

(1 − η)s =M∑

m,n=0

ξtm(1 − ηn)s ψmn(ξ, η),



which implies that ξt/(1 − η)s ∈ GM (T0). Therefore, we have span{SM} ⊂ GM (T0).On the other hand, it is easy to check the set SM is linear independent. Recalling that dim(GM (T0)) =

(M + 1)2, we must have GM (T0) = span{SM}, i.e.,

GM (T0) = span{

ξiηj , 0 ≤ i, j, i + j ≤ M ; ξt

(1 − η)s ,1 ≤ s ≤ M , s ≤ t ≤ M}

.

However, we should point out that the finite element space GM (T0) is not invariant under rotations and re-flections of the triangle because the space contains more resolution in one corner of the triangle than in theother two corners, but the polynomial spaces generated by other choices of basis functions including the Dubinerpolynomials are invariant under the same operations of the triangle.

Remark 3.7 Depending on the Jacobians of the transformations (3.4), one may choose to directly approximateQ rather than Q̂. Specifically, we have observed that approximating Q could result in better accuracy when theJacobian J of the transformation is degenerate. In this case, similar to Eq. (2.3), Q can be approximated by

Q(ξ, η, t) ≈ QN (ξ, η, t) =N∑

j=1

Qj(t)βj(ξ, η), (3.18)

where Qj(t) are again time-dependent expansion coefficients. Then we can approximate Q̂ by

Q̂(ξ, η, t) = JQ(ξ, η, t) ≈ JQN (ξ, η, t) = JN∑

j=1

Qj(t)βj(ξ, η). (3.19)

Substituting the above approximation for Q̂ into Eq. (2.4), we can write

N∑j=1

(m̄ij

dQj(t)dt

+ m̄ξijFξ,j(t) + m̄ηijFη,j(t)

)+

∫∂K

βiF ·nds = (JS, βi) , i = 1, · · · , N, (3.20)

where Fj = F(Qj) = (Fξ,j ,Fη,j), and the local mass matrix and the local derivative matrices become

M̄ = (m̄ij), m̄ij =∫

K

Jβiβjdξ, (3.21)

M̄ξ = (m̄ξij), m̄ξij =

∫K

J∂βi∂ξ

βjdξ, (3.22)

M̄η = (m̄ηij), m̄ηij =

∫K

J∂βi∂η

βjdξ. (3.23)

The disadvantage of this approximation scheme is that different elements will have different mass matrices.But more importantly, the mass matrices might not be diagonal even though the basis {βi} is orthogonal. Inthis case, inverses of the mass matrices have to be calculated, which might be ill-conditioned for high orderpolynomial basis functions. However, for the non-polynomial nodal basis, if the integrals are approximated byquadrature, every local mass matrix is still numerically diagonal as∫∫

T0

J(ξ, η)ψmn(ξ, η)ψij(ξ, η)dξdη =18

∫∫R0

J(a, b)φm(a)φn(b)φi(a)φj(b)(1 − b)dadb

≈ 18

M∑s,t=0

ωsωtJ(τs, τt)φm(τs)φn(τt)φi(τs)φj(τt)(1 − τt)

=18δmiδnjωmωn(1 − τn)J(τm, τn).



3.4 Spectral derivative matrices

When we solve Eq. (2.5) or Eq. (3.20) by explicit methods, the maximum time step size allowed for numericalstability is always a major concern which, generally speaking, shall depend on the eigenvalues of the derivativematrices as the results of the Galerkin approximation to the derivative operators ∂/∂ξ and ∂/∂η. For rectangles,the orthogonal nodal basis (3.1) will produce the standard derivative matrices as in the Galerkin Legendre spectralmethod [15]. Therefore, the eigenvalues of the derivative matrices Mξ and Mη are well-known which grow at arate of O(M2) [15]. For triangles, the derivative matrix Mξ produced by the orthogonal nodal basis (3.10) can beeasily shown to share in principle the same property as that produced by the orthogonal nodal basis on rectangles.To do so, let us denote βi = ψmn(ξ, η) and βj = ψst(ξ, η) be two nodal basis functions on the triangle T0. Thenthe derivative matrix Mξ in Eq. (2.7) becomes

mξij =∫∫

T0

∂ψmn(ξ, η)∂ξ

ψst(ξ, η)dξdη

=∫∫

R0

(∂qmn(a, b)

