Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis

Transcription

1 Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis T. Tran Q. T. Le Gia I. H. Sloan E. P. Stephan Abstract Radial basis functions are used to define approximate solutions to boundary integral equations on the unit sphere. These equations arise from the integral reformulation of the Laplace equation in the exterior of the sphere, with given Dirichlet or Neumann data, and a vanishing condition at infinity. Error estimates are proved. Numerical results supporting the theoretical results are presented. 1 Introduction Throughout this paper we denote by S the unit sphere in R 3, i.e., S := {x R 3 : x = 1}, and by B e the exterior of the sphere, i.e., B e := {x R 3 : x > 1}, where x denotes the Euclidean norm in R 3. We consider the Laplace equation with either a Dirichlet boundary condition or else a Neumann boundary condition U = 0 in B e, (1.1) U = U D on S, (1.2) ν U = Z N on S, (1.3) where ν = / ν denotes differentiation in the direction of the outward unit normal ν, and the vanishing condition at infinity for both the Dirichlet and Neumann cases is U(x) = O(1/ x ) as x. (1.4) School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia. Thanh.Tran@unsw.edu.au, qlegia@maths.unsw.edu.au, I.Sloan@unsw.edu.au Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, Germany. stephan@ifam.uni-hannover.de 1

2 The solutions of these problems can be represented in terms of spherical harmonics. A spherical harmonic of order l on S is the restriction to S of a homogeneous harmonic polynomial of degree l in R 3. The space of all spherical harmonics of order l is the eigenspace of the Laplace-Beltrami operator S corresponding to the eigenvalue λ l = l(l + 1). The dimension of this space being 2l + 1 (see e.g. [7, page 4]), one may choose for it an orthonormal basis {Y l,m } l m= l. The collection of all the spherical harmonics Y l,m, m = l,..., l and l = 0, 1,..., forms an orthonormal basis for L 2 (S). Let (r, θ, ϕ) be the spherical coordinates of a point x = (x 1, x 2, x 3 ) R 3, where r = x, and θ and ϕ are the polar and azimuthal angles, so that x 1 = r sin θ cos ϕ, x 2 = r sin θ sin ϕ, x 3 = r cos θ. For any function v L 2 (S), its associated Fourier series, v = v l,m Y l,m (θ, ϕ), where v l,m = S v(θ, ϕ)y l,m (θ, ϕ)dσ, (1.5) converges in L 2 (S). Here dσ is the element of surface area. For more details on spherical harmonics, the reader is referred to Müller s book [7]. It is well-known that if the Dirichlet data U D has an expansion as a sum of spherical harmonics U D (θ, ϕ) = (U D ) l,m Y l,m (θ, ϕ), then (see [9, Theorem 2.5.1]) the Dirichlet problem (1.1), (1.2) and (1.4) has the unique solution 1 U(r, θ, ϕ) = (UD ) r l+1 l,m Y l,m (θ, ϕ). (1.6) Similarly, if Z N (θ, ϕ) = (Z N ) l,m Y l,m (θ, ϕ), then (see [9, Theorem 2.5.2]) the Neumann problem (1.1), (1.3) and (1.4) has the unique solution U(r, θ, ϕ) = 1 (l + 1)r l+1 (ZN ) l,m Y l,m (θ, ϕ). (1.7) The spherical harmonic basis functions in (1.6), (1.7) are global. In contrast, in this paper we intend to use spherical basis functions obtained from compactly supported radial basis functions, which are better able to capture local properties of the solutions. Our intent is to consider the above boundary value problems as 2

