COLLOCATION SOLUTIONS TO PSEUDODIFFERENTIAL EQUATIONS OF NEGATIVE ORDERS ON THE SPHERE USING SPHERICAL RADIAL BASIS FUNCTIONS T. D. PHAM AND T. TRAN Abstract. Spherical radial basis functions are used to define approximate solutions to pseudodifferential equations of negative orders on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the collocation method. A salient feature of our approach in this paper is a simple error analysis for the collocation method using the same argument as that for the Galerkin method.
1. Introduction Pseudodifferential equations on the sphere have important applications in geodesy, oceanography, and meteorology [3, 10]. Demands for efficient solutions to these equations are increasing when more and more global data are collected by satellites. In this paper, we study the use of spherical radial basis functions to find approximate solutions to pseudodifferential equations of negative orders. The use of spherical radial basis functions results in meshless methods which, over the past few years, become more popular [12, 13]. These methods are alternative to finite-element methods. The advantage of spherical radial basis functions is in the construction of the finite-dimensional subspace; it is independent of the dimension of the geometry. In particular, when scattered data are given, they can be used as centres to define the spherical radial basis functions without resorting to interpolation. 1
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T. D. PHAM AND T. TRAN
Collocation solutions to pseudodifferential equations on the sphere by using spherical radial basis functions have been studied by Morton and Neamtu [5]. Error bounds have later been improved by Morton [4]. These papers emphasise operators of positive orders, and the crux of the analysis therein is the transformation of the collocation problem to a Lagrange or Hermite interpolation problem. In this paper, we shall consider the collocation method for solving pseudodifferential equations of negative orders. These equations arise for example when one solves, in the exterior of the sphere, the Dirichlet problem with the Laplacian; see [3, 10]. The novelty of our work compared to [4, 5] is not only at the negativity of the orders of the operators, but also at the analysis of the error of the approximation. It is well-known that in general the collocation method is easier to implement but elicits a more complicated analysis than the Galerkin method. Since the spherical radial basis functions to be used in the approximation are defined from a reproducing kernel, we can view the collocation method as the Galerkin method applied to another pseudodifferential operator. Therefore, we can use the same error analysis for the Galerkin method to study errors in the collocation approximation. In Section 2 we introduce the definition of pseudodifferential operators in terms of spherical harmonics and describe in detail the problem to be solved. Section 3 is devoted to the introduction of spherical radial basis functions and their approximation properties. The main result of the paper is presented in Section 4. Throughout this paper, C denotes a generic positive constant which may take different values at different occurrences.
2. The problem In this paper we denote by S the unit sphere in R3 , i.e., S := {x ∈ R3 : kxk = 1}. For s ∈ R, the Sobolev space H s on S is defined as
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usual, see e.g. [7], s
0
H := {v ∈ D (S) :
∞ X ` X
(` + 1)2s |b v`,m |2 < ∞},
`=0 m=−`
where D0 (S) is the space of distributions on S. The space H s is equipped with the following norm and inner product: ∞ X ` X
kvks :=
!1/2 (` + 1)2s |b v`,m |2
`=0 m=−`
(2.1) hv, wis :=
∞ X ` X
b`,m . (` + 1)2s vb`,m w
`=0 m=−`
When s = 0 we write h·, ·i instead of h·, ·i0 , which is in fact the L2 -inner product on S. We note that (2.2)
| hv, wis | ≤ kvks kwks
∀v, w ∈ H s , ∀s ∈ R,
and (2.3)
kvks = sup
hv, wi s+t
w∈H t w6=0
2
kwkt
∀v ∈ H s , ∀s, t ∈ R.