∂a

∂a

∂ξ+

∂qmn(a, b)∂b

∂b

∂ξ

)qst(a, b)

(1 − b)8

dadb

=12

∫∫R0

∂qmn(a, b)∂a

qst(a, b)dadb,

which turns out to be exactly the same as the corresponding derivative matrix for the orthogonal nodal basis onthe square R0 except for a factor of 1/2. On the other hand, the derivative matrix Mη in Eq. (2.8) can be writtenas

mηij =∫∫

T0

∂ψmn(ξ, η)∂η

ψst(ξ, η)dξdη

=∫∫

R0

(∂qmn(a, b)

∂a

∂a

∂η+

∂qmn(a, b)∂b

∂b

∂η

)qst(a, b)

(1 − b)8

dadb

= −14

∫∫R0

(1 + a)∂qmn(a, b)

∂aqst(a, b)dadb +

14

∫∫R0

(1 − b)∂qmn(a, b)∂b

qst(a, b)dadb,

whose spectral property is therefore not clear and will be a subject of further research.If we choose to directly approximate Q rather than Q̂, then the property of the consequent derivative matrices

M̄ξ and M̄η will depend not only on the choice of the basis but also the underlying geometry of the problembeing solved.

Nevertheless, it has been heuristically claimed in [4] and partially analyzed in [8] that, for an orthogonal basison a triangle constructed by a direct product like (3.9), the explicit time step size has a O(1/M4) limitation dueto the concentration of a lot of resolution in a single corner of the triangle, while for the Dubiner orthogonalpolynomial basis the explicit time step size has only a O(1/M2) limitation. To demonstrate such scaling, weconsider the spectrum of the inverse of the mass matrix M times the derivative matrix Mξ or Mη. And our nu-merical analysis seems to show that the largest eigenvalue of the inverse of the mass matrix times each derivativematrix does scale as O(M4) if using the non-polynomial nodal basis but as only O(M2) if using the Dubinerpolynomial basis, which is in support of the above heuristic argument. However, at this point it appears to beimpossible to have a conclusive mathematical analysis of the spectrum of the inverse of the mass matrix timesthe derivative matrix and thus the time step restriction of the non-polynomial nodal basis, which therefore shallconstitute a separate topic of future research.

4 Numerical results

In this section, we shall give some illustrative examples which demonstrate the exponential convergence propertyof the orthogonal nodal basis on triangles. The application of the basis to the numerical simulation of opticalcoupling by whispering gallery modes between microcylinders shall be presented as well.



4.1 Approximation properties of the orthogonal nodal basis on triangles

Because the approximation space spanned by the orthogonal non-polynomial nodal basis GM (T0) contains thecomplete polynomial space PM (T0), we expect the standard spectral accuracy of spectral methods [14, 15] overtriangles. To show such an approximation property, the first example we consider is the following oscillatorywave function on the square R1 = [0, 1] × [0, 1]

u(x, y) =12

(sin

(2πk

(x +

715

))+ cos

(2πk

(y +

215

))), (4.1)

and in Fig. 3 we display the contour plot of this function with the wave number k = 5.

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

AB

C D

Fig. 3 The contour plot of the wave function (4.1) with the wave number k = 5.