3 prototypes for more complicated problems for which explicit solutions may not be available. We shall propose a solution process in which the boundary value problems are reformulated in terms of boundary integral equations on S, the solutions of which are then approximated by spherical basis functions. In the next section we introduce the integral reformulation of the boundary value problems (1.1) (1.4) into boundary integral equations. Section 3 discusses the use of radial basis functions to define the finite dimensional subspace used in the approximation. The key result of the paper (Theorem 3.7) presents the approximation property in different Sobolev norms (including non-integral and negative norms) of this subspace. This result is employed in Section 4 to prove the convergence of the approximation schemes. Numerical experiments are presented in Section 5 which support the theoretical results. 2 Boundary integral equations and weak formulations The single-layer potential S and the double-layer potential D are defined by S v(x) = 1 1 v(y) 4π x y dσ y and Dv(x) = 1 v(y) 1 4π ν y x y dσ y, S for x B e. Associated with these potentials, we define the following boundary integral operators Sv(x) = 1 1 v(y) 4π S x y dσ y Dv(x) = 1 v(y) 1 4π S ν y x y dσ y D v(x) = 1 1 v(y) 4π ν x S x y dσ y Nv(x) = 1 v(y) 1 4π ν x ν y x y dσ y, for x S. It is well-known that for all s R, the mappings S : H s 1/2 (S) H s+1/2 (S), D : H s+1/2 (S) H s+1/2 (S), S S N : H s+1/2 (S) H s 1/2 (S), D : H s 1/2 (S) H s 1/2 (S) are bounded operators. (If we replace S by Γ, the boundary of a Lipschitz domain in R 3, then the above operators are bounded when s ( 1/2, 1/2).) The traces and normal derivatives on S of S and D are given by (see e.g. [4, 9], noting that the limits are taken from the exterior of S) (S v) S = Sv and ν (S v) = 1 2 v + D v, if v H 1/2 (S), 3

4 and (Dv) S = 1 2 v + Dv and ν(dv) = Nv if v H 1/2 (S). If U H 1 loc (B e) satisfies (1.1) and (1.4), then using the single-layer and doublelayer potentials, and Green s theorem we can represent U as (see [6, Theorems 7.12 & 8.9]) U = D(U S ) S ( ν U) in B e, (2.1) allowing us to compute U from a knowledge of both U S and ν U. In fact, by taking the trace on both sides of (2.1) we obtain, after rearranging the equation, S( ν U) = 1 2 U S + D(U S ) on S. Similarly, by taking the normal derivative of both sides of (2.1) we find N(U S ) = 1 2 νu + D ( ν U) on S. Therefore, the Dirichlet problem (1.1), (1.2) and (1.4) is equivalent to Sz = f on S, where f = 1 2 U D + DU D, (2.2) and the Neumann problem (1.1), (1.3) and (1.4) is equivalent to Nu = g on S, where g = 1 2 Z N + D Z N. (2.3) Due to (2.1), the solution U of the Dirichlet problem can be computed from the solution z of (2.2) by U = DU D S z, and the solution of the Neumann problem can be computed from the solution u of (2.3) by U = Du S Z N. Equation (2.2) is a weakly singular integral equation and equation (2.3) is a hypersingular integral equation. In the following we will design efficient algorithms to solve these equations. We note that S and N are pseudo-differential operators of order 1 and 1, respectively. They have the following representations in terms of spherical harmonics (see [9, page 122]): Sv = Nv = 1 2l + 1 v l,my l,m, (2.4) 4 l(l + 1) 2l + 1 v l,my l,m. (2.5)

5 In the remaining of this section we shall define weak solutions to (2.2) and (2.3). For s R, the Sobolev space H s (S) is defined as usual (see e.g. [9]) with norm and Hermitian product given by ( ) 1/2 v s := (l + 1) 2s v l,m 2 (2.6) and We note that and v s1 = v, w s := (l + 1) 2s v l,m ŵ l,m. v, w s v s w s v, w H s (S), s R, (2.7) v, w s 1 +s 2 2 sup w H s 2 (S) w s2 v H s 1 (S), s 1, s 2 R. (2.8) w 0 It is well-known [6, 9] that S : H 1/2 (S) H 1/2 (S) and N : H 1/2 (S)/R H 1/2 (S) are bijective, implying that (or see (2.4) and (2.5)) (2.2) has a unique solution for all f H 1/2 (S), and (2.3) has a unique solution up to a constant for all g H 1/2 (S). Defining the bilinear forms a S (v, w) := Sv, w 0 v, w H 1/2 (S) and a N (v, w) := Nv, w 0 v, w H 1/2 (S), we seek weak solutions to equations (2.2) and (2.3) respectively as follows: and u H 1/2 (S) : We note from (2.4) and (2.5) that z H 1/2 (S) : a S (z, v) = f, v 0 v H 1/2 (S), (2.9) S u(x) dσ x = 0 and a N (u, v) = g, v 0 v H 1/2 (S). (2.10) a S (v, v) v 2 1/2 v H 1/2 (S) and a N (v, v) v 2 1/2 v H 1/2 (S)/R. (2.11) In the following we shall approximate the solutions of the above equations by using spherical basis functions. These functions are defined via positive definite kernels. 5

