Numerical analysis of the MFS for certain harmonic problems

Vol. 38, N 3, 2004, pp. 495–517 DOI: 10.1051/m2an:2004023

NUMERICAL ANALYSIS OF THE MFS FOR CERTAIN HARMONIC PROBLEMS

Yiorgos-Sokratis Smyrlis

and Andreas Karageorghis

Abstract. The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace’s equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem which is applicable even in the cases when the MFS matrix might be singular. We prove the convergence of the method for analytic boundary data and perform a stability analysis of the method with respect to the distance of the singularities from the origin and the number of degrees of freedom.

Finally, we test the algorithm numerically.

Mathematics Subject Classification. Primary 65N12, 65N38; Secondary 65N15, 65T50, 65Y99.

Received: July 29, 2003.

1. Introduction

The Method of Fundamental Solutions is a relatively new boundary method for the solution of certain elliptic boundary value problems. The formulation of the method as a numerical technique was first proposed by Mathon and Johnston [11]. The MFS is a meshless method which is very easy to implement and avoids integration on the boundary. The method has been applied to a variety of physical problems in fluid mechanics, acoustics, electromagnetism and elasticity. Comprehensive surveys of the applications of the MFS and related methods can be found in the recent survey articles [3, 4, 6]. Error estimates and a convergence analysis of the MFS for circular harmonic problems appear in [8, 9]. Applications of the MFS can also be found in the books by Kolodziej [10], Golberg and Chen [5] and Doicu, Eremin and Wriedt [2].

In [12] we considered the approximation of the solution of Laplace’s equation in the disk Ω ={x∈R² : |x|<

} subject to the Dirichlet boundary conditionu=f. Although this is a very specific problem, it enables us to identify applicability as well as error and stability features of the method which are present in more general cases.

Keywords and phrases. Method of fundamental solutions, boundary meshless methods, error bounds and convergence of the MFS.

1 Department of Mathematics and Statistics, University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus.

e-mail:smyrlis@ucy.ac.cy; andreask@ucy.ac.cy

c EDP Sciences, SMAI 2004

In the MFS, the solutionuis approximated by the harmonic function u_N(c,Q;P) =

N j=1

c_jk(P, Q^α_j), P∈Ω, (1.1)

where c= (c1, c2, . . . , c_N)^T andQis a 2N-vector containing the coordinates of the singularities (sources) Q^α_j, j = 1, . . . , N, which lie outside Ω. The functionk(P, Q) is a fundamental solution of Laplace’s equation given by

k(P, Q) =− 1

2πlog|P−Q|, (1.2)

where |P−Q|is the distance between P andQ. The singularitiesQ^α_j are fixed on the boundary∂Ω of a disk˜ Ω =˜ {x∈R² :|x|< R}concentric to Ω, withR > . A set of collocation points {P_i}^N_i=1 is placed on∂Ω. If P_i= (x_Pi, y_Pi), then we take

x_Pi =cos2(i−1)π

N , y_Pi=sin2(i−1)π

N , i= 1, . . . , N. (1.3)

IfQ^α_j = (x_Qα j, y_Qα

j), then x_Qα

j =Rcos2(j−1 +α)π N , y_Qα

j =Rsin2(j−1 +α)π

N , j= 1, . . . , N, (1.4)

where the positions of the sources differ by an angle^2πα_N from the positions of the boundary points and 0≤α <1.

In the caseα= 0, we thus have arotationof the singularities with respect to the boundary points (see [8, 12]).

This rotation improves the accuracy of the method whenR−1 (see Figs. 2 and 4).