To demonstrate the exponential convergence of the orthogonal nodal basis on triangles and to compare thebehaviors of the nodal basis and the Dubiner basis, we first decompose the square R1 into two triangular elementsas shown in Fig. 3, and then approximate the oscillatory function u(x, y) with the wave number k = 2 byinterpolation in terms of the nodal basis and the Dubiner basis with different expansion orders, respectively.When expanding the function in terms of the Dubiner polynomials, we evaluate it at the same collocation pointsas defined in (3.11), and then solve a linear least square problem to find expansion coefficients. As shown inFig. 4 are the L∞ approximation errors plotted with respect to the expansion order. The exponential convergenceof both bases are clearly observed as indicated by the asymptotic linear behavior of the curves on this lin-log plot.In addition, the results also suggest that the nodal basis appears to be more accurate than the Dubiner basis, whichactually coincides with the fact that the finite element space spanned by the Dubiner basis PM (T0) is contained inthat spanned by the nodal basis GM (T0). On the other hand, however, since most of the extra resolution comparedwith the Dubiner basis is isolated to a single corner of each triangle, the orthogonal non-polynomial nodal basisonly gives a small decrease in the error of the Dubiner polynomial basis.

Next, we shall study the number of points per wavelength required to approximate the oscillatory functionu(x, y) to certain degree of accuracy by the orthogonal non-polynomial nodal basis. For this purpose, we definethe number of points per wavelength used Nppw as

Nppw =√

2M + 1

k, (4.2)

where the factor√

2 reflects that the number of total collocation points (and thus the number of total nodal basisfunctions) is 2(M + 1)2 =

(√2(M + 1)

)2. We investigate two cases, requiring the L∞ approximation errors

being less than 2% and 0.02%, respectively, and the results are shown in Fig. 5. One can note in particular thatabout 5 to 6 points per wavelength are sufficient to approximate the oscillatory wave with high wave numbers todegree of accuracy of 1%.



8 10 12 14 16 18 20

10−8

10−6

10−4

10−2

100

Expansion Order

Err

or

Orthogonal Nodal BasisDubiner Basis

Fig. 4 Exponential convergence of the nodal ba-sis and the Dubiner basis on triangles in the L∞norm for the approximation of the wave func-tion (4.1) with the wave number k = 2.

0 2 4 6 8 10 124

5

6

7

8

9

10

11

12

Number of Waves

Num

ber

of P

oint

s pe

r W

avel

engt

h

Error: < 2%Error: < 0.02%

Fig. 5 Average number of points per wavelengthneeded to approximate the wave function (4.1)when using the orthogonal nodal basis on trian-gles.

4.2 Scattering by single dielectric cylinder

In this test, we shall consider a typical electromagnetic scattering problem as illustrated in Fig. 6 (a), i.e., scatter-ing by a dielectric cylinder in free space with a TM wave excitation. Maxwell’s equations (2.1), together with theboundary condition that at the cylindrical boundary the tangential components of the fields should be continuous,can be used to solve the problem.

(ε , µ )2

(ε , µ )1 1

2

y

x

a) b)

Fig. 6 a) Illustration of scattering by single dielectric cylinder; b) The computational mesh used for simulatingscattering by single dielectric cylinder with the exact boundary condition.

We assume that the cylinder is illuminated by a time-harmonic incident plane wave of the form

Eincz = e−i(k1x−ωt), H incy = −e−i(k1x−ωt),

where k1 = ω√

µ1�1 is the propagation constant for homogeneous, isotropic free space medium. In this case, theexact solution of the problem is given in [16] and is also available in [17].

We would like to verify the exponential convergence of the nodal basis for solving the above scatteringproblem. To this end, the cylinder radius is set as a = 0.6, and the computational domain is chosen asΩ = [−1, 1] × [−1, 1], which is subdivided into quadrilateral and triangular elements as shown in Fig. 6 (b).We should point out that although for this sample scattering problem one doesn’t have to use triangular elements,we employ them so that we can develop more general codes for simulating scattering of not only one cylinder,multiple cylinders of the same size, but also multiple cylinders of different sizes. Regarding the boundary condi-tion at the artificial boundary of the computational domain, we simply use the boundary values obtained by theexact solution so that we can measure the approximation errors and thus investigate the convergence property ofthe nodal basis. In practical simulations when boundary values are not available, absorbing boundary conditionssuch as PML boundary conditions should be used [18]-[20].