6 3 Approximation by spherical basis functions The finite dimensional subspaces that we shall use in our approximation are defined by positive definite kernels on S and spherical basis functions. 3.1 Positive definite kernels and native spaces A continuous function Φ : S S C is called a positive definite kernel on S if it satisfies (i) Φ(x, y) = Φ(y, x) for all x, y S; (ii) for every set of distinct points {x 1,..., x M } on S, the M M matrix A with entries A i,j = Φ(x i, x j ) is positive semi-definite. If the matrix A is positive definite then Φ is called a strictly positive definite kernel; see [12, 16]. We shall define the kernel Φ in terms of a univariate function φ : [ 1, 1] R, Φ(x, y) = φ(x y) x, y S. If φ has a series expansion in terms of Legendre polynomials P l, where φ(t) = 1 4π then due to the addition formula [7, 9] m= l the kernel Φ can be represented as (2l + 1) φ(l)p l (t), (3.1) l=0 1 φ(l) = 2π φ(t)p l (t) dt, (3.2) 1 Y l,m (x)y l,m (y) = 2l + 1 4π P l(x y) x, y S, (3.3) Φ(x, y) = φ(l) Y l,m (x)y l,m (y). (3.4) This kernel is called a zonal kernel. In [2], a complete characterisation of strictly positive definite kernels is established: the kernel Φ is strictly positive definite if and only if φ(l) 0 for all l 0, and φ(l) > 0 for infinitely many even values of l and infinitely many odd values of l; see also [12] and [16]. In the following we shall assume that φ(l) > 0 for all l 0. 6

7 The native space associated with φ is defined by N φ := {v D (S) : v 2 φ = v l,m 2 φ(l) < }, where D (S) is the space of distributions defined on S. This space is equipped with an inner product and a norm defined by v, w φ = v l,m ŵ l,m φ(l) and v φ = ( ) 1/2 v l,m 2. φ(l) If the coefficients φ(l) for l = 0, 1,... satisfy c 1 (l + 1) 2τ φ(l) c 2 (l + 1) 2τ (3.5) for some positive constants c 1 and c 2, and some τ R, then the native space N φ can be identified with the Sobolev space H τ (S), and the corresponding norms are equivalent. In particular, if τ > 1 then the series (3.1) converges pointwise and N φ C(S), which is essentially the Sobolev embedding theorem. 3.2 Spherical basis functions and approximation properties Let = {x 1,..., x M } be a set of data points on the sphere. Two important parameters characterising the set are the mesh norm h and separation radius q, defined by h := sup y S min θ(x i, y) and q := 1 1 i M 2 min θ(x i, x j ), i j where θ(x, y) := cos 1 (x y). The spherical basis functions Φ i, i = 1,..., M, associated with and the kernel Φ are defined by Φ i (x) := Φ(x, x i ) = φ(l)y l,m (x i )Y l,m (x). (3.6) We note that if (3.5) holds then Φ i H s (S) for all s satisfying s < 2τ 1. Let V φ := span{φ 1,..., Φ M }. (3.7) In the following we will assume that (3.5) holds for some τ > 1 so that V φ N φ = H τ (S) C(S). We now study the approximation property of V φ as a subspace of Sobolev spaces. The following lemma shows the boundedness of the interpolation operator in the native space. 7

