Applications of boundary integral equation methods ... - UMM Directory

Journal of Computational and Applied Mathematics 104 (1999) 89–110

Applications of boundary integral equation methods in 3D electromagnetic scattering 1 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

G.C. Hsiao ∗ , R.E. Kleinman, D.-Q. Wang Received 17 June 1998

Abstract This paper is concerned with the application of boundary integral equation method to the electromagnetic scattering of a perfect conductor in the three dimensional space. A collocation method is employed for the magnetic eld integral equation and error estimates are derived. Far- eld patterns and radar cross sections are computed for various wave numbers in the case of sphere. Numerical experiments are compared to those ontained from the Mie series method in order to c 1999 Published by Elsevier Science B.V. All rights reserved. verify the predicted theoretical results. Keywords: Maxwell’s equations; Scattering; Boundary integral equation; Far- eld pattern

1. Introduction Boundary integral equation methods have played a major role in the numerical solution of Maxwell’s equations for over three decades. The exact formulation of the boundary integral equations for the physical problems in exterior domains is especially suitable for electromagnetic scattering. Compared to the classical Mie series method, it allows more exible geometry of the obstacles and a variety of incident waves. While tremendous strides have been taken over the years, many problems remain intractable due to the large size of the scattering objects and the oscillation of the kernels in the integral equations. One of these problems concerns the accuracy of approximate solutions. In a recent series of papers [4–7, 9, 18], an attempt has been made in the context of integral equations for scalar two dimensional problems. ∗

Corresponding author. Fax: +1-302-831-4511; e-mail: [email protected] (G.C. Hsiao) This work was supported by the Air Force Oce of Scienti c Research, Air Force Material Command, USAF, under Grant F9620-96-1-10039. The views and conclusions contained herein are those of authors and should not be intepreted as necessarily representing the ocial policies or endorsements, either expressed or implied, of the Air Force Oce of Scienti c Research or the US government. 1

c 1999 Published by Elsevier Science B.V. All rights reserved. 0377-0427/99/$ - see front matter PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 2 4 - 6

90

G.C. Hsiao et al. / Journal of Computational and Applied Mathematics 104 (1999) 89–110

Very little rigorous analysis for electromagnetic scattering in three dimensions is available. We only know of the method of residual analysis in Sobolev spaces [8] and covolume schemes for time dependent Maxwell’s equations [12, 13]. Our goal in this paper is to provide a rigorous proof of convergence in terms of mesh sizes for the magnetic eld integral equations (MFIE) applied to the electromagnetic scattering of a perfect conductor in the three-dimensional space. The corresponding analysis for the transmission problems will be given in a subsequent report. The contents of this paper are as follows: we begin Section 2 with an integral formulation for electromagnetic scattering of a perfect conductor object. Following that is a section on numerical discretizations of the MFIE. The numerical scheme used is the collocation method. Section 4 contains the main result, an error estimate for the numerical approximation. In the last section, Section 5, numerical experiments are performed and compared to those obtained from the Mie series method. These numerical results verify the error estimate in the previous sections.

2. Integral representations Let − be a convex bounded domain in R3 with twice continuously dierentiable boundary and unit normal n pointing to the exterior domain. Let + denote the exterior of − . Then if (E+ (x); H+ (x)) and (E− (x); H− (x)) denote, respectively, electromagnetic eld in + and − , the source free Maxwell’s equations [3] are curl E+ = ikZH+ ;

curl H+ = −ikY E+

in + ;

(2.1)

curl E− = ikZH− ;

curl H− = −ikY E−

in −

(2.2)

and E+ (x), H+ (x) satisfy a Silver–Muller radiation condition

1 x E+ (x) × ; + H+ (x) = o |x| |x|

|x| → ∞

(2.3)

uniformly p for all directions x=|x|. In (2.1) and (2.2), k denotes the wave number and Z = 1=Y = = is the intrinsic impedance of the medium. The constants and denote, respectively, electric permittivities and magnetic permeabilities occupying the mediums. By a representation theorem due to Stratton and Chu [16], solutions of (2.1)–(2.3) can be represented as the electromagnetic eld generated by a combination of surface sources. More precisely, we have the following. Theorem 2.1. Let (E+ ; H+ ) ∈ (C 1 ( + ) ∩ C( + ))3 and (E− ; H− ) ∈ (C 1 ( − ) ∩ C( + ))3 be a solution of (2.1)–(2.3). Then 1 4

