Trefftz type method for 2D problems of electromagnetic scattering from inhomogeneous bodies.

Trefftz type method for 2D problems of electromagnetic scattering from inhomogeneous bodies. S. Yu. Reutsiy Magnetohydrodynamic Laboratory, P. O. Box 136, Mosovsi av.,199, 61037, Kharov, Uraine. e mail: reutsiy@synet.harov.com Abstract A new numerical method for scattering from inhomogeneous bodies is presented. The cases of E and H polarizated incident wave scattered by an infinite 2D cylinder are considered. The scattered field is looed for in two different domains. The first one is a bounded region inside the scattering body with an inhomogeneous permittivity ε(x, y). The second one is an unbounded homogeneous region outside the scatterer. An approximate solution for the scattered field inside the scatterer is looed for by applying the QTSM technique. The method of discrete sources is used to approximate the scattered field in the unbounded region outside the scattering body. A comparison of the numerical and analytic solutions is performed. 1 Introduction Recently various numerical methods have been developed to compute electromagnetic scattering by bodies of irregular form. Boundary methods: the extended boundary condition method (EBCM) [1], [2] or T matrix method [3], the methods based on the generalized multipole technique (GMT) [4], [5] and the method of discrete sources (MDS) [6], [7] are the fastest and most powerful tools in this field. In this paper a new numerical method of the Trefftz type is presented for this goal. We consider the 2D problem of scattering of a plane, electromagnetic wave by a penetrable continuously inhomogeneous dielectric cylinder. The axis of the cylinder is along OZ. We denote the section of the cylinder in the XY plane as Ω. The region surrounding the cylinder is free space. It is assumed throughout that all fields are taen to be harmonic and that the time dependence, taen to be e iωt, has been factored out. Suppose also that the magnetic permeability constant µ is a fixed constant everywhere: µ = µ 0. Under these assumptions Maxwell s system is: roth = i 0 εe, rote = i 0 H. (1) 1

It is written in dimensionless form. Here E and H are vectors of electric and magnetic fields correspondingly, ε is the relative permittivity, the wave number 0 = 2πl s /λ, λ is the wave length of an incident field and l s is the scaling length ( e.g. a typical size of a scatterer). The relative permittivity is a smooth enough complex valued function inside Ω. And it is taen to be 1 outside the scatterer (free space). The both cases of polarization of the incident wave E polarization: E (inc) = {0, 0, E (inc) z }, H (inc) = {H (inc) x, H (inc) y, 0} and H polarization: E (inc) = {E (inc) x, E y (inc), 0}, H (inc) = {0, 0, H z (inc) } are considered. Under the assumptions listed before the scattered field {E (s), H (s) } has the same polarization as the incident one. Let us use the following notations: u (i) E z (s) (x, y), E polarization (x, y) = (x, y) Ω H z (s) (x, y), H polarization u (e) (x, y) = E z (s) (x, y), E polarization H z (s) (x, y), H polarization (x, y) R 2 \ Ω Using Maxwell s system (1) one can get the systems of the scalar equations: in the E polarization case and [ + 2 0 ] u (e) (x, y) = 0, (x, y) R 2 \ Ω (2) [ ] + 0ε 2 (i) (x, y) u (i) (x, y) = 0, (x, y) Ω (3) [ ( 1 x ε (i) (x, y) [ + 2 0 ] u (e) (x, y) = 0, (x, y) R 2 \ Ω (4) x ) + ( y 1 ε (i) (x, y) y in the case of the H polarized incident wave. The boundary conditions are: E polarization : u (i) u (e) = u (inc), H polarization : u (i) u (e) = u (inc), ) ] + 0 2 u (i) (x, y) = 0, (x, y) Ω (5) u (i) u(e) 1 u (i) ε (i) u(e) = u(inc) = u(inc) on Ω. (6) on Ω. (7) Here denotes the derivative in the direction of the normal vector n = (n x, n y ) of Ω and we denote u (inc) =E z (inc), or H z (inc). We also suppose that the exterior field u (e) is an outgoing cylindrical wave at a large distance from the scatterer: u (e) 1 r exp( i 0 r), r, r = x 2 + y 2. (8) 2