8 Lemma 3.1 The interpolation operator I : C(S) V φ defined by I v(x j ) = v(x j ), j = 1,..., M, v C(S), (3.8) is well-defined, and is a bounded operator in N φ. Proof. Finding I v satisfying (3.8) entails, on writing I v = M i=0 c iφ i, solving the system M c i Φ i (x j ) = v(x j ), j = 1,..., M. i=1 This has a unique solution due to the positive definiteness of the matrix (Φ i (x j )) M i,j=1. Moreover, for any continuous function v there holds v, Φ j φ = so that (3.8) is equivalent to v l,m φ(l)yl,m (x j ) φ(l) = v(x j ), I v, Φ j φ = v, Φ j φ, j = 1,..., M. Hence implying I v v 2 φ = I v v, I v v φ = I v v, v φ I v v φ v φ, I v v φ v φ v N φ. (3.9) This proves that I is a bounded operator in N φ. The following proposition is a consequence of a non-trivial result proved in [5]. Proposition 3.2 For any σ > 1, if w H σ (S) satisfies w := {x 1,..., x M } S then the following estimate holds where c may depend on t. w t ch σ t w σ, 0 t σ, Proof. It is proved in [5, Theorem 3.3] that w = 0 implies w 0 ch σ w σ. = 0 for some set This inequality together with w σ w σ and a standard interpolation argument yields the required estimate. (For this interpolation argument, the reader is referred to e.g. [1, 4, 6]. A simple proof for the case of Sobolev spaces on the sphere can be found in [3].) A direct consequence of the above proposition and (3.9) is the following error estimate when v H τ (S) is approximated by its interpolant I v. 8

9 Corollary 3.3 Assume that (3.5) holds for some τ > 1. If v H τ (S) then for any t [0, τ] there holds I v v t ch τ t v τ. Proof. This is a result of Proposition 3.2 (taking w = I v v), of (3.9) and the equivalence of the native space norm and H τ -norm. When v is smoother, the error bound can be improved using the technique developed in [11] and [15], which is modified in [13] for hybrid approximation using radial basis functions and polynomials. The crucial tool is the isomorphism defined in the following lemma. Lemma 3.4 Assume that (3.5) holds for some τ > 0. For s R, let T : H s τ (S) H s+τ (S) be defined by T ψ := φ(l) ψ l,m Y l,m. Then (i) T is an isomorphism satisfying (ii) For any ψ H τ (S) and ξ N φ there holds T ψ s+τ ψ s τ ψ H s τ (S); (3.10) T ψ, ξ φ = ψ, ξ 0, (3.11) i.e., T is the adjoint of the embedding operator of the native space N φ H 0 (S); into Proof. It is clear that, under the assumption (3.5), T is well-defined for all ψ H s τ (S), and has an inverse given by T 1 η = η l,m φ(l) Y l,m. Properties (3.10) (3.11) follow from (3.5), the definitions of various inner products and Sobolev norms, noting that (T ψ) l,m = φ(l) ψ l,m. With the help of the above lemma, the error bound for the interpolation can be improved. Proposition 3.5 Assume that (3.5) holds for some τ > 1. For any s, t R satisfying 0 t τ s 2τ, if v H s (S) then the following estimate holds min{s t, 2(τ t)} I v v t Ch v s. 9

10 Proof. Consider first the case when v H 2τ (S). By Lemma 3.4, we can write v as v = T ψ for some ψ H 0 (S). It follows from Proposition 3.2 and the equivalence of the H τ -norm and φ -norm that I v v 2 0 ch 2τ I v v 2 φ = ch 2τ I v v, I v v φ. Using the orthogonality property of the projection I we have I v v, I v v φ = v I v, v φ, and therefore, with the help of (3.10) and (3.11) (taking s = τ), implying I v v 2 0 ch 2τ v I v, T ψ φ ch 2τ v I v, ψ 0 ch 2τ I v v 0 ψ 0 ch 2τ I v v 0 v 2τ, I v v 0 ch 2τ v 2τ. (3.12) Now assume that v H s (S), τ s 2τ. We can write s as s = (1 θ)τ + 2θτ (3.13) for some θ [0, 1]. More precisely, θ = (s τ)/τ. t = t/(1 θ) to have If 0 t 2τ s we define Corollary 3.3 gives 0 t τ and t = (1 θ)t + 0 θ. (3.14) I v v t ch τ t v τ. By using interpolation, we deduce from the above estimate and (3.12), noting (3.13) and (3.14), I v v t c ( h 2τ ) θ (h τ t We note that s t 2(τ t) for t [0, 2τ s]. If 2τ s < t τ, we write ) 1 θ v s = ch τ+θτ t v s = ch s t v s. (3.15) t = (1 θ )τ + θ (2τ s), where θ = (τ t)/(s τ). By interpolation there holds I v v t I v v θ 2τ s I v v 1 θ τ. 10