Z "

#

eikR eikR eikR ikZ ny × H± (y) + 5y ny · E± (y) − 5y × (ny × E± (y)) dsy R R R

=(x)E+ (x) + ((x) − 1)E− (x);

x ∈ ± ;

(2.4)


1 4

Z "

#

eikR eikR eikR −ikY ny × E± (y) + 5y ny · H± (y) − 5y × (ny × H± (y)) dsy R R R

=(x)H+ (x) + ((x) − 1)H− (x); where

(x) :=

91

1; 0;

x ∈ ± ;

(2.5)

x ∈ + ; x ∈ − ;

x; y ∈ R3 ; R := |x − y| is the length of x − y; ny is the unit normal at y on We say a function f(x) is Holder continuous on that |f(x) − f(y)|6c|x − y| ;

directed into + .

if there exist constants c and , 0 ¡ ¡ 1, such

x; y ∈ :

The classical jump conditions [3] state that if n · E± are Holder continuous on Z

1 eikR 5y ny · E± (y) dsy x→ ± 4 R Z eikR 1 = − 5y ny · E± (y) dsy ± 12 nx (nx · E± (x)); 4 R lim

(2.6)

R

where − denotes Cauchy principal value integral. A similar result holds for n · H± . Moreover, if n × E± are Holder continuous on , then Z

1 eikR 5y × ny × E± (y) dsy x→ ± 4 R Z eikR 1 × ny × E± (y) dsy ± 12 nx × (nx × E± (x)) = − 5y 4 R lim

(2.7)

and a similar resultRholds for n × H± . Here, the notation x → ± means x ∈ ± and x → . Lastly, integrals (eikR =R) (y) dsy are continuous as x → ± with no jumps and the resulting boundary integrals have weakly singular kernels. These jump conditions remain valid in dierent senses with weaker conditions on densities, see [8] for details. Combining these limiting values with the representations (2.4) and (2.5) and using the vector identity n × (n × a) = (n · a)n − (n · n)a; we obtain the boundary integral equations Z "

#

1 eikR eikR eikR − ikZ (ny × H± (y)) + 5y (ny · E± (y)) − 5y × (n × E± (y)) dsy 4 R R R = ± 12 E± (x);

(2.8)

92


Z "

#

1 eikR eikR eikR − −ikY (ny × E± (y)) + 5y (ny · H± (y)) − 5y × (n × H± (y)) dsy 4 R R R = ± 12 H± (x)

(2.9)

for x ∈ . In a scattering problem (E− ; H− ) are identi ed with the known incident eld (E inc ; H inc ) while (E+ ; H+ ) are identi ed with the scattering eld (E s ; H s ). Adding the two forms of (2.8) and (2.9) and denoting by (E; H ) the total eld (E inc + E s ; H inc + H s ) we have Z "

#

1 eikR eikR eikR (ny × H (y)) + 5y (ny · E(y)) − 5y × (n × E(y)) dsy − ikZ 4 R R R = 12 E(x) − E inc (x);

(2.10)

Z "

#

eikR eikR eikR 1 − −ikY (ny × E(y)) + 5y × (n × H (y)) dsy (ny · H (y)) − 5y 4 R R R = 12 H (x) − H inc (x):

(2.11)

For perfect conductors we use the boundary conditions on n×E =0

and

n · H = 0;

(2.12)

and perform the obvious operations with nx and use the following identity: ny · 5y × F(y) = − 5ty ·(ny × F(y)) to obtain the well-known integral equations Z "

#

eikR eikR t 1 ikZ nx × − (ny × H(y)) − 2 5y 5y ·(ny × H(y)) dsy 4 R k R = − nx × Einc (x); nx − × 4

Z

5y

(2.13)

eikR × (ny × H(y)) dsy R

= 12 nx × H(x) − nx × Hinc (x);