As described in Section 2, each physical element in the subdivided domain shall be mapped individually ontothe reference square R0 or the reference triangle T0 by an isoparametric transformation, which in general canbe arrived by the blending function method [1]. Next we only give the transformation required to map thosecurvilinear triangles shown in Fig. 6 (b) onto the reference triangle T0. For more detailed description of theblending function method as well as more examples the readers may consult [1].

C (x3,y

3) A (x1,y1)

B (x2,y

2)

A (0,0) B (1,0)

C (0,1)

ξ

η

x=χ(ξ)

y=π(ξ)

Fig. 7 Transformation between the reference triangle T0 and a curvilinear triangle ∆ABC.

Each triangular element in Fig. 6 (b) has one curved boundary so we need a transformation that can mapthe reference triangle T0 onto a triangle with one curved boundary. Suppose that the curved boundary AB isparametrizable as shown in Fig. 7 by (χ(ξ), π(ξ)) such that χ(0) = x1, χ(1) = x2, π(0) = y1 and π(1) = y2.Then the transformation can be written as

x = (1 − ξ − η)x1 + ξx2 + ηx3 + (χ(ξ) − (1 − ξ)x1 − ξx2) 1 − ξ − η1 − ξ ,

y = (1 − ξ − η) y1 + ξy2 + ηy3 + (π(ξ) − (1 − ξ)y1 − ξy2) 1 − ξ − η1 − ξ .

We consider a situation in which �1 = µ1 = 1, i.e., the material exterior to the cylinder is assumed to bevacuum. The permittivity and the permeability of the cylinder are set as �2 = 9 and µ2 = 1, respectively, andthe angular frequency ω = 2π. Figure 8 shows the contours of the three computed field components at the timet = 1 by using the orthogonal nodal basis with M = 12, as well as the global L∞ errors for the three componentsfor several expansion orders. Once again we include corresponding convergence analysis results for the Dubinerbasis. All results clearly illustrate the spectral convergence of the two bases, and not surprisingly again that thenodal basis is a little more accurate than the Dubiner basis when high order expansions are employed.

4.3 Optical coupling by whispering gallery modes between microcylinders

Finally, we shall apply the DSEM with the orthogonal nodal bases to study optical coupling by whisperinggallery modes between two microcylinders, the building blocks of novel photonic waveguides: Coupled Res-onator Optical Waveguide (CROW) [21]. Such waveguides have applications in optical buffering and delay linesin controlling the speed of light propagations, and therefore constitute topics of great interest to the internationalphotonic community. While it is theoretically well-known that in CROW devices waveguiding is provided bythe weak coupling of evanescent whispering gallery modes in the individual microresonators, in this example weshall numerically demonstrate that the successful optical coupling between microcylinders can be achieved.

Whispering gallery modes (WGMs) are electromagnetic resonances traveling in dielectric media of circularsymmetric structures such as microcylinders, microdisks and microspheres. In the case of a dielectric cylinder,the WGMs were first studied by Lord Rayleigh trying to understand the acoustic waves clinging to the dome ofSt. Paul’s Cathedral [22]. In this paper, we shall consider electromagnetic WGMs of a circular dielectric cylinderof radius a and infinite length with dielectric constant �1 and magnetic permeability µ1, which is embedded in aninfinite homogeneous medium of material parameters �2 and µ2. With respect to a cylindrical coordinate system(r, θ, z), for a time factor exp(−iωt), the components of the magnetic field H = (Hr,Hθ,Hz) and the electric



−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

Hx(x,y,1.0)

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

a)2 3 4 5 6 7 8 9 10

10−5

10−4

10−3

10−2

10−1

100

Expansion Order

Err

or: L

∞(H

x)


b)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

Hy(x,y,1.0)

−4

−3

−2

−1

0

1

2

3

4

c)2 3 4 5 6 7 8 9 10

10−5

10−4

10−3

10−2

10−1

100

Expansion Order

Err

or: L

∞(H

y)


d)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

Ez(x,y,1.0)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

e)2 3 4 5 6 7 8 9 10

10−5

10−4

10−3

10−2

10−1

100

Expansion Order

Err

or: L

∞(E

z)


f)