11 It follows from Corollary 3.3, noting v τ v s, that I v v τ c v s. Therefore, with the help of (3.15) (with t = 2τ s) we infer I v v t ch 2θ (s τ) v s = ch 2(τ t) v s. We finish the proof by noting that 2(τ t) < s t for 2τ s < t τ. Remark 3.6 We note that the estimate proved in the above proposition is only optimal when 0 t 2τ s and τ s 2τ. The convergence analysis for the approximate solutions to (2.9) and (2.10) requires the approximation property of V φ in a larger range of Sobolev norms than those considered in the above proposition, namely error bounds are sought when the approximated function v belongs to H s (S) for s < τ (e.g. s = 1/2 in the case of (2.10) and s = 1/2 in the case of (2.9)). The following theorem states our most general approximation result for V φ. Theorem 3.7 Assume that (3.5) holds for some τ > 1. For any s, t R satisfying t τ, 2τ s 2τ, and t s, if v H s (S) then there exists η V φ such that v η t Ch µ v s, (3.16) where µ = min{s t, 2(τ t), 2τ + s }, and where the constant C is independent of v and h. Proof. We prove the result by considering different cases of values of s and t. Case 1: 0 t τ s 2τ We note that in this case µ = min{s t, 2(τ t)}, and the result is in fact Proposition 3.5. In the following cases, it is easy to see that s t 2(τ t) and thus µ = min{s t, 2τ + s }. Case 2: 0 t s < τ Let L = 1 h. We define for each v H s (S) a polynomial of degree L by P L v = L v l,m Y l,m. 11

12 With η = I P L v we have v η 2 t 2 v P Lv 2 t + 2 P Lv I P L v 2 t 2 (l + 1) 2t v l,m 2 + ch 2(τ t) P L v 2 τ = 2 l=l+1 m= l l=l+1 m= l + ch 2(τ t) (l + 1) 2(t s) (l + 1) 2s v l,m 2 L l=1 m= l (l + 1) 2(τ s) (l + 1) 2s v l,m 2 c(l + 1) 2(t s) v 2 s + cl2(τ s) h 2(τ t) v 2 s, where in the second step we have used the result given in Case 1. Since L h 1 (L + 1) 1 h, we deduce (3.16) with µ = s t. Case 3: t < 0 s 2τ Let P 0 : H 0 (S) V φ be the projection defined by and P 0 v V φ : P 0v, w 0 = v, w 0 w V φ. Then P 0 v is the best approximation of v from V φ in the H0 -norm, thus the results proved in Cases 1 and 2 give If 2τ t < 0 we have If t < 2τ then P 0 v v 0 ch σ v σ v H σ (S), 0 σ 2τ. v P 0 v, w P 0 v v t = sup 0 v P 0 v, w P 0 w = sup 0 w H t (S) w t w H t (S) w t w P 0 w 0 v P 0 v 0 sup w H t (S) w t ch s v sh t. Case 4: 2τ s < 0 and t s Let P s : H s (S) V φ be defined by P 0 v v t P 0 v v 2τ ch 2τ+s v s. P s v, w s = v, w s w V φ. It is easily seen that P s v v s v s. (3.17) 12