A pseudodifferential operator L on the sphere is defined by (2.4)
Lv =
∞ X ` X
b v`,m Y`,m L(`)b
`=0 m=−`
for all v ∈ D0 (S) where, for ` = 0, 1, 2, . . . and m = −`, . . . , `, the functions Y`,m are spherical harmonics of degree ` (see e.g., [6]), and Z vb`,m = v(x) Y`,m (x) ds S
are the Fourier coefficients of v with respect to spherical harmonics. b The sequence (L(`)) `≥0 is referred to as the spherical symbol of L. b = 0}. Then Let K(L) := {` : L(`) ker L = span{Y`,m : ` ∈ K(L), m = −`, ..., `}. Denoting M := dim ker L, we assume that 0 < M < ∞.
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T. D. PHAM AND T. TRAN
The operator L is said to be of order 2α for some α ∈ R if there exist positive constants C1 and C2 such that (2.5)
b ≤ C2 (` + 1)2α C1 (` + 1)2α ≤ L(`)
∀` ∈ / K(L).
Under this assumption one can prove that L : H s+α → H s−α is a bounded operator for all s ∈ R. The following pseudodifferential operators are commonly seen; see [7]. (i) The Laplace-Beltrami operator is an operator of order 2 and b has as symbol L(`) = `(` + 1). This operator is the restriction of the Laplacian on the sphere. (ii) The hypersingular integral operator (without the minus sign) is b = `(`+1)/(2`+1). an operator of order 1 and has as symbol L(`) This operator arises from the boundary-integral reformulation of the Neumann problem with the Laplacian in the interior or exterior of the sphere. (iii) The weakly-singluar integral operator is an operator of order −1 b = 1/(2` + 1). This operator arises from and has as symbol L(`) the boundary-integral reformulation of the Dirichlet problem with the Laplacian in the interior or exterior of the sphere. The problem we are solving in this paper is posed as follows. Problem: Let L be a pseudodifferential operator of order 2α with α ≤ 0. Given, for some σ ∈ R, g ∈ H σ−α satisfying gb`,m = 0 for all ` ∈ K(L) and m = −`, . . . , `, find u ∈ H σ+α satisfying Lu = g, (2.6) µi (u) = ai ,
i = 1, . . . , M,
where, for i = 1, . . . , M , ai ∈ R are given numbers and µi ∈ (H σ+α )∗ , i.e., µi are continuous linear functionals on H σ+α . These functionals are assumed to be unisolvent with respect to ker L, namely, for every v ∈ ker L, if µi (v) = 0 for i = 1, . . . , M , then v = 0.
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The following lemma which is proved in [5] yields the existence and uniqueness of the solution to (2.6). Lemma 2.1. There exists a unique solution u ∈ H σ+α to (2.6), which can be written as (2.7)
u = u0 + u1
where u0 ∈ ker L
and
u1 ∈ (ker L)⊥ H σ+α .
Here, (ker L)⊥ H σ+α is the orthogonal complement of ker L as a subspace of the Hilbert space H σ+α . Moreover, u1 satisfies Lu1 = g in H σ−α , and in particular (2.8)
hLu1 , vi = hg, vi
∀v ∈ H α−σ .
In the following we will find an approximate solution to (2.6) in terms of spherical radial basis functions using the collocation method.
3. Spherical radial basis functions Spherical radial basis functions are defined from positive-definite kernels. 3.1. Positive-definite kernels. 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 {x1 , . . . , xN } on S, the N × N matrix A with entries Ai,j = Φ(xi , xj ) is positive-semidefinite. If the matrix A is positive-definite then Φ is called a strictly positivedefinite kernel; see [9, 14]. We define the kernel Φ in terms of a univariate function φ : [−1, 1] → R: Φ(x, y) = φ(x · y) ∀x, y ∈ S.