In the version of the MFS with fixed singularities, the coefficients care determined so that the boundary condition is satisfied at the boundary points{P_i}^N_i=1:

u_N(c,Q;P_i) =f(P_i), i= 1, . . . , N. (1.5) This yields a linear system of the form

G^αc = f, (1.6)

for the coefficientsc, where the elements of the matrixG^α are given by G^α_i,j=− 1

2πlog|P_i−Q^α_j|, i, j= 1, . . . , N. (1.7) ClearlyG^αis a circulant matrix, see [1]. More precisely G^α= circ(g1(α), . . . , g_N(α)) where

g_j(α) = − 1

2πlogP₁−Q^α_j

= − 1 4πlog

R²−2Rcos

2π(j+α−1) N

+²

, (1.8)

j= 1, . . . , N. ThusG^αis diagonalizable

G^α = U^∗DU, where D = diag(λ1(α), . . . , λ_N(α)),

with eigenvalues

λ_j(α) = N k=1

g_k(α)ω^{(k−1)(j−1)} (1.9)

whereω= e^2πi/N and j= 1, . . . , N. The corresponding normalized eigenvectors are ξ_j = 1

√N (1, ω^j−1, ω^2(j−1), . . . , ω^{(N−1)(j−1)})^T, j= 1, . . . , N,

and they form an orthonormal basis ofC^N. The matrixU is therefore unitary (UU^∗=I) and its conjugate is

U^∗ = 1

√N









1 1 1 · · · 1

1 ω ω² · · · ω^N−1 1 ω² ω⁴ · · · ω^2N−2

... ... ... ...

1 ω^N−1 ω^2(N−1) · · · ω^{(N−1)(N−1)}







 .

The matrix U is known as the Fourier matrix.

Let ζ,η=_N

k=1ζ_kη¯_k be the complex inner product ofζ,η∈C^N. Any vectorv∈C^N, can be expressed as v=_N

k=1v,ξ_kξ_k and henceG^αv=_N

k=1λ_k(α)v,ξ_kξ_k. In particular, whenG^αis nonsingular (G^α)⁻¹v =

N k=1

1

λ_k(α)v,ξ_kξ_k. (1.10)

In Katsurada and Okamoto [9], the authors prove the convergence of the unrotated MFS for Laplace’s equation in a disk, subject to analytic Dirichlet boundary data. The extension to non-analytic boundary data and to the rotated case is studied by Katsurada [8]. Our approach is different as it is based on the study of the eigenvalues of the matrix G^α. We first identify the cases when the matrix G^α is singular, we then develop an efficient algorithm for the solution of the problem and, using the information we have obtained about the eigenvalues, we prove the convergence of the method. Finally, from the properties of the eigenvalues, we develop a stability analysis for the method with respect toR.

This paper is structured as follows: in Section 2 we study the invertibility of the matrixG^αand isolate the eigenvalues which might vanish. In Section 3 we propose a modified algorithm which avoids these troublesome eigenvalues. In Section 4 we prove the exponential convergence of the MFS approximate solution u_N to the exact solutionufor analytic boundary data asN tends to infinity. In Section 5 we provide a stability result, namely we demonstrate that when the radius R, roundoff error is generated. The resulting deterioration in accuracy when R is large, which is contrary to the theoretical predictions, has often been reported in the literature. Finally in Section 6 we test the method numerically on two examples for which the exact solution in known.

2. Properties of the eigenvalues

In this section we investigate the properties of the eigenvalues of the matrixG^α. We divide these eigenvalues in three groups:

(i) the “first” eigenvalueλ1;

(ii) in the caseN is even, the “middle” eigenvalueλN

2+1; and (iii) the remaining eigenvalues.

In the sequel we shall assume, without loss of generality, that the radius of the disk is= 1. Also, because of symmetry, we shall restrict the parameterαin the interval [0,¹₂].

Combining (1.8) with (1.9) we obtain forj= 1, . . . , N that λ_j(α) = − 1

4π N k=1

e2πi(j−1)(k−1)

N log

R²+ 1−2Rcos

2π(k+α−1) N

· (2.1)

Clearly,

N k=1

sin

2π(j−1)(k−1) N

log

R²+ 1−2Rcos

2π(k−1) N

= 0, (2.2)

which leads to the following result:

Lemma 2.1. Whenα= 0, the eigenvalues λ_j(0),j= 1, . . . , N, ofG^α are real and λ_j(0) = λ_N−j+2(0), j= 2, . . . ,

N+ 1 2

· Whenα= 0, the eigenvalueλ1(α)is real, and so isλN

2+1(α), in the caseN is even. The remaining eigenvalues are complex with

λ_j(α) = ¯λ_N−j+2(α), j= 2, . . . ,

N+ 1 2

·

2.1. The first eigenvalue

We are particularly interested in the points where the first eigenvalue vanishes.