2 Numerical algorithm The method presented belongs to the group of boundary methods when an approximate solution is looed for in the form of a linear combination K u (x, y) = q Ψ (x, y), where the trial functions Ψ (x, y) satisfy exactly the corresponding partial differential equation (PDE) but do not necessary satisfy the boundary conditions which are imposed on the solution. As far as the homogeneous exterior region R 2 \Ω is considered one can use a version of the method of discrete sources (MDS) a very effective technique for scattering problems in homogeneous medium developed in the last few years and which is gaining more and more interest because of its many practical applications. The description of the recent development of MDS and other references can be found in [7]. An approximate solution is looed for in the form of a linear combination of Green s functions which satisfy the radiation condition at infinity: K e u (e) (x, y) = q (e) ) K e H(2) 0 ( 0 (x x ) 2 + (y y ) 2 = q (e) Ψ(e) (x, y). Here H (2) 0 denotes the Hanel function of the second ind and zero order. The source points (x, y ) are placed inside the scatterer Ω and q (e) To write the similar approximate solution are the free parameters. K i u (i) (x, y) = q (i) Ψ(i) for inhomogeneous region Ω a new numerical technique developed (x, y) (9) in [8], [9], [10] is used. First, it assumes applying the embedding procedure to the irregular solution domain. In this wor it is assumed that the solution domain Ω may be embedded in the square Ω 0 = [ 1, 1] [ 1, 1]. Of course, if this is not the case originally, then appropriate translation and scaling operations may be performed to mae it so. Let us consider each polarization separately. 2.1 E polarization According to the embedding idea the initial differential operator L (i) = + 2 0 ε(i) (x, y) is replaced with L (i) = + 2 0 ε(i) (x, y). Here ε (i) (x, y) is a truncated series ε (i) (x, y) = +N n,m= N G nm e iπ(nx+my), (x, y) Ω 0, (10) which approximates ε (i) (x, y) for (x, y) Ω. A numerical technique called the C expansion procedure proposed by Smelov is used to get this approximation. With more details this technique is described in [8], [9], [10]. 3

The trial functions Ψ (i) (x, y) are taen as solutions of the PDE: ] L (i) Ψ (i) [ = + 0 ε 2 (i) (x, y) Ψ (i) = I M,l(x ζ ), (11) where I M,l (x ζ) is a δ-shaped source function which essentially differs from zero only inside some neighborhood of the source point ζ =(ξ, η), i.e. this is an approximate Dirac s δ function. However, this is an infinitely differentiable function in the form of truncated series over the trigonometric system exp[iπ(nx + my)]: I M,l (x ζ) = M n,m= M C n,m (ζ) exp[iπ(nx + my)], (12) where the regularization coefficients are: C n,m (ζ) = 1 4 r n(m, l) r m (M, l) exp[ iπ(nξ + mη)], (13) r 0 (M, l) = 1, r n (M, l) = σ l n(m), σ n (M) = sin nπ (M + 1) / nπ (M + 1). Here σ n (M) are so called the Lanczos sigma factors. Note that the regularization coefficients r n (M, l) coincide with σ l n(m) only in the case of the trigonometric expansion. The general method of the regularization procedure is presented in [11]. An approximate solution of (11) is looed for in the same form of a truncated series: Ψ (i) M (x, y) = n,m= M U () n,me iπ(nx+my). (14) Integrating over Ω 0 with the weight functions e iπ(nx+my), n, m = 0, ±1,..., ±M one gets: π 2 (n 2 + m 2 )U () nm + 2 0 i n, j m N These equations can be rewritten in the form: with Introducing the (2M + 1) 2 vectors: M i,j=1 G n i,m j U () i,j A i,j n,mu () i,j = C n,m () A i,j n,m = π 2 (n 2 + m 2 )δ i,n δ j,m + 2 0G n i,m j. = C n,m (ζ )=C () n,m. (15) U = {U () () () () M, M,..., U M,+M,..., U +M, M,..., U +M,+M }, C = {C () M, M,..., C() M,+M,..., C() +M, M,..., C() +M,+M } 4