Fig. 8 Scattering by single dielectric cylinder with material parameters µ1 = µ2 = �1 = 1 and �2 = 9. Onthe left are contours of the computed field components at the time t = 1 by using the orthogonal nodal basiswith M = 12, and on the right are the exponential convergence analysis results for both the nodal basis and theDubiner basis. a) Ez(x, y, 1); b) L∞(Ez); c) Hx(x, y, 1); d) L∞(Hx); e) Hy(x, y, 1); and f) L∞(Hy).

field E = (Er, Eθ, Ez) of the WGMs are given by the following equations [23, 24]

Hr =(

annk2

µωλ2rGn(λr) + bn

ih

λG′n(λr)

)Fn,

Hθ =(

anik2

µωλG′n(λr) − bn

nh

λ2rGn(λr)

)Fn, (4.3)

Hz = bnGn(λr)Fn,

Er =(

anih

λG′n(λr) − bn

µωn

λ2rGn(λr)

)Fn,

Eθ = −(

annh

λ2rGn(λr) + bn

iµω

λG′n(λr)

)Fn, (4.4)

Ez = anGn(λr)Fn,



where Fn = exp(inθ + ihz− iωt) with h being the axial propagation constant. The function Gn ≡ Jn for r < aand H(1)n for r > a, where Jn is the Bessel function of the first kind and H

(1)n is the Hankel function of the first

kind. Prime denotes differentiation with respect to the argument λr. Also, for r < a, k = k1 = ω√

�1µ1, λ = λ1where λ21 = k

21 − h2, and for r > a, k = k2 = ω

√�2µ2, λ = λ2 where λ22 = k

22 − h2.

The coefficients an and bn are determined by the boundary condition that, at the cylindrical boundary r = a,the tangential components of the fields are continuous. For a nontrivial solution, the axial propagation constant hshall satisfy the following eigenvalue equation [23, 24](

µ1u

J ′n(u)Jn(u)

− µ2v

H(1)′n (v)

H(1)n (v)

)(k21µ1u

J ′n(u)Jn(u)

− k22

µ2v

H(1)′n (v)

H(1)n (v)

)= n2h2

(1v2

− 1u2

)2, (4.5)

where u = λ1a and v = λ2a. For a given mode number n, Eq. (4.5) does not have a unique solution and theelectromagnetic WGMs are represented by solutions of Eq. (4.5) when n is of the order of λ1a. Note that themode number n is also the number of maxima in the field intensity in the azimuthal direction and is thus calledthe azimuthal number of the WGMs.

In order to investigate optical coupling by WGMs between microcylinders, we shall turn to Maxwell’s equa-tions. For a WGM with the axial propagation constant h, the magnetic field H = (Hx,Hy,Hz) and the electricfield E = (Ex, Ey, Ez) in a rectangular coordinate system (x, y, z) may be expressed as

H(x, y, z, t) = H(x, y, t)eihz, E(x, y, z, t) = E(x, y, t)eihz.

Substituting them into the three-dimensional Maxwell’s equations

µ∂H∂t

= −∇× E, � ∂E∂t

= ∇× H,

we obtain the following hyperbolic system of equations in matrix form

∂Q∂t

+ A(�, µ)∂Q∂x

+ B(�, µ)∂Q∂y

= S, (4.6)

where

Q =(

µH�E

), S =

⎛⎜⎜⎜⎜⎜⎜⎝

ihEy−ihEx

0−ihHyihHx

0

⎞⎟⎟⎟⎟⎟⎟⎠

,

and

A(�, µ) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 00 0 0 0 0 − 1�0 0 0 0 1� 00 0 0 0 0 00 0 1µ 0 0 00 − 1µ 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, B(�, µ) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 1�0 0 0 0 0 00 0 0 − 1� 0 00 0 − 1µ 0 0 00 0 0 0 0 01µ 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The model considered here is a system of two identical circular dielectric cylinders of infinite length in contact.The radiuses of the cylinders are r1 = r2 = 0.5775, and the material parameters are �1 = 10.24 and µ1 = 1while the external medium is vacuum. Then it can be shown that WGMs exist in such cylinders. In fact, bysetting the angular frequency ω = 2π, and the azimuthal number n = 8, we find that the eigenvalue equation(4.5) has a solution h = 6.80842739 between k1 = 6.4π and k2 = 2π, and the corresponding lossless WGM isdenoted by WGM8,1,0.