13 If 2(s τ) t 2s so that 0 2s t 2τ then for any w H 2s t (S) the result proved in Case 3 ensures the existence of η w V φ satisfying Using (2.7) and (2.8) we then deduce P s v v t = In particular, we have w η w s ch s t w 2s t. P s v v, w sup s P s v v, w η w = sup s w H 2s t (S) w 2s t w H 2s t (S) w 2s t ch s t P sv v s ch s t v s. P s v v 2s ch s v s. The result for 2s < t s is obtained by interpolation, noting (3.17). If t < 2(s τ) then P s v v t P s v v 2(s τ) ch 2τ s v s. The theorem is proved. The next section details how approximate solutions to (2.9) and (2.10) are sought in finite dimensional spaces of the type defined in this section. 4 Galerkin approximations The finite dimensional space V φ defined in (3.7) depends on the univariate function φ. In this section we shall choose appropriate functions φ for each of the two equations (2.9) and (2.10). 4.1 The hypersingular integral equation For the approximation of (2.10) we use radial basis functions suggested by Wendland [14, page 128]. First we define a smoothing operator I on the space C K [0, ) of continuous functions in [0, ) with compact supports by I : C K [0, ) C K [0, ), Iv(r) = For any non-negative integer m, let { (1 r) m+2, 0 < r 1, ρ m (r) = 0, r > 1, r sv(s) ds, r 0. and ρ m (r) = I m ρ m (r), r 0. 13

14 We define φ N (t) = ρ m ( 2 2t), t [ 1, 1], (4.1) and denote by Φ N i, i = 1,..., M, the corresponding spherical basis functions; see (3.6). These functions are locally supported radial basis functions. Strict positive definiteness is proved in [15, Theorem 9.13]. It is proved in [8, Proposition 4.6] that φ N (l) satisfies (3.5) with τ N = m + 3/2. (4.2) In Figure 1 we plotted φ N (l)/l 2m+3 to observe the asymptotic behaviour of φ N (l) for m = 0, 1, 2, 3, with φ N (l) computed by the MATLAB function quadl which uses an adaptive Lobatto quadrature. Effects of integration errors are seen for l in the hundreds and m = 2, m= m= ratio 11 ratio degree 7 x 105 m= degree 8 x 107 m=3 6 6 ratio 5 ratio degree degree Figure 1: Asymptotic behaviour of φ N (l), degree = l, ratio = φ N (l)/l 2m+3 For given = {x 1,..., x M } S, let V N := V φ N. We will solve (2.10) approximately by solving instead u V N : u (x) dσ x = 0 and a N (u, v ) = g, v 0 v V N. (4.3) S 14

15 The conformity condition V N H 1/2 (S) requires τ N > 3/4, and the approximation property (Theorem 3.7) requires τ N > 1. Both conditions hold for m = 0, 1, 2,.... By using (2.5) and (3.3), we obtain the following formula to compute the entries of the stiffness matrix resulting from equation (4.3): a N (Φ N i, ΦN j ) = l(l + 1) 2l + 1 φ N (l) 2 l=0 = 1 4π m= l Y l,m (x i )Y l,m (x j ) l(l + 1) φ N (l) 2 P l (x i x j ). (4.4) l=0 The right-hand side in (4.3) is computed by using (2.3), noting D = S/2 (see [9, page 122]), as follows: g, Φ N i 0 = = ( ) (Z N ) 2(2l + 1) l,m φn (l)y l,m (x i ) l 2l + 1 (Z N ) l,m φn (l)y l,m (x i ). (4.5) Theorem 3.7 yields the following a priori error estimate. Theorem 4.1 Let φ N be defined by (4.1) for some non-negative integer m, and let τ N = m + 3/2. If u is the solution to (2.10) satisfying u H s (S), 1/2 s 2τ N, and u the solution to (4.3) then min{s 1/2, 2τ 1} u u 1/2 Ch u s. Proof. The condition u(x) dσ x = u (x) dσ x = 0 S S yields û 0,0 = (u ) 0,0 = 0, implying (together with (2.11)) u u 2 1/2 a N(u u, u u ) = a N (u u, u w ) c u u 1/2 u w 1/2 for all w V N. This implies u u 1/2 c inf u w 1/2, w V N and the required estimate is now a consequence of Theorem