they can be presented as a set of (2M + 1) 2 (2M + 1) 2 linear systems with the same matrix and different vectors of the right hand side: ÂU = C, = 1,..., K i. (16) It is important to note that the coefficients A i,j n,m of the matrix are obtained in an analytic way without any numerical integration. This is due to the extension of the initial irregular domain Ω to the square Ω 0 and replacing the initial complex permittivity ε (i) (x, y) with the truncated series ε (i) (x, y). Solving these systems one gets the trial functions for expansion (9). Note that using these trial functions means that in fact the Helmholtz equation is solved with the nonzero right hand side: K i [ + 0 ε 2 (i) (x, y)]u (i) = q (i) I M,l(ζ ). (17) When the source points ζ are removed from Ω inside Ω 0, then the right hand side of (17) is a small value for (x, y) Ω. So, we introduce an additional error in an approximate solution when these basis functions are used. But as the numerical experiments have shown, this error can be of the same level as the one due to the approximate satisfaction of the boundary condition. 2.2 H polarization Here we begin with an approximation of the function ν(x, y) = 1/ε (i) (x, y). It is assumed that it can be approximated by the expansion similar to (10): ν(x, y) = +N n,m= N G nm e iπ(nx+my), (x, y) Ω 0. (18) The trial functions Ψ (i) (x, y) are taen now as solutions of the PDE (cf. (11)): [ ( ν(x, y) ) + ( ν(x, y) ) ] + 0 2 Ψ (i) x x y y = I M,l(x ζ ). (19) An approximate solution of (19) is looed for in the same form as the one in the E polarization case (see (14)). By integrating over Ω 0 with the weight functions e iπ(nx+my), n, m = 0, ±1,..., ±M one gets: π 2 i n, j m N (in + jm)g n i,m j U () i,j + 2 0U () n,m = C () n,m. (20) It can be rewritten in the form of a linear system lie (16) with respect to the unnowns U () i,j. The coefficients of the matrix now are: A i,j n,m = π 2 (in + jm)g n i,m j + 2 0δ i,n δ j,m. 5

Note that, as above, when the approximation (18) is nown, then the coefficients can calculated in an analytic way without any numerical integration. Having the two linear combinations K e u (e) (x, y) = q (e) Ψ(e) (x, y), u(i) (x, y) = K i q (i) Ψ(i) (x, y) (21) we get the free parameters q (e), q(i) as a solution of an over determined system of algebraic equations. The system is obtained from the boundary conditions (6) or (7) using the collocation procedure at the collocation points (x j, y j ), j = 1,..., N c distributed uniformly on Ω. The resulting linear system with 2 N c equations and K e + K i unnowns q (e), q(i) procedure. is solved by the least squares 3 Numerical examples To test the algorithm described above the axially symmetric problems are considered. Let Ω be a dis of radius a centered at the point (0, 0) in the XY plane and let (r, ψ) be the polar coordinates: x = r cos ψ, y = r sin ψ. (22) All the calculations were performed with a = 0.5. Following [12] the source points are placed as far as possible from the boundary Ω. So, they are uniformly distributed on the circles of the radiuses r (i) = 0.95 and r (e) = 0.1 for approximation of the inner and outer solution correspondingly. Without loss of generality, the incident field is taen in the form of a plane wave propagating along the X axis: u inc (x) = exp(i 0 x) = exp(i 0 r cos ψ) = J 0 ( 0 r) + 2 i m J m ( 0 r) cos mψ. (23) Here J m (... ) denotes the Bessel function of the first ind and m th order. 1) Constant permittivity. The relative permittivity inside Ω is taen to be constant ε (i) = const. In this case both systems (2), (3) and (4), (5), written in polar coordinates, are: 2 u (e) r 2 + 1 r u (e) r + 1 r 2 2 u (e) ψ 2 + 2 0u (e) = 0, r > a, (24) 2 u (i) r 2 + 1 r u (i) r The boundary conditions, e.g. for H polarization, are: + 1 r 2 2 u (i) ψ 2 + 2 0ε (i) u (i) = 0, r < a. (25) 6