We will investigate the optical energy transport by WGMs from one cylinder to the other. To this end, in oursimulation we assume that initially there exists a WGM in the left cylinder and no fields exist inside the rightcylinder. More specifically, the exact values of WGM8,1,0 in the left cylinder are taken as the initial conditionsin the entire computational domain except for the inside of the right cylinder, where a zero field is initialized. Inaddition, to assure that the initial field satisfies the interface condition on the surface of the right cylinder, in ournumerical test we also assume that there exist surface currents over the surface of the right cylinder of the form

Jm(x, t) = J0m(x)e−αt, Je(x, t) = J0e(x)e

−αt, (4.7)

where the constant α > 0 is chosen so that the surface currents become negligible after a short period of time,and J0m and J

0e are calculated from the initial fields E(x, 0) and H(x, 0) as

J0m(x) = n ×(E+(x, 0) − E−(x, 0)) , J0e(x) = n × (H+(x, 0) − H−(x, 0)) .

For such boundary currents, the numerical normal flux in the DSEM can be written as [25]

(F · n)− =

⎛⎜⎝ n ×

(Y E−n×H)−+(Y E+n×H)+−JeY −+Y + − Y

+

Y −+Y + Jm

−n × (ZH+n×E)−+(ZH−n×E)++JmZ−+Z+ + Z+

Z−+Z+ Je

⎞⎟⎠ ,

for the − side of the surface and

(F · n)+ =

⎛⎜⎝ n ×

(Y E−n×H)−+(Y E+n×H)+−JeY −+Y + +

Y −Y −+Y + Jm

−n × (ZH+n×E)−+(ZH−n×E)++JmZ−+Z+ − Z−

Z−+Z+ Je

⎞⎟⎠ ,

for the + side of the surface. Here Z± and Y ± are the local impedance and admittance, respectively, and aredefined as Z± = 1/Y ± = (µ±/�±)1/2.

Regarding the boundary condition on the boundary of the computational domain, since most of the electromag-netic energy of a WGM is confined inside the cylinder and fields decay fast away from the cylindrical boundary,a simple matched layer (ML) technique introduced in [26] will be sufficient and is thus used in our simulation.

The computational mesh for this simulation is similar to that shown in Fig. 6 (b), except here we have twotouching cylinders. The surrounding ML layer has a width of d = r = 0.5775. The expansion order M = 10,while the constant α = 10 in Eq. (4.7). To demonstrate the dynamics of the optical energy transport by WGMsfrom the left cylinder to the right cylinder, in Fig. 9 we show the snapshots of the Ez component at four differenttimes. The initial state of the system is represented by a counterclockwise circulating wave, i.e., the fundamentalmode WGM8,1,0 in the left cylinder. The four sequential snapshots Fig. 9 (a)-(d) then illustrate the generation ofa clockwise WGM in the right cylinder due to the optical coupling and the phase matching, and thus confirm theoptical energy transport from the left cylinder to the right cylinder. More discussions can be found in [25].

4.4 Explicit time step size

As discussed earlier, when we solve Eq. (2.5) or Eq. (3.20) by an explicit method, the explicit time step restrictionfor numerical stability is always a major concern which generally speaking shall depend on the eigenvalues of thederivative matrices. And as heuristically claimed in [4], partially analyzed in [8] and supported by our numericalexperiments, for an orthogonal basis on a triangle constructed by a direct product like (3.9), the explicit time stepsize has a O(1/M4) limitation due to the concentration of a lot of resolution in a single corner of the triangle andis therefore considered unacceptable, while for the Dubiner orthogonal polynomial basis the explicit time stepsize has only a O(1/M2) limitation. However, in practical situations the explicit time step size may also dependon some other factors. For instance, if we choose to approximate Q instead of Q̂ in a DSEM as described inRemark 3.7, the eigenvalues of the derivative matrices M̄ξ and M̄η depend on not only the choice of the basisfunctions but also the underlying geometry of the problem being solved. Moreover, in practice the expansionorder M normally will not be taken very large so a O(1/M4) limitation on the time step size could be stillacceptable.