Proposition 2.2. The first eigenvalue λ1(α) of the matrix G^α is given by λ1(α) = − 1

4πlog

R^2N −2R^Ncos(2πα) + 1

. (2.3)

Proof. SinceG^αis circulant,i.e., G^α= circ (g1(α), g2(α), . . . , g_N(α)) , the eigenvalues ofG^αare given by (1.9).

In particular,

λ1(α) = N k=1

g_k(α). (2.4)

From (1.8), we have

λ1(α) = − 1 4πlog

_N

k=1

R²+ 1−2Rcos

2π(k+α−1) N

·

From [7] (p. 34)

n−1 k=0

x²−2xycos

α+2kπ n

+y²

= x²ⁿ−2xⁿyⁿcosnα+y²ⁿ, (2.5)

from which (2.3) follows.

From (2.3) we obtain the following corollary:

Corollary 2.3. The eigenvalue λ1(α) vanishes if and only if α=₂¹_πcos⁻¹ R^N

2

. We observe the following:

Remarks 2.1.

(i) An immediate consequence ofCorollary 2.3is that ifR^N >2 (or equivalently R >2^N1 ), the eigenvalue λ1(α) cannot vanish.

(ii) In order for λ1(α) to vanish we need 2 cos 2πα > 1,i.e., 0< α < ¹₆ or ⁵₆ < α <1. Since because of symmetry we are only considering values of α∈[0,¹₂), it follows that the eigenvalueλ1(α) could only vanish for values ofα∈[0,¹₆].

2.2. The middle eigenvalue λ

2+1

(α) when N is even

2+1(α) in the case N = 2M is even, is of particular interest. Its corre- sponding normalized eigenvector is

ξ^N₂+1 = 1

N^1/2(1,−1,1,−1, . . . ,1,−1)^T, which isolates from the boundary data (f1, . . . , f_N)^T the term

1 N^1/2

N k=1

(−1)^k−1f_k

ξN 2+1, corresponding tonoise.

Proposition 2.4. If N is even, then the eigenvalueλN

2+1(α)is given by λN

2+1(α) = − 1

4πlog R^N−2R^N/2cosαπ+ 1

R^N −2R^N/2cos(α+ 1)π+ 1· (2.6)

Proof. Let N = 2M. We know that the eigenvalues ofG^α are given by (1.9); therefore λN

2+1(α) = − 1

4π N k=1

(−1)^k−1log

R²−2Rcos2π(k+α−1)

N + 1

= − 1 4π

M n=1

log

R²−2Rcos

2πn−1 M +απ

M

+ 1

+ 1 4π

M m=1

log

R²−2Rcos

2πm−1

M +(α+ 1)π M

+ 1

.

From (2.5) we get that λN

2+1(α) =− 1 4πlog

R^2M −2R^Mcosαπ+ 1 + 1

4πlog

R^2M−2R^Mcos(α+ 1)π+ 1

=− 1

4πlog R^N −2R^N/2cosαπ+ 1

R^N −2R^N/2cos(α+ 1)π+ 1·

A direct application of this theorem yields the following corollary:

Corollary 2.5. ForN even and R >1, the eigenvalue λN

2+1(α) vanishes if and only if α= ¹₂. [We assume that α∈[0,1).]

2.3. The eigenvalues λ

(α), j = 1, and j =

+ 1, when N is even

λ_j(α) = N 4π

∞ m=0

e^{−i2πN(mN+j−1)α}

(mN+j−1)R^mN+j−1+ e^{i2πN(mN+N−j+1)α} (mN+N−j+1)R^mN+N−j+1

. (2.7)

In particular whenN is even

λN

2+1(α) = N 2π

∞ m=0

cos(2m+ 1)απ mN+^N₂

R^mN+N2 · (2.8)