u (i) (a, ψ) u (e) (a, ψ) = J 0 ( 0 a) + 2 i m J m ( 0 a) cos mψ, (26) 1 u (i) u(e) ε (i) (a, ψ) r r (a, ψ) = 0J 0( 0 a) + 2 0 i m J m( 0 a) cos mψ (27) for all 0 ψ 2π. Here we denote J m(z) = dj m (z)/dz. This problem has an analytic solution in the form of expansions over the orthogonal system of the functions cos mψ: where H (2) m u (e) (r, ψ) = u (i) (r, ψ) = m=0 m=0 A m H (2) m ( 0 r) cos mψ, r > a, (28) ) B m J m ( ε (i) 0 r cos mψ, r < a, (29) denotes the Hanel function of the second ind and m th order. Using orthogonality of the cos mψ basis one gets a 2 2 linear system for each pair A m, B m. We denote the resulting solution as u (e) anlt and u(i) anlt and use it to compare with the numerical solution u(e) num and u (i) num. Note that in this case the trial functions can be found in the analytical way. Indeed, as it follows from (18) and so, eq. (20) can be easily resolved Suppose 1/ε (i), n = m = 0; G n,m = 0, otherwise. U () n,m = ε (i) C () n,m ε (i) 2 0 π2 (n 2 + m 2 ). ε (i) 2 0 π 2 ( n 2 + m 2) 0, n, m = 1, 2,... To estimate the accuracy of the calculations we use the maximal relative error. For the internal solution it is defined as: e in = max n=1,...,n t { (i) u num(x (i) n, y n (i) ) u (i) anlt (x(i) n, y n (i) ) u (i) num(x (i) n, y n (i) ) + u (i) anlt (x(i) n, y n (i) ) The similar value e out is used to compare u (e) num and u (e) anlt. We use N t }. (30) = 100 test points x (i) n, y n (i) which are distributed uniformly inside Ω to compute e in. To compute e out we choose N t = 100 test points uniformly located on the circles of radius r m = a(m + 1), m = 1,..., 10 outside Ω. These errors are placed in Table 1 for different numbers of harmonics M and degrees of freedom (DOF) 7

K i = K e (we tae them equal). We also compute the errors e bcd and e bcn in approximation of the boundary conditions: e bcd = 1 N c N c j=1 [ u (i) j u (e) j ] 2, u (inc) j ebcn = [ 1 N c 1 u N c ε (i) j=1 (i) j u (e) j (inc) u j ] 2, (31) where summation is performed using all the collocation points. Table 1: H polarization. The constant permittivity ε (i) = 1.2, the wave number 0 = 1. The maximal relative errors e in, e out inside and outside the scatterer. DOF M = 10 M = 20 e in e out e bcd e bcn e in e out e bc e bcn 10 1.5 10 3 3.8 10 2 7.6 10 3 2.6 10 2 1.6 10 3 3.6 10 2 8.0 10 3 2.7 10 2 20 2.9 10 4 4.4 10 3 9.8 10 5 3.0 10 4 3.6 10 5 1.0 10 4 8.7 10 5 1.5 10 4 30 3.0 10 4 4.6 10 3 1.7 10 5 8.0 10 5 2.1 10 7 1.1 10 6 5.9 10 7 1.4 10 6 This example demonstrates two different inds of error in the approximate solution. The error of the first ind is caused by approximate satisfaction of the boundary condition on Ω. It is inherent in all boundary methods and it depends on the DOF which are used to approximate the boundary data. For a small number of harmonics M, increasing of the DOF decreases the boundary error e bc. However, this does not improve the accuracy of approximation of the scattered field. This can be explained by using of the source functions I(x ξ) instead of the Dirac functions δ(x ξ). In this case we do not have enough harmonics to obtain the appropriate condensed ones. When M increases, I(x ξ) approaches the delta function. As a result, the errors in computation of the scattered field have the same level as the ones in the boundary conditions. 2) Variable permittivity, E polarization. The relative permittivity inside Ω is taen as ε (i) (x, y) = 1 + β 2 (x 2 + y 2 ) 1 + β 2 r 2, β = const. (32) Here eqs. (2), (3) tae the form: 2 u (i) r 2 + 1 r u (i) r + 1 r 2 2 u (i) ψ 2 + 2 0(1 + β 2 r 2 )u (i) = 0, r < a, (33) 2 u (e) r 2 + 1 r u (e) r + 1 r 2 2 u (e) ψ 2 + 2 0u (e) = 0, r > a. (34) 8