16 4.2 The weakly singular integral equation The same family of univariate functions defined in (4.1) can be used to define the finite dimensional subspace in the approximation of (2.9). However, for the purpose of preconditioning to be studied in a future paper, we choose to relate φ S to φ N by (cf. (3.1)) φ S (t) = 1 (2l + 1)(l + 1) φ N (l)p l (t), (4.6) 4π l=0 and denote by Φ S i, i = 1,..., M, the corresponding spherical basis functions. It is clear that φ S (l) satisfies (3.5) with τ S = τ N 1/2 = m + 1; see (4.2). Letting V S := V φ S = span{φs 1,..., ΦS M }, we approximate the solution u of (2.9) by z V S : a S (z, v ) = f, v v V S. (4.7) The resulting stiffness matrix has entries given as (cf. (4.4)) a S (Φ S i, Φ S (l + 1) 2 j ) = 2l + 1 φ N (l) 2 Y l,m (x i )Y l,m (x j ) l=0 = 1 4π m= l (l + 1) 2 φ N (l) 2 P l (x i x j ). l=0 The right-hand side of (4.7) is computed by using (2.2), noting D = S/2 (see [9, page 122]), ( f, Φ S i = 1 ) 2 1 (l + 1) (U D ) 2(2l + 1) l,m φn (l)y l,m (x i ) = (l + 1) 2 2l + 1 (U D ) l,m φn (l)y l,m (x i ). (4.8) A priori error estimates similarly to those in Theorem 4.1 can be proved. Theorem 4.2 Let φ N be defined by (4.1) for some positive integer m, and let τ S = m + 1. If z is the solution to (2.9) satisfying z H s (S), 1/2 s 2τ S, and z the solution to (4.7), then z z 1/2 Ch s+1/2 z s. Proof. We note that approximation property requires m > 0. Since z z 2 1/2 a S (z z, z z ), the proof is similar to that of Theorem 4.1 and is therefore omitted. 16

17 Figure 2: Scattered points in equally partitioned areas 5 Numerical experiments In this section, we present numerical results obtained from experiments with the set of scattered points generated by a simple algorithm [10] which partitions the sphere into equal areas; see Figure 2. We chose the sets of points carefully so that the mesh norms h of different sets are reasonably different in order to easily observe the order of convergence. The sets of points we used have number of points M = 20, 30, 40, 50, 100, 500, and Note that these experiments are purely for observing the order of convergence. In a following paper, we will experiment with sets of real data points obtained by satellites. 5.1 The Neumann problem We solved the exterior Neumann problem (1.1), (1.3) and (1.4) with a boundary data given by Z N (x) = 0.5x 3 1 (1.25 x 3 ) 3/2, 17

18 so that the exact solution is U(x) = 1 x p with p = (0, 0, 0.5). Here x = (x 1, x 2, x 3 ). An approximation to (2.10) will be sought and compared to the exact solution which, due to (2.1) and (2.3), is given by u = U S. The spherical basis functions Φ N i, i = 1,..., M, are defined by (3.6) using the univariate function φ N given by (4.1) with m = 0, 1, 2. The coefficients φ N (l), l = 1,..., 500, are computed by the MATLAB function quadl which uses an adaptive Lobatto quadrature. The errors are approximated by u u 1/2 ( 500 l=1 m= l By using the identity (see e.g. [9, page 20]) ) 1/2 (l + 1) (u ) l,m û l,m 2. (1 2r cos θ + r 2 ) 1/2 = r l P l (cos θ), r < 1, l=0 we can easily obtain û l,m = 1 2 l Y l,m(n), n = (0, 0, 1). On the other hand, (u ) l,m = φ N (l) M c i Y l,m (x i ). i=1 Due to the addition formula (3.3), there holds (l + 1) (u ) l,m û l,m 2 = m= l (l + 1)(2l + 1) 4π φ N (l) 2 l 1 M i=1 ( ( φ M N (l)) 2 c i c j P l (x i x j ) c i P l (x i n) l It is expected from our theoretical result (Theorem 4.1) that the order of convergence for the H 1/2 -norm of the error is 2(m+1). The estimated orders of convergence (EOC) shown in Tables 1 3 appear to agree with our theoretical results, even though the orders seem to drop for large values of M when m = 2. i,j=1 ). 18