(2.14)

where 5ty · denotes the surface divergence. Both (2.13) and (2.14) are vector equations for the unknown surface current ny ×H(y). The rst kind equation (2.13), the electric eld integral equation (EFIE), involves an integral operator with a Cauchy singularity and a surface divergence while the integral operator in the second kind equation (2.14), the magnetic eld integral equation (MFIE), is no longer Cauchy singular but only weakly singular as shown in [3]. These well-known EFIE and MFIE are usually attributed to Maue [14] but known much earlier [15]. In this report, we consider the numerical algorithm for the MFIE (2.14). Denote the surface current by J := n × H; then(2.14) can be rewritten as ( 12 I + K)J = n × Hinc ;

(2.15)


93

where the operator K is de ned as KJ :=

nx × 4

Z

5y

eikR × J(y) dsy : R

(2.16)

The mapping properties of K are discussed in detail in [8, 3]. In particular, we notice that 1 KJ (x) = 4 1 = 4 1 = 4

Z

!

eikR × J (y) dsy nx × 5y R

Z " Z "

eikR eikR − nx · 5y nx · J (y) 5y R R

!

#

J (y) dsy #

@ eikR eikR + (nx − ny ) · J (y) 5y J (y) dsy : R @nx R

The added term ny · J (y) is 0, since J (y) is a tangential vector. A simple computation shows that |nx − ny |6L|x − y|;

x; y ∈ ;

and |nx · (x − y)|6L|x − y|2 ;

x; y ∈

and is bounded since where L ¿ 0 depends on the curvature l of has a weakly singular kernel. Thus we have the well-known result.

is a C 2 surface. Hence K

Theorem 2.2. K is a linear compact operator from L2 ( ) to L2 ( ). It is likely that (2.15) is not uniquely solvable for certain values of the wave number k. However, there are a variety of techniques available to resolve this problem. So we only consider the case when (2.15) is uniquely solvable. The next theorem states that 12 I + K an isomorphism if (2.15) has an unique solution. Theorem 2.3. Suppose that (2.15) is uniquely solvable. Then there exist constants c1 , c2 ¿ 0 such that c1 ||( 12 I + K)J ||L2 ( ) 6||J ||L2 ( ) 6c2 ||( 12 I + K)J ||L2 ( ) :

(2.17)

The left inequality of (2.17) follows directly from the continuity of K. The right inequality of (2.17) follows from the Banach theorem [17] since 12 I + K is one-to-one, onto and continuous. We note that K is compact. Thus uniqueness implies existence by the classical Fredholm alternative. 3. Discretization of magnetic ÿeld integral equation In this section the boundary surface of a perfect conductor obstacle is approximated by triangulated meshes. On the resulting meshes, the magnetic eld integral equation (MFIE) (2.15) is

94


Fig. 1. 128 triangular patches approximating the unit sphere.

discretized. This will lead to a detailed analysis and numerical implementations in the following sections. Assume that is approximated by piecewise planes h and the intersection of the half-spaces bounded by these planes forms a polyhedron approximation of the obstacle domain − with all their vertices lying on . Assume that h consists of piecewise triangles parametrized by the maximum side length, which is generically denoted by h, and assume that the aspect ratio of radii of circumsribing circles and inscribed circles of all the individual triangle patches are bounded above and below as h approaches 0. It follows from analysis in the appendix that these triangulation meshes h approximate in the order of h if is twice continously dierentiable. From this some important properties of the integral operator on are preserved on h , which is the key to the results which follow. We will illustrate this in detail in the next section. The N patches of the triangulated meshes are assumed to be numbered sequentially in some convenient way and the individual patch is denoted by n . The centroid and the diameter of nth patch are denoted by xn and hn , respectively. We also denote the area of nth patch by sn . In C3N , we introduce the inner product (·; ·)W de ned by N X

(u; C)W :=

n=1;n ∈

(un ; Cn )sn

(3.1)

h

and denote the resulting inner product space by U and the associated norm by ||u||W :=

p

(u; u)W ;

(3.2)


95

where in (3.1) (·; ·) is the inner product in C3 . The inner product and norm thus de ned are associated with the discrete L2 inner product and norm on h . To discretize (2.15) on h , we approximate the surface current J (x) = n(x) × H (x) by piecewise constant function J˜ , which takes a complex vector Jm in mth triangle patch, and evalute both sides of (2.15) at xk , k = 1; : : : ; N: The resulting collocation scheme is as follows 1 J 2 k