The boundary conditions are: u (i) (a, ψ) u (e) (a, ψ) = J 0 ( 0 a) + 2 i m J m ( 0 a) cos mψ, u (i) u(e) (a, ψ) r r (a, ψ) = 0J 0( 0 a) + 2 0 i m J m( 0 a) cos mψ for all 0 ψ 2π. Similar to the example considered above this problem has an analytic solution in the form of expansion over the same system cos mψ: u (e) (r, ψ) = A m H m (2) ( 0 r) cos mψ, r > a, (35) u (i) (r, ψ) = m=0 B m V m (r 0, β) cos mψ, r < a, (36) m=0 where V m (r 0, β) is a solution of the equation [r 2 d2 dr 2 + r d ] dr m2 + 0(1 2 + β 2 r 2 )r 2 V m (r 0, β) = 0 bounded at the point r = 0. It can easily be verified that in this case V m (r/ 0 0, β) satisfies the equation [r 2 d2 dr 2 + r d ] dr m2 + (1 + β2 0 2 r 2 )r 2 V m (r/ 0 0, β) = 0. So, with a fixed m it depends on the only parameter α = β/ 0. We denote a solution of the equation bounded at the point ρ = 0 as Z m (ρ α). So we get [ρ 2 d2 dρ 2 + ρ d ] dρ m2 + (1 + α 2 ρ 2 )ρ 2 Z m = 0 (37) V m (r 0, β) = Z m ( 0 r β 0 ). The function Z m (ρ α) is looed for in the form of a power series: Z m (ρ α) = ρ m n=0 Substituting in (37) we get the recursive formula: b n (m, α)ρ 2n. (38) b 0 b 1 = 4(m + 1), b n = b n 1 + α 2 b n 2 4n(m + n) (39) with the free parameter b 0. Note that the value α = 0 corresponds to the scattering by a homogeneous circular cylinder. In this case the functions V m should be replaced by the Bessel s functions of the first ind J m. This can be provided by taing b 0 = 1 m!2 m (40) 9

because in this case Z m (ρ 0) = J m (ρ). (41) Numerical experiments show the rapid converges of (38) at least for ρ, α satisfying 0 ρ 0.5, 0 α 5. The summation is truncated as b n (m, α) becomes less than 10 18. We use finite sums with the number of terms equal to 20 in evaluation (35), (36). This provides a truncating error less than 10 10 in all of the calculations performed. The data in Table 2 were obtained for the wave number 0 = 2. The coefficient β = 1. In all of the calculations performed we use N = 6 (see (10)). This provides the maximal absolute error e a 10 6 in the approximation of ε (i) (x, y) inside Ω. Table 2: E polarization. Scattering by an inhomogeneous cylinder with the relative permittivity ε (i) (x, y) = 1 + x 2 + y 2 ; the wave number 0 = 2; the number of harmonics M = 20. error DOF= 10 DOF= 15 DOF= 20 e in 3.5 10 3 2.2 10 4 1.1 10 5 e out 4.7 10 3 2.4 10 4 1.7 10 5 3) Variable permittivity. H polarization. The relative permittivity inside Ω is given in (32). In this case eqs. (4), (5), written in polar coordinates, are: 2 u (e) r 2 + 1 r u (e) r + 1 r 2 2 u (e) ψ 2 + 2 0u (e) = 0, r > a, (42) 2 u (i) r 2 + 1 β2 r 2 u (i) r (1 + β 2 r 2 ) r + 1 2 u (i) r 2 ψ 2 + 0(1 2 + β 2 r 2 )u (i) = 0, r < a (43) with the boundary conditions: for all 0 ψ 2π. u (i) (a, ψ) u (e) (a, ψ) = J 0 ( 0 a) + 2 i m J m ( 0 a) cos mψ, 1 u (i) u(e) 1 + β 2 a 2 (a, ψ) r r (a, ψ) = 0J 0( 0 a) + 2 0 i m J m( 0 a) cos mψ Looing for the approximate solution inside the scatterer in the form (36) one gets the equation for V m : r 2 ( 1 + β 2 r 2) d 2 V m dr 2 + r ( 1 β 2 r 2) dv m dr + ( 1 + β 2 r 2) [ 2 0r 2 ( 1 + β 2 r 2) m 2] V m = 0 10