19 5.2 The Dirichlet problem We also solved the exterior Dirichlet problem (1.1), (1.2) and (1.4) with boundary data 1 U D (x) = (1.25 x 3 ). 1/2 The exact solution is given by U(x) = 1 x p with p = (0, 0, 0.5), and hence, due to (2.1) and (2.2), the approximate solution to (2.9) is compared to z(x) = ν U(x) = 1 + x p x p 3 = (0.5x 3 1) (1.25 x 3 ) 3/2. The spherical basis functions Φ S i, i = 1,..., M, are defined by (3.6) with φ S given by (4.6) where φ N is defined as in Subsection 5.1. The errors are approximated by z z 1/2 ( 500 l=1 m= l (z ) 1/2 ) l,m ẑ l,m 2. l + 1 The coefficients in the spherical harmonics expansion of the exact solution z are and of the approximate solution z are The addition formula yields m= l (z ) l,m ẑ l,m 2 l + 1 ẑ l,m = l l Y l,m (n), (z ) l,m = (l + 1) φ N (l) = (l + 1)(2l + 1) 4π + φ N (l) 2 l 1 M i=1 M c i Y l,m (x i ). i=1 ( ( φ M N (l)) 2 c i c j P l (x i x j ) c i P l (x i n) l Our theoretical result (Theorem 4.2) requires m > 0 and an order of convergence of 2m + 5/2 is shown in the H 1/2 -norm. We carried out the experiment for m = 0, 1, 2, and observed some agreement between the estimated orders of convergence and our theoretical result (Tables 4 6). The case when m = 0 is the borderline case; the theory holds for all positive real values of m. 19 i,j=1 ).

20 Acknowledgement This work was initiated during Stephan s visit to the University of New South Wales, and further developed during Sloan s and Tran s visits to the University of Hannover. The support of the School of Mathematics and Statistics (UNSW) and IFAM (University of Hannover) is acknowledged, as too is the support of the Australian Research Council under its Centres of Excellence Program. References [1] J. Bergh and J. Löfström. Interpolation Spaces: An Introduction. Springer- Verlag, Berlin, [2] D. Chen, V. A. Menegatto, and. Sun. A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 131 (2003), [3] K. Hesse and I. H. Sloan. Optimal lower bounds for cubature error on the sphere S 2. J. Complexity, 21 (2005), [4] R. Kress. Linear Integral Equations. Springer-Verlag, New York, [5] Q. T. LeGia, F. J. Narcowich, J. D. Ward, and H. Wendland. Continuous and discrete least-square approximation by radial basis functions on spheres. J. Approx. Theory, 143 (2006), [6] W. McLean. Strongly Elliptic Systems and Boundary Integral Equations. CUP, Cambridge, [7] C. Müller. Spherical Harmonics, volume 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, [8] F. J. Narcowich and J. D. Ward. Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal., 33 (2002), [9] J.-C. Nédélec. Acoustic and Electromagnetic Equations. Springer-Verlag, New York, [10] E. B. Saff and A. B. J. Kuijlaars. Distributing many points on a sphere. Math. Intelligencer, 19 (1997), [11] R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68 (1999),

21 [12] I. J. Schoenberg. Positive definite function on spheres. Duke Math. J., 9 (1942), [13] I. H. Sloan and A. Sommariva. Approximation on the sphere using radial basis functions plus polynomials. Advances in Comput. Math., (to appear). [14] H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Comp. Math., 4 (1995), [15] H. Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, [16] Y. u and E. W. Cheney. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116 (1992),

22 M h u u 1/2 EOC Table 1: Errors in the H 1/2 -norm with m = 0 M h u u 1/2 EOC Table 2: Errors in the H 1/2 -norm with m = 1 M h u u 1/2 EOC Table 3: Errors in the H 1/2 -norm with m = 2 22

23 M h u u 1/2 EOC Table 4: Errors in the H 1/2 -norm with m = 0 M h u u 1/2 EOC Table 5: Errors in the H 1/2 -norm with m = 1 M h u u 1/2 EOC Table 6: Errors in the H 1/2 -norm with m = 2 23