˜ J˜ (xk ) = nk × H inc (xk ): +K

(3.3)

˜ is de ned as In (3.3) nk denotes the unit normal of kth patch and K ˜ J˜ (xk ) := nk × K 4

Z

K (xk ; y) × J˜ dsy ;

x ∈ k ;

(3.4)

h

and K (x; y) := 5y

eikR (1 − ikR)eiR (x − y): = R R3

˜ J˜ (xk ), i.e., ˜ J˜ as the vector in C3N whose kth component is K We denote K N X ˜ J˜ (xk ) = nk × K 4 l=1

Z l

K (xk ; y) × Jl dsy :

The linear system (3.3) has 3N unknowns and 3N equations. The following theorem shows that (3.3) is uniquely solvable provided (2.15) has a unique solution. Theorem 3.1. Assume that (2.15) has a unique solution. Then for h small enough, there exists c ¿ 0 such that ˜ J ||W ¿c||J ||W ; || 12 J + K

(3.5)

where J is any vector in C3N . Proof. Even though h is a piecewise linear boundary instead of twice continuously dierentiable, Theorem 2.3 and (2.17) still hold if h is small. This follows from the approximation of h to . For details, see the estimates in the appendix. So by (2.17) there exists c1 ¿ 0 such that ||( 21 I + K)J||L2 (

h)

¿c1 ||J||L2 (

h)

;

for any complex vector function J which is constant J m in each triangle patch m . So ||J||L2 (

h)

= ||J ||W :

(3.6)

96


On kth patch k N X nk ˜ KJ − KJ (xk ) = × 4 l=1

=

N Z X l=1

l

Z l

(K (x; y) − K (xk ; y)) × J l dsy

[(nk − nl ) · Jl (K (x; y) − K (xk ; y))

−(K (x; y) − K (xk ; y)) · nk J l ] dsy By Theorem A.1 in the appendix, we have |nk − nl ||K (x; y) − K (xk ; y)|6 |(K (x; y) − K (xk ; y)) · nk |6

L ∈ L1y (l ) |x − y|

L ∈ L1y (l ); |x − y|

where L depends on the curvature of shows

and the aspect ratio of the meshes. A simple computation

lim(K (x; y) − K (xk ; y)) = 0 a:e: x; y:

h→0

Collecting all these and applying dominant convergence theorem twice, we conclude that for any ¿ 0, there exists h1 such that when h ¡ h1 we have ||KJ − K˜ J ||L2 (

h)

˜ L2 ( 6||J||

h)

= ||J ||W :

(3.7)

Thus we obtain a constant c2 ¿ 0 such that ˜ J ||W = || 1 J + K ˜ J ||L2 ( || 12 J + K 2 ¿ || 12 J

+ KJ||L2 (

h) h)

˜ J ||L2 ( − ||KJ − K

h)

¿ c2 ||J ||W if h is small enough. The unique solvability of (3.3) now follows from Theorem 3.1, since the homogeneous system has only the trivial solution. 4. Error estimate In this section an error estimate will be given for the collocation scheme (3.3). We will show that the error under the appropriate norm is of the rst order in the mesh size h. First we introduce the error vector e ∈ C3N e := (e1 ; : : : ; eN );

(4.1)


97

where each ek ∈ C3 is de ned on the kth triangle patch such that ek := Jk − J (xk ):

(4.2)

Next apply (2.15) on 1 J (xk ) 2

+

nk × 4

h

Z

and evalute both sides at xk :

K (xh ; y) × J (y) dsy = nk × E inc (xk ):

(4.3)

h

By subtracting (4.3) from (3.3), we obtain 1 (Jk 2

−

N X nk × − J (xk )) + 4 l=1 N X nk × 4 l=1

Z l

Z l

K (xk ; y) × Jl dsy

K (xk ; y) × J (y) dsy = 0:

Or, 1 e 2 k

+

=

N X nk × 4 l=1

N X nk × 4 l=1

Z

Z

l

l

K (xk ; y) × el dsy

K (xk ; y) × (J (y) − J (xl )) dsy :

(4.4)