Proof. Let F(R, ϑ) =−₄¹_πlog(R²−2Rcosϑ+ 1). Then, from [7] we have F(R, ϑ) = − 1

4πlog(R²−2Rcosϑ+ 1) = − 1

2πlogR+ 1 2π

∞ n=1

cosnϑ

nRⁿ · (2.9)

Then forj= 2, . . . , N, we have λ_j(α) =

N k=1

exp

i2π

N(j−1)(k−1)

F

R,2π

N(k−1 +α)

= 1

2π ∞ n=1

1 nRⁿ

_N

k=1

cos 2π

N(j−1)(k−1)

cos 2π

Nn(k−1 +α)

(2.10) +i 1

2π ∞ n=1

1 nRⁿ

_N

k=1

sin 2π

N(j−1)(k−1)

cos 2π

Nn(k−1 +α)

·

We are going to calculate the above two sums. We first observe that:

N k=1

cos 2π

Nk

cos 2π

Nn(+α)

=1 2cos

2π Nnα

C_n−^N +C_n^N₊

, (2.11)

N k=1

sin 2π

Nk

cos 2π

Nn(+α)

=1 2sin

2π Nnα

C_n−^N −C_n^N₊

, (2.12)

where

C_κ^N = N j=1

cos 2π

Nκj

=

N if κ≡0 modN

0 if κ≡0 modN. (2.13)

Combining (2.10), (2.11) and (2.12), and replacing theC_κ^N according to (2.13), we obtain λ_j(α) = 1

4π ∞ n=1

e^−i2πNnα

nRⁿ C_n−j^N ₊₁+ 1 4π

∞ n=1

e^i2πNnα

nRⁿ C_n^N_+j−1

= N

4π ∞ m=0

e^{−i2πN(mN+j−1)α}

(mN+j−1)R^mN+j−1 + N 4π

∞ m=0

e^{i2πN(mN+N−j+1)α}

(mN+N−j+ 1)R^mN+N−j+1·

Formula (2.8) is an immediate consequence of (2.7).

In particular, whenα= 0 we have the following formula for the eigenvalues Corollary 2.7. Forα= 0and j= 1, we have

λ_j(0) = N 4π

∞ m=0

1

j−1+mN · 1

R^j−1+mN + 1

N−j+1+mN · 1 R^N−j+1+mN

· (2.14)

In particular, forN even andj= ^N₂ + 1, we have λN

2+1(0) = N 2π

∞ m=0

1

N

2 +mN · 1

R^N2+mN· (2.15)

Finally forj = [^N₂] + 1, . . . , N, we have λ_j(0) =λ_N−j+2(0). We also have the following two corollaries:

Corollary 2.8. For allj = 2, . . . , N we have λ_j(0)>0.

In particular, sinceλ1(0) = 0 if and only ifR= 2^N1 (see Cor. 2.3), we also have the corollary:

Corollary 2.9. The matrix G⁰ is nonsingular if and only ifR= 2^N1.

Next we shall prove that the eigenvaluesλ_j(α), forα∈[0,¹₂] andj= 2, . . . ,_N+1

2

, never vanish. In particular, we have the following result:

Lemma 2.10. For α∈[0,¹₂) and j= 2, . . . ,_N+1

2

, we have Re λ_j(α)>0.

Proof. Let r_j(α) =Re λ_j(α). The proof of the lemma is a consequence of the following two properties of the r_j:

(i) for all α∈[0,¹₂] and j= 2, . . . ,_N+1

2

, we have that ^dr_d^j_α^(α) <0 . Therefore, for α∈[0,¹₂], ther_j(α) are decreasing functions in the interval [0,¹₂];

(ii) for all j= 2, . . . ,_N+1

2

, we have thatr_j(¹₂)>0 . Therefore,r_j(α)≥r_j(¹₂)>0.