Ez(x,y,2):

a)

Ez(x,y,6):

b)E

z(x,y,8):

c)

Ez(x,y,10):

d)

Fig. 9 Optical energy transport by WGMs between two microcylinders. The initial state of the system isrepresented by a counterclockwise circulating wave in the left cylinder. The four sequential snapshots at thetimes t = 2, 6, 8 and 10 illustrate the generation of a clockwise WGM in the right cylinder due to the resonantoptical coupling and the phase matching.

To numerically compare the explicit time step sizes for the non-polynomial nodal basis and the Dubiner poly-nomial basis in our applications, we find the “almost optimal” time step sizes for these two bases in thoseapplications described in Sections 4.2 and 4.3, and the results are displayed in Fig. 10. Here we call a stepsize X.X × 10−s “almost optimal” when the simulation is stable for ∆t = X.X × 10−s but unstable for∆t = (X.X + 0.1) × 10−s. As can be seen, for large expansion orders both bases will have to use very smalltime step sizes, but there is no significant difference between the “almost optimal” step sizes for the two basesexcept for a factor of around 1.5. And also as indicated in Fig. 10, for the above two applications it appearsthat the maximum explicit step sizes decrease at a rate faster than O(1/M2) for both bases. It should be pointedout that the faster-than-O(M2) growth rate of the explicit time step size for the polynomial basis contradictsmost previous work [8]. This difference could be caused by the approximation of Q instead of Q̂ in the DSEM,but a full-blown analysis of the difference is not available at this time and it could be another subject of furtherresearch.

4 6 8 10 12 140

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Expansion Order M

M2 ∆

t


a)4 6 8 10 12 14

0

0.005

0.01

0.015

0.02

0.025

Expansion Order M

M2 ∆

t


b)

Fig. 10 Non-quadratic decay of the explicit time step size for both the nodal basis and the Dubiner polynomialbasis on triangles in applications as described in Sections 4.2 and 4.3. a) Scattering by a dielectric cylinder; b)Optical coupling between two microcylinders.



4.5 Computational efficiency

In order to make a fair comparison of the effectiveness of the non-polynomial nodal basis and the Dubinerpolynomial basis, we measure the computational cost of each basis in terms of the CPU time used to simulatethe scattering by a single cylinder. We first investigate the computational cost per time step for each basis. Whenprogramming, we have made every effort to identify and optimize those components that are using most of theCPU time, including evaluating most if not all integrals and computing mass matrix inverses in the preprocessingphase. Conducting our numerical experiments on a Pentium Xeon 2.4GHz processor and 2GB of RAM, we havefound that most of the CPU time is used to (1) compute the resolved normal fluxes and (2) compute those slopevectors in the Runge-Kutta method. For part (1), both bases use almost the same amount of CPU time because itis primarily determined by the number of element edges. For part (2), however, the situation is a bit complicatedand it really depends on the problem being solved. For instance, when simulating the scattering of a singledielectric cylinder, if we employ the total-field formulation in which the source term S in Eq. (2.1) is zero, thepolynomial basis will use less CPU time to compute those slope vectors because all integrals can be computed inpreprocessing. On the other hand, if we employ the scattered-field formulation, then the nodal basis will use lessCPU time since in this case we have to evaluate the right-hand side integral in Eq. (2.5) or Eq. (3.20) at each timestep.

For example, the CPU time per time step for the simulation of the scattering by a single cylinder is shownin Table 1. As mentioned earlier we use both quadrilateral and triangular elements, and in this test the sameexpansion order M = 14 is used for the approximation on both elements. It can be seen from the table that it willtake about 0.11-0.12s to calculate the resolved normal fluxes and 0.02-0.023s to calculate slopes for quadrilateralsfor either basis. If we use the total-field formulation, it will take about 0.01s and 0.026s to compute slopes fortriangles by the polynomial basis and the non-polynomial nodal basis, respectively. If we use the scattered-fieldformulation, however, it will take about 0.082s for the polynomial basis but only 0.028s for the non-polynomialnodal basis to compute the slopes for triangular elements.