The main result in this section is the following theorem: Theorem 4.1. Denote by J ∈ C3N the solution of (3.3) and by J (x) ∈ (H 1 ( of (2.15) on h . Then we have the estimate ||J − J (1) ||W 6Kh||H ||(H 1 (

h ))

h )∩C(

h ))

3

the solution

3

where J (1) ∈ C3N is deÿned as Jk(1) := J (xk )

on the kth patch:

Proof. Let us rst estimate the right-hand side of (4.4): N X nk × 4 l=1

=

Z

K (xk ; y) × (J (y) − J (xk )) dsy

l

N 1 X [nk · (J (y) − J (xl ))K (xk ; y) − nk · K (xk ; y)(J (y) − J (xk ))] dsy 4 l=1

N 1 X = 4 l=1

Z

l

[(nk − nl ) · (J (y) − J (xl ))K (xk ; y) − nk · K (xk ; y)(J (y) − J (xl ))] dsy :

(4.5)

98


The last step follows since J (y) − J (xl ) is perpendicular to the unit normal nl of the lth patch. By the estimates in the appendix, we obtain |nk − nl ||K (xk ; y)|6 |nk · K (xk ; y)|6

L ; |xk − y|

L ; |xk − y|

where L depends on the curvature of |

N X nk × 4 l=1

Z l

and the aspect ratio of the triangular meshes

K (xk ; y) × (J (y) − J (xl )) dsy |6

N 1 X 4 l=1

Z l

h.

Thus

L1 |(J (y) − J (xl ))| dsy : |xk − y|

Let J˜ (y) := |J (y) − J (xl )|; and Ak :=

N Z X L1 |J˜ (y)| l=1

l

|xk − y|

y ∈ l

dsy :

Then by the Cauchy–Schwartz inequality, A2k 6

Z h

L1 dsy |xk − y|

Since (1=|xk − N X k=1

y|) ∈ L1y (

A2k sk 6 L2

Z

6 L3

h

k=1

Z Z Z

h

L1 |J˜ (y)|2 dsy : |xk − y|

h ), N X

h

6 L2

Z

h

!

sk |J˜ (y)|2 d sy |xk − y|

d s(x) |J˜ (y)|2 d sy |x − y|

|J˜ (y)|2 d sy : h

R

Here we use a result in [3] that h (dsx =|x − y|) is Holder continuous of order , 0 ¡ ¡ 1 in y so is a bounded function in y. By a standard approximation result in nite element methods (see e.g., [2, 12, 13]), we obtain Z

|J˜ (y)|2 dsy 6L4 h2 |J |2(H 1 ( h

h ))

3

;

(4.6)

where L4 depends only on the aspect ratio. Combining all these results yields ˜ W 6L5 h2 |J |2(H 1 ( || 12 e + Ke||

h ))

3

:

The theorem and (4.5) is then followed by (3.5) in Theorem 3.1.

(4.7)


Fig. 2. 8–32 Patches.


99

100





101

Table 1 L2 error between the collocation and Mie methods Err n

Mesh n = 0

Mesh n = 1

Mesh n = 2

Mesh n = 3

k = 0:5 k = 1:0 k = 2:0

8:283 · 10−2 4:788 · 10−1 1:825 · 100

2:123 · 10−2 9:965 · 10−2 5:536 · 10

3:096 · 10−3 1:041 · 10−2 1:007 · 10−1

3:090 · 10−4 2:934 · 10−3 2:465 · 10−2

5. Numerical results In this section we describe the implementation and the numerical results of the collocation algorithm in previous sections for solving the surface current n × H from the boundary integral equation (2.15). We consider the electromagnetic scattering by a prefect conductor occupying an unit ball {(x; y; z) : 2 x + y2 + z 2 61}. The incident wave propagates in the positive direction of z-axis: s

inc

E (x; y; z) = −

ikz ˆ e x;

H inc (x; y; z) = eikz y; ˆ

(5.1)

where x, ˆ yˆ are, respectively, unit vectors in the direction of x- and y-axis. The resulting scattering eld can be computed by the Mie series which uses spherical harmonics and Bessel functions to represent the electromagnetic eld in an in nite series, see [10, 11, 1]. The Mie series package by Warren J. Wiscombe (at ftp : ==emlib:jpl:nasa:gov=pub=miev:tar:Z) is used to compute the far- eld pattern of the magnetic eld. The far- eld pattern or the scattering amplitude H∞ (r) ˆ or H∞ (; ) is de ned on the unit sphere such that (see [3]) H s (x) =