Proof of (i):

Clearly

r_j(α) = N 4π

∞ m=0

cos_2π

N(mN+j−1)α

(mN+j−1)R^mN+j−1 + cos_2π

N(mN+N−j+ 1)α (mN+N−j+1)R^mN+N−j+1

·

Differentiating with respect toαwe obtain r_j(α) = −1

2 ∞ m=0

sin_2π

N(mN+j−1)α

R^mN+j−1 +sin_2π

N(mN+N−j+ 1)α R^mN+N−j+1

·

Letz=^ei2π_R^{N α}; then r_j(α) =−1

2Im _∞

m=0

z^mN+j−1+z^mN+N−j+1

=−1 2Im

(z^j−1+z^N−j+1)(1−z^N)

|1−z^N|²

· It thus suffices to show that Im

(z^j−1+z^N−j+1)(1−z^N)

>0,for α∈[0,¹₂] . After some simple calculations we get

Im

(z^j−1+z^N−j+1)(1−z^N)

=Im

e^{i2πN(j−1)α}

R^j−1 +e^{i2πN(N−j+1)α}

R^N−j+1 1−e^−i2πNNα R^N

(2.16)

= sin 2π

N(j−1)α 1

R^j−1+ 1 R^2N−j+1

+sin

2π

N(N−j+1)α 1

R^N−j+1+ 1 R^N+j−1

· Both the arguments of the sines, in the above expression, lie in (0, π) and therefore both sin_2π

N(j−1)α and sin_2π

N(N−j+1)α

are positive. Thus the right hand side of (2.16) is positive, which concludes the proof of (i).

Proof of (ii):

Letγ= 1/R, then we have r_j

1 2

= N

4π ∞ m=0

cos_N^π(mN+j−1)

(mN+j−1)R^mN+j−1 + cos_N^π(mN+N−j+1) (mN+N−j+1)R^mN+N−j+1

= N

4πcosπ(j−1) N

∞ m=0

γ^2mN+j−1

(2mN+j−1)+ γ^{(2m+2)N−j+1} ((2m+2)N−j+1)

−N

4πcosπ(j−1) N

∞ m=0

γ^{(2m+1)N+j−1}

((2m+1)N+j−1)+ γ^{(2m+1)N−j+1} ((2m+1)N−j+1)

·

Clearly cos^π(j−_N¹⁾>0 ; thus it suffices to show that for everym >0 we have γ^2mN+j−1

2mN+j−1 + γ^{(2m+2)N−j+1}

(2m+2)N−j+1 > γ^{(2m+1)N−j+1}

(2m+1)N−j+1+ γ^{(2m+1)N+j−1} (2m+1)N+j−1, or equivalently

1

2mN+j−1 − γ^N

(2m+1)N+j−1 > γ^N−2j+2

1

(2m+1)N−j+1− γ^N (2m+2)N−j+1

.

Sinceγ^N−2j+2<1, it is sufficient to show that 1

2mN+j−1 − γ^N

(2m+1)N+j−1 > 1

(2m+1)N−j+1− γ^N

(2m+2)N−j+1, or equivalently

mN−2j+2

(2mN+j−1)((2m+1)(N−j+1) > γ^N mN−2j+2

((2m+1)N+j−1)((2m+2)(N−j+1),

which is clearly true. This concludes the proof of property (ii) and the theorem.

This leads to the central result of this section:

Theorem 2.11. The matrixG^α is singular if and only if (i) α= ₂¹_πcos⁻¹

R^N 2

, in which case the first eigenvalue λ1(α) vanishes;

or

(ii) α= ¹₂, whenN is even. In this case the eigenvalue λN

2+1(α)vanishes.

Remarks 2.2.

(i) From Remark 2.1(ii) and Theorem 2.11 it follows that, if α∈₁

6,¹₂

, then detG^α= 0;

(ii) for every R >1,α∈[0,¹₂), for sufficiently largeN, the matrixG^α is nonsingular. In particularG^α is nonsingular forN ≥[_log^{log 2}_R] + 1;

(iii) more detailed proofs of the results presented in this section may be found in [13].