Table 1 CPU time in seconds per time step for the simulation of the scattering by a single cylinder.

Total-field Scattered-fieldDubiner Nodal Dubiner Nodal

Normal fluxes 0.110 0.121 0.117 0.120Slopes for quadrilaterals 0.020 0.020 0.023 0.023Slopes for triangles 0.010 0.026 0.082 0.028

Next we consider the total CPU time used to simulate the scattering by a single cylinder to the final timet = 1, and the scattered-field formulation is employed since for scattering problem it is often more convenient.Conducted on the same computer, the used CPU time of both bases for different expansion orders is reported inTable 2, where the number in parenthesis represents the ratio of the CPU time of the polynomial basis to that ofthe non-polynomial nodal basis. As can be seen for the same time step size ∆t, the CPU time of the Dubinerbasis is slightly larger than that of the nodal basis. Moreover, this difference broadens with increased expansionorders, which can be partially understood within the fact that the computational cost to evaluate the right-handside integral in Eq. (2.5) or Eq. (3.20) under the nodal basis is two orders less than that under the Dubiner basis.However, if the stability constraint is taken into account, that is, if we use the corresponding “almost optimal”time step size for each basis, we have found that the computational cost of the Dubiner basis is around 80% ofthat of the nodal basis.

As our concluding remark, we believe the orthogonal nodal basis on triangles has its applications even thepotential O(1/M4) limitation on the explicit time step size, especially in situations that the computational timeis not a major concern. As discussed in Section 3, the orthogonal nodal basis will produce diagonal local massmatrices no matter what approximation scheme is chosen. And compared with the Dubiner polynomial basis,the non-polynomial nodal basis is very much easier to implement. For example, when both quadrilateral andtriangular elements are involved, the codes for handling the quadrilateral and triangular elements are almostidentical if the orthogonal nodal bases are used in approximating the solution on both types of elements and thevalues of the solution at the collocation points defined in (3.11) are used in calculating integrals over triangularelements.



Table 2 Total CPU time in seconds for the simulation of the scattering by a single cylinder.

Expansion ∆t1 ∆t2 Nodal Dubiner Dubinerorder (∆t = ∆t1) (∆t = ∆t1) (∆t = ∆t2)

4 2.0E−3 2.5E−3 46 46 (1.00) 38 (0.83)6 6.1E−4 8.0E−4 213 220 (1.03) 169 (0.79)8 2.4E−4 3.2E−4 738 788 (1.07) 593 (0.80)

10 1.1E−4 1.5E−4 2179 2396 (1.10) 1767 (0.81)12 6.0E−5 8.5E−5 5393 6049 (1.12) 4264 (0.79)14 3.4E−5 5.0E−5 12884 14604 (1.13) 10218 (0.79)

5 Conclusion

In this paper, we have proposed and analyzed an orthogonal non-polynomial nodal basis on triangles whichresults in diagonal mass matrices for DSEMs over unstructured grids. By applying this basis to approximateboth oscillatory functions and the solution of electromagnetic scattering by single dielectric cylinder, we havedemonstrated the exponential convergence of the orthogonal non-polynomial nodal basis. Meanwhile, the sameidea can be easily extended to three-dimensional problems for tetrahedral elements or any other elements whichcan be obtained by a similar “collapsing ” mapping from the standard cube.

Acknowledgements The authors appreciate the insightful and productive comments from all reviewers. Also for the workreported in this paper, S. Deng would like to acknowledge the support of the Army Research Office through the AppliedMathematics Research Center at Delaware State University (grant number: DAAD19-01-1-0738), and W. Cai would like tothank the support of the National Science Foundation (grant numbers: DMS-0408309, CCF-0513179) and the Department ofEnergy (grant number: DEFG0205ER25678).

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