1 eik |x| H∞ (r) ˆ +O ik|x| |x|

;

|x| → ∞:

(5.2)

Here rˆ = x=|x| = (sin cos ; sin sin ; cos ) (066, 0662) is a vector in the direction of x on the unit sphere and H s (x) denotes the scattering eld. It can be shown that H∞ (; ) is orthogonal to rˆ so that it can be decomposed into ˆ H∞ (; ) = H (; )Âˆ + H (; ); ˆ ˆ are two orthogonal unit vectors that are in the tangent plane of the unit sphere at r. where Â, ˆ In this numerical experiment we will compare H and H obtained from the Mie series with those from our collocation algorithm. To solve the MFIE (2.15) by the collocation method, we approximate the unit sphere by triangular patches (see Fig. 1). These triangular patches are generated by iterations. We start with eight triangular patches by connecting six points (±1; 0; 0), (0; ±1; 0), (0; 0; ±1) on the unit sphere. Each triangular patch is then divided into four smaller triangles by connecting the midpoints of each side. The new vertices of triangles are then projected into the unit sphere and four new triangular patches are generated from each old patch in the previous iteration. The aspect ratio of these triangular patches can be shown to be bounded above and below as h approaches 0. The rst four iterations are presented in Figs. 2–5.

102


Fig. 6. Plot of |S1 ()| and |S2 ()| for k = 0:5

Fig. 7. Plot of |S1 ()| and |S2 ()| for k = 0:5


Fig. 8. Plot of |S1 ()| and |S2 ()| for k = 1.


103

104


Second-order quadrature rule is used to approximate the integrals and the Gauss–Seidel iterative scheme is used to solve the linear system (3.3) in order to save the memory load of storing the dense matrix. The surface current J (x) thus obtained is used to compute the scattering eld H s (x) 1 H (x) = − 4 s

Z

5y h

eikR × J (y) dsy R

(5.3)

and the far- eld pattern H∞ (; ) is then computed by using a second-order quadrature rule to approximate the following integral: H∞ (; ) = −

rˆ × 4

Z

ˆ e−ikr·y J (y) dsy :

(5.4)

h

We computed the far- eld patterns for the following wave numbers k = 0:5, k = 1:0 and k = 2:0. The computatuions are performed on the triangular meshes corresponding to the rst four iterations of mesh re nement in Figs. 2–5. In most literature S1 () and S2 () are computed where S1 () :=

H (; ) ; cos

(5.5)

S2 () :=

H (; ) : sin

(5.6)

It can be shown [1] that S1 () and S2 () are independent of the angle . In the numerical experiment Sm (), m = 1; 2, are computed at 39 dierent angles l = l=39, l = 0; : : : ; 38. Denote by SmM () and Smn (), m = 1; 2, the solution from the Mie series and the collocation scheme on the triangular meshes corresponding to the iteration n=0; : : : ; 3, respectively (see Figs. 2–5). Then the corresponding discrete L2 errors between the Mie series solution and the collocation schemes are de ned by v ! u 38 u X n Err := t (S1n (l ) − S1M (l ))2 + (S2n (l ) − S2M (l ))2 =38;

n = 0; : : : ; 3:

l=0

The results in Table 1 show that the errors decrease in the average order O(h2 ) as the meshes get re nement each time. Figs. 6–11 are the corresponding plots of |Smn ()| and |SmM ()| for m = 1; 2 and for n = 0; : : : ; 3. Here the solution by the Mie series is represented by the solid line, while the numerical solutions for n = 0 by the dot line, n = 1 by the star line, n = 2 by the plus line and n = 3 by the circle line. We also present here the three-dimensional radar cross section. We recall by de nition that the bistatic radar cross section (; ) is de ned by [1] (; ) := lim 4r 2 r→∞

|H s |2 ; |H i |2

where H s and H i represent the scattered and incident magnetic eld, respectively. It can be shown [1] that, for an incident magnetic eld of unit magnetic eld, the bistatic radar cross section can also be represented by (; ) =

4 |S(; )|2 : k2




105

106


Fig. 12. RCS plot of (; ) for k = 2 with n = 1; 2.