3. Implementation

3.1. Expression of u

in terms of the eigenvalues and eigenvectors of G

u_N(P) = c,l = (G^α)⁻¹f,l, where

l = l(P) = − 1

2π(log|P−Q1|, . . . ,log|P−Q_N|)^T. (3.1) Since the{ξ_k}k=1,...,N form an orthonormal basis ofC^N, for nonsingularG^α we have

(G^α)⁻¹f = (G^α)⁻¹ N k=1

f,ξ_kξ_k = N k=1

1

λ_k(α)f,ξ_kξ_k. Thus using (1.10) we obtain

u_N(P) = (G^α)⁻¹f,l = N k=1

1

λ_k(α)f,ξ_kl,ξ_k. (3.2)

Remark 3.1. By using Fast Fourier Transforms (FFT), the quantities λ_k(α),f,ξ_k andl(P),ξ_k, fork= 1, . . . , N, can be evaluated at a cost ofO(NlogN) operations.

3.2. Description of the proposed algorithm

We will suggest a modification of the method which avoids the presence of the inverses of possible zero eigenvalues in (3.2). Clearly, since the only possibly vanishing eigenvalues are λ1 and λN

2+1, if the boundary condition vectorf does not contain the corresponding eigenvectorsξ1andξ^N

2+1(i.e. f,ξ1=f,ξ^N

2+1= 0 ), then the troublesome terms do not appear in (3.2), and thus the MFS can be used without modification. To treat the general boundary condition vectorf, whenN is even, we decompose it as follows:

f = f⁰+f¹+f², where

f⁰ = f− f,ξ1ξ1− f,ξ^N

2+1ξ^N

2+1, f¹ = f,ξ1ξ1, f² = f,ξ^N

2+1ξ^N

2+1,

and

ξ1 = 1

√N (1,1, . . . ,1), ξN

2+1 = 1

√N (1,−1,1,−1, . . . ,1,−1).

We shall solve the boundary value problems, corresponding to f, for= 0,1,2, separately.

• For= 0 the boundary value problem is solved with the MFS yielding the approximate solution u⁰_N(P) =

2≤k≤N k=^N₂+1

1

λ_k(α)f,ξ_kl,ξ_k. (3.3)

• For= 1 the vector describing the boundary is a constant multiple of theconstantvector (1,1, . . . ,1) . Since the constant functions are harmonic, the most natural choice for a solution in this case is the constant solution

u¹_N(P) = 1

√N f,ξ1· (3.4)

• For= 2 the vectorf² is a constant multiple of

(1,−1,1,−1, . . . ,1,−1).

An appropriate solution in this case, i.e., a harmonic function the boundary values of which at the points{P_k}^N_k=1, coincide with f², is a suitable multiple of

Re z^N2 = Re(x+iy)^N2. The appropriate solution is thus

u²_N(P) = 1

√N f,ξ^N₂+1Re z^N2. (3.5)

Summarizing, the approximate solution of the modified MFS is taken to be

u_N(P) = u⁰_N(P) +u¹_N(P) +u²_N(P). (3.6) In the case whenN is odd the partu²_N is not present.

This algorithm can thus be performed inO(NlogN) operations.

For α∈₁

6,¹₂

, the matrixG^αis nonsingular and therefore the algorithm can be applied without subtracting the two troublesome terms.

4. Convergence of the MFS for analytic boundary data

In this section we shall show that the (unmodified) MFS approximationu_N converges uniformly exponentially fast in the || · ||_∞−norm to the exact solutionuof the Dirichlet problem in the unit disk Ω, provided that the boundary data f = u|_∂Ω are analytic or equivalently u can be extended as a harmonic function in an open regionD containing Ω ={(x, y)|x²+y²≤1}. In particular, we assume that for someβ >0

B(0,1+β) = {(x, y)|(x²+y²)^1/2≤1+β} ⊂ D . (4.1)

If we expressf as a Fourier series f(ϑ) =

k∈Zfˆ_ke^ikϑ,then (4.1) implies that

||f||1+β =

k∈Z

|fˆ_k|²(1 +β)^2|k|

¹₂

< +∞. (4.2)

In particular, we have that for every k∈Z

|fˆ_k| ≤ ||f||1+β(1 +β)^−|k|. (4.3) Let f_N be the Discrete Fourier Interpolantof f corresponding to the values off at ϑ_j = ²_N^πj, j = 1, . . . , N. Namely forN even (the case when N is odd can be dealt with similarly)

f_N(ϑ) =

N2

k=−^N₂+1

ϕ_ke^ikϑ,

where the coefficients ϕ_k∈C, k=−^N₂ + 1, . . . ,^N₂, are chosen so that f_N and f agree at the points {ϑ_j|j = 1, . . . , N}. Clearly ifω= e^2πiN , then