Denoting z = βr, c = 2 0/β 2 one gets: z 2 ( 1 + z 2) d 2 V m dz 2 + z ( 1 z 2) dv m dz + ( 1 + z 2) [ cz 2 ( 1 + z 2) m 2] V m = 0. (44) An approximate solution of (44) bounded at z = 0 is sought in the form of the series: V m (z, c) = b n (m, c) z m+2n. (45) The following recurrence formulae are obtained using the Mathematica 3 pacage: n=0 b n = cb n 3 + 2cb n 2 + [ 4 ( n 2 3n + 2 ) + c + (4n 6) m ] b n 1, 4n (m + n) b 0 = 1 2 m m!, b 1 = b 0 (2m c) 4(m + 1), b 2 = 2b 0c + b 1 (c + 2m). 8 (m + 2) As the numerical experiments shown the series (45) converge at least for z, c satisfying z 1, c 5. The summation is truncated as b n (m, c) becomes less than 10 18. The unnown coefficients A m, B m are obtained as a solution of the sequence of 2 2 linear systems following from orthogonality of the cos mψ basis. In Table 3 the analytic solution is used to examine the one obtained by the numerical algorithm presented in the previous section. The wave number is: 0 = 1 and ν(x, y) = 1 + 1.5(x 2 + y 2 ). The values e in and e out are the same as the ones in Table 1. The data presented correspond to the same number of harmonics M = 20 in approximate solutions and the two different approximations of ν(x, y): with N = 1 and N = 5. Table 3: H polarization. Scattering by an inhomogeneous cylinder with ν(x, y) = 1+1.5(x 2 +y 2 ); the wave number 0 = 1; the number of harmonics M = 20. DOF N = 1 N = 5 e in e out e in e out 10 4.3 10 3 6.2 10 1 9.6 10 4 3.3 10 2 20 4.2 10 3 6.2 10 1 1.7 10 5 2.9 10 4 30 4.2 10 3 6.2 10 1 1.1 10 7 1.5 10 5 One can see that for N = 1 corresponding to 9 terms expansion (18) increasing of DOF from 10 to 30 does not increase the exactness of the approximate solution. This means that the main 11

error in the solution is caused by the error in approximation of ν(x, y). This is the third ind of error of approximation in the method presented. When N increases, the differential operator of (19) approaches the initial one in (5) and the total error begin to decrease with the growth of DOF. 4 Conclusion The method presented can be regarded as a generalization of MDS onto the problems of scattering by electrical inhomogeneous bodies. When an accurate approximation of the complex permittivity ε (i) ( or of the inverse value 1/ε (i) ) is nown in the form of truncated Fourier series, then it provides solution of the scattering problem with a high precision. The method presented can be extended to 3D scattering. At least it seems to be quite simple in the scalar acoustic case. These extensions and 3D electromagnetic scattering are topics of the further investigations. References [1] T. Wriedt and A. Doicu. Formulations of the extended boundary condition method for three dimensional scattering using the method of discrete sources. J. Modern Opt., 45:199 213, 1998. [2] T. Wriedt and U. Comberg. Comparison of computational scattering methods. J. Quant. Spectrosc. Radiat. Transfer, 60:411 423, 1998. [3] M. I. Mishcheno, L. D. Travis, and D. W. Macowsi. T matrix computations of light scattering by nonspherical particles: a review. J. Quant. Spectrosc. Radiat. Transfer, 55:535 575, 1996. [4] Ch. Hafner. The Generalized Multipole Technique for computational electromagnetics. Artech House Boos, Boston, 1990. [5] N. Kuster. Multiple Multipole Method for simulating EM problems involving biological bodies. IEEE Transactions on Biomedical Engineering, 40:611 620, 1993. [6] Y. Leviatan and A. Boag. Analysis of electromagnetic scattering using a Current Model Method. Comput. Phys. Comm., 68:311 335, 1991. [7] A. Doicu, Y. Eremin, and T. Wriedt. Acoustic and Electromagnetic Scattering Analysis using Discrete Sources. Academic Press, San Diego, 2000. 12

[8] S. Reutsiy and B. Tirozzi. Trefftz Spectral Method for elliptic equations of general type. Comp. Ass. Mech. and Eng. Sci., 8(4):629 644, 2001. [9] S. Reutsiy and B. Tirozzi. A new boundary method for electromagnetic scattering from inhomogeneous bodies. J. Quant. Spectrosc. Radiat. Transfer, 72:837 852, 2002. [10] S. Reutsiy. A boundary method of Trefftz type with approximate trial functions. Engineering Analysis with Boundary Elements, 37:341 353, 2002. [11] S. Reutsiy and B. Tirozzi. Quasi Trefftz spectral method for separable linear elliptic equations. Comp. Math. with Applications, 37:47 74, 1999. [12] A. Bogomolny. Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal., 22:644 669, 1985. 13