Fig. 13. RCS plot of (; ) for k = 2 with n = 1; 2.


Fig. 14. RCS plot of (; ) for k = 2 with n = 3 and Mie series.

Fig. 15. RCS plot of (; ) for k = 2 with n = 3 and Mie series.

107

108


Figs. 12–15 are the corresponding plots of (; ) for k = 2 when the iteration n = 1; 2; 3 are used. Appendix Since h is not twice continuously dierentiable, we may not directly use the results of [3] in some estimates of previous sections. In this appendix we give the following two estimates that will be needed for our purpose. Theorem A.1. Denote the centroid of the kth patch k by xk and y ∈ l . Then |nk − nl |6C|xk − y|;

(A.1)

|nk · (nk − y)|6C|xk − y|2 ;

(A.2)

where the constant C depends on the curvature of lemma is needed to prove (A.1) and (A.2)

and the aspect ratio of

h.

First the following

Lemma A.2. In the triangular patch k |nk − n(y)|6Chk ; where n(y) is the unit normal of of k .

(A.3) at y and the projection of y onto

h

is in k , hk is the diameter

The proof of (A.3) is a routine application of Taylor expansion and omitted here. The constant C in (A.3) depends on the curvature of and the aspect ratio of k . To this end,we need the following. Deÿnition. De ne the lifting operator L on h such that for any y ∈ l , we have Ly ∈ and Ly − y ⊥ l . The lifting operator L is well de ned since is convex. By Taylor expansion we can show that if y ∈ l |Ly − y|6Chl ;

(A.4)

where C depends on the curvature of

. Now we are in a position to prove Theorem A.1.

Proof of (A.1). First we claim that |Lxk − Ly|6C|xk − y|:

(A.5)

To verify (A.5), we see that (i) If |xk − y|6 13 max(hk ; hl ), then |Lxk − Ly| 6 |Lxk − xk | + |Ly − y| + |xk − y| 6 |xk − y| + C(hk + hl ) 6 C1 |xk − y|:

by (A:4)


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(ii) If |xk − y|6 13 hk , then both xk and y are in the kth triangular patch and L is a Lipschitz operator in k , so (A.5) follows. Now from (A.5) and [3] we obtain |nk − nl | 6 |nk − n(L(xk ))| + |n(L(xk )) − n(L(y))| + |n(L(y)) − nl | 6 C(hk + hl ) + |L(xk − y)| 6 C(hk + hl ) + C1 |xk − y|: From this (A.1) follows. Proof of (A.2). Since any point in a given triangular patch is the convex combination of its vertices, we can nd vertices x0 and y0 in the patch k and l , respectively, such that |x0 − y0 |6C|xk − y|

(A.6)

for y ∈ l and C62. Since xk − x0 ⊥ nk and y − y0 ⊥ nl |nk · (xk − y)| = |nk · (x0 − y)| 6 |(nk − nl ) · (x0 − y)| + |nl · (x0 − y)| 6 C|xk − y|2 + |nl · (x0 − y0 )|

(A:1)

6 C|xk − y| + |nl − n(y0 )||x0 − y0 | + |n(y0 ) · (x0 − y0 )|: Since x0 , y0 ∈ , |n(y0 ) · (x0 − y0 )|6C1 |x0 − y0 |2 :

(A.7)

Combining all these, we have |nk · (xk − y)|6C(|xk − y|2 + hl |xk − y|): Now after using a scale change argument among hk , hl and |xk − y|, we conclude |nk · (xk − y)|6C|xk − y|2 : References [1] J.J. Bowman, T.B.A. Senior, P.L.E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes, NorthHolland, Amsterdam, 1969. [2] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [3] D. Colton, R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. [4] M. Feistauer, G.C. Hsiao, R.E. Kleinman, Asymptotic and a posterior estimates for boundary element solutions of hypersingular integral equations. SIAM J. Numer. Anal. 33 (1996) 666–685. [5] G.C. Hsiao, R.E. Kleinman, Feasiable error estimates in boundary element methods, in C.A. Brebbia, M.S. Ingber,(Eds.), Boundary Element Technology VII, Computational Mechanics Publications, Southampton Boston, 1991, pp. 875–886.

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