ϕ_k = 1 N

N j=1

ω^−kjf(ϑ_j) = 1 N

N j=1

ω^−kj

∈Z

fˆe^iϑj

= 1

N N j=1

ω^−kj

∈Z

fˆω^j

=

∈Z

fˆ



1 N

N j=1

ω^−kjω^j





=

m∈Z

fˆ_k+mN. (4.4)

Thus

|ϕ_k| ≤

m∈Z

|fˆ_k+mN| ≤ ||f||1+β

m∈Z

(1 +β)^−|k+mN|

≤ 2||f||1+β(1 +β)^−|k|

+∞

n=0

(1 +β)^−nN = 2||f||1+β(1 +β)^−|k| 1 1−(1 +β)^−N

≤ 2(1 +β)

β ||f||1+β(1 +β)^−|k| = M1||f||1+β(1 +β)^−|k|, (4.5) withM1depending only onβ. This interpolant converges tof exponentially fast, with respect to the supremum norm, asN tends to infinity. In fact

|f_N(ϑ)−f(ϑ)| =

k∈Z

fˆ_ke^ikϑ−

N2

k=−^N₂+1

ϕ_ke^ikϑ ≤

N2

k=−^N₂+1

|fˆ_k−ϕ_k|+

|k|≥^N₂

|fˆ_k|

≤ 2

|k|≥^N₂

|fˆ_k| ≤ 2(1 +β)

β · ||f||1+β·(1 +β)^−N2. (4.6)

Let us denote byu(·;h) the solution of the boundary value problem ∆u= 0 in Ω

u=h on ∂Ω

and byu_N(·;h) its MFS approximation. Note thatu_N depends also onR and the angular parameterα. Since both u(·;h) andu_N(·;h) depend linearly onhwe have that

||u_N(·;f)−u(·;f)||∞≤||u(·;f−f_N)||∞+||u_N(·;f−f_N)||∞+||u_N(·;f_N)−u(·;f_N)||∞. (4.7) Clearly, sincef andf_N agree at the boundary points, the middle term u_N(·;f−f_N) is identically zero. In what follows we shall show that, each of the remaining two terms on the right hand side of (4.7), decays exponentially fast asN tends to infinity.

A. The term ||u(·;f−f_N)||_∞

Sinceuis harmonic, it satisfies the maximum principle in Ω

||u(·;f −f_N)||_∞ = sup

ϑ∈[0,2π]

|(f −f_N)(ϑ)|.

Using (4.6) we obtain that

||u(·;f−f_N)||_∞ ≤ 2(1 +β)

β · ||f||1+β·(1 +β)^−N2 . (4.8) B. The term ||u_N(·;f_N)−u(·;f_N)||∞

Let f_N(ϑ) =^N₂

k=−^N₂+1ϕ_ke^ikϑ. Then u(·;f_N) =

N2

k=−^N₂+1

ϕ_ku

·; e^ikϑ

and u_N(·;f_N) =

N2

k=−^N₂+1

ϕ_ku_N

·; e^ikϑ .

Thus

||u_N(·;f_N)−u(·;f_N)||_∞ ≤

N2

k=−^N₂+1

|ϕ_k| ·u_N

·; e^ikϑ

−u

·; e^ikϑ

∞. (4.9)

The observation which enables us to obtain the desired bound, is the fact that the right hand side of (3.2) reduces to a single term when h= e^ikϑ, which is easy to treat.

Lemma 4.1. We have the following estimates for ||u_N(·; e^ikϑ)−u(·; e^ikϑ)||_∞: (i) Case I, k= 0: ForR >1 and every N∈N

||u_N(·; 1)−u(·; 1)||_∞ ≤ 8

NlogR·R^N; (4.10)

(ii) Case II,0<|k| ≤N/2: ForR >1 and sufficiently large N

||u_N(·; e^ikϑ)−u(·; e^ikϑ)||_∞ ≤ 16

1− 1

√R ₋1

· |k|

(N− |k|)· 1

R^N−2|k|· (4.11)

Proof. Case I. k= 0

In this caseh≡1,h= (1,1, . . . ,1), u≡1 and

h,ξ_j = √

N ifj= 1 0 otherwise.

Thus from (3.2) we obtain

u_N(x, y; 1) = N j=1

1

λ_jh,ξ_jl(x, y),ξ_j =

√N

λ1 l(x, y),ξ1.

On the other hand if (x, y) = (rcosφ, rsinφ) andr <1, we have

l(rcosφ, rsinφ),ξ1 = 1

√N N j=1

l_j(rcosφ, rsinφ),

where l= (l₁, . . . , l_N) (see (3.1)) and if we set ϑ_j,α= ²_N^π(j−1 +α) we obtain

l_j(rcosφ, rsinφ) = −1

4πlog(R²−2rRcos(ϑ_j,α−φ) +r²)

= − 1

2πlogR+ 1 2π

∞ m=1

1 m

r R

_m

cosm(ϑ_j,α−φ).

Imitating the proof of Theorem 2.6 we obtain

l(rcosφ, rsinφ),ξ1=−

√N 2π

logR+

∞ m=1

1 mN

r R

_mN cos

mN

φ−2π

Nα

· (4.12)

Similarly, for 0<|k|< N/2 , we have l(rcosφ, rsinφ),ξ_k+1 =

√N 4π

∞ m=0

r R

_k+mN e^{−i(k+mN)(φ−2πNα)}

k+mN +

r R

_N−k+mN e^{i(N−k+mN)(φ−2πNα)} N−k+mN

· (4.13)

Consequently, using (4.12) and (2.9), we have

u_N(x, y; 1) =

√N

−

√N

2π logR+

√N 2π

∞ m=1

1 mN

r R

_mN

cos(mN(φ−2π Nα))

− 1

4πlog(R^2N −2R^Ncos(2πα) + 1)

=

logR−^∞

m=1

1 mN

r R

_mN

cos(mN(φ−2π Nα)) 1

2N log(R^2N −2R^Ncos(2πα) + 1)

= 1 +

logR−^∞

m=1

r^mNcos(φ−2mπα) mNR^mN

−

logR−^∞

m=1

cos(2mπα) mNR^mN

logR−^∞

m=1

cos(2mπα) mNR^mN

·

Sinceu_N−uis harmonic andu(·; 1)≡1, using the maximum principle we have

||u_N(·; 1)−u(·; 1)||∞ = max

ϑ∈[0,2π]|u_N(cosϑ,sinϑ; 1)−1|

≤ max

ϑ∈[0,2π]

1 N

∞ m=1

1

mR^mN · |cos(mNφ−2mπα))−cos(2mπα)|

logR− ∞ m=1

1 mNR^mN

≤ 2 1

N ∞ m=1

1 R^mN

1

2logR = 4

NlogR(R^N−1) ≤ 8 NlogR·R^N,

for N sufficiently large,i.e. there exists someN1 =N1(R), such that the above holds forN ≥N1. (Indeed, eventually 1−_R¹N ≥ ¹₂ and _∞

m=1 1

mNR^mN ≤ ¹₂logR.) Case II. 0<|k|< ^N₂

Let us first treat the case k >0 . Then for h= e^ikϑ, combining (5.2) and (4.13) we obtain u_N

rcosφ, rsinφ; e^ikϑ

= 1

λ_k+1h,ξ_k+1l(rcosφ, rsinφ),ξ_k+1=

re^iφk 1+S

T

,

where

T = ∞ m=0

1 R

_mN

e^−imN2πNα k+mN +

1 R

_N−2k+mN e^{i(N−2k+mN)2πNα} N−k+mN

and

S = ∞ m=0

r R

_mN e^{imN(φ−2πNα)} k+mN +

r R

_N−2k+mN e^{−i(N+mN)(φ−2πNα)} N−k+mN

−T.