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T. D. PHAM AND T. TRAN
Assume that φ has a series expansion in terms of Legendre polynomials P` , ∞ 1 X b (2` + 1)φ(`)P φ(t) = ` (t), 4π `=0
(3.1) where
1
Z b = 2π φ(`)
φ(t)P` (t) dt. −1
By using the well-known addition formula for spherical harmonics [6, 7], (3.2)
` X
Y`,m (x) Y`,m (y) =
m=−`
2` + 1 P` (x · y) ∀x, y ∈ S, 4π
we can write (3.3)
Φ(x, y) =
∞ X `=0
b φ(`)
` X
Y`,m (x)Y`,m (y).
m=−`
Remark 3.1. The kernel Φ is called a zonal kernel. In [2], a complete characterisation of strictly positive definite kernels is established: the b ≥ 0 for all ` ≥ 0, kernel Φ is strictly positive definite if and only if φ(`) b > 0 for infinitely many even values of ` and infinitely many and φ(`) odd values of `; see also [9] and [14]. b > 0 for all ` ≥ 0, so that In the following we shall assume that φ(`) the kernel Φ is strictly positive-definite. The native space associated with φ is defined by ∞ X ` X |b v`,m |2 < ∞}. Nφ := {v ∈ D (S) : b φ(`) 0
`=0 m=−`
This space is equipped with an inner product and a norm defined by (3.4) ∞ X ` X b`,m vb`,m w hv, wiφ = b φ(`)
and kvkφ =
`=0 m=−`
∞ X ` X |b v`,m |2 b φ(`) `=0 m=−`
b for ` = 0, 1, . . . satisfy If the coefficients φ(`) (3.5)
b ≤ c2 (` + 1)−2τ c1 (` + 1)−2τ ≤ φ(`)
!1/2 .
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for some positive constants c1 and c2 , and some τ ∈ R, then the native space Nφ can be identified with the Sobolev space H τ , and the corresponding norms are equivalent. In particular, if τ > 1 then the series (3.1) converges pointwise and Nφ ⊂ C(S), which is essentially the Sobolev embedding theorem. It is noted that for any v ∈ C(S) there holds
(3.6)
∞ X ` X b vb`,m φ(`)Y `,m (x) = hv, Φ(·, x)iφ v(x) = b φ(`)
∀x ∈ S.
`=0 m=−`
Therefore, Φ is a reproducing kernel of the Hilbert space Nφ . In the next section we shall use the reproducing kernel Φ to define the spherical radial basis functions, which in turn define our finitedimensional subspace for the approximation. 3.2. Spherical radial basis functions and approximation properties. Let X = {x1 , . . . , xN } be a set of data points on S. An important parameter characterising the set X is the mesh norm hX defined by hX := sup min cos−1 (xj · y). y∈S 1≤j≤N
The spherical radial basis functions Φj , j = 1, . . . , N , associated with X and the kernel Φ are defined by (3.7)
Φj (x) := Φ(x, xj ) =
∞ X ` X
b φ(`)Y `,m (xj )Y`,m (x),
`=0 m=−`
so that d b (Φ j )`,m = φ(`) Y`,m (xj ),
` = 0, 1, 2, . . . , m = −`, . . . , `.
We note that if (3.5) holds then (3.8)
Φj ∈ H s
∀s < 2τ − 1.
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T. D. PHAM AND T. TRAN
Moreover, for any continuous function v defined on S, it follows from (3.6) that v(xj ) = hv, Φj iφ ,
(3.9)
j = 1, . . . , N.
The finite-dimensional subspace to be used in our approximation is now defined by VXφ := span{Φ1 , . . . , ΦN }. We note that if τ > 1, then VXφ ⊂ Nφ = H τ ⊂ C(S). We recall here the approximation property of VXφ as a subset of Sobolev spaces. Proposition 3.2 (Theorem 3.7 in [11]). Assume that (3.5) holds for some τ > 1. For any s, t ∈ R satisfying t ≤ s ≤ 2τ and t ≤ τ , if v ∈ H s then there exists η ∈ VXφ such that kv − ηkt ≤ ChνX kvks ,
(3.10)
where ν = min{s − t, 2(τ − t), 2τ + |s|}, and where the constant C is independent of v and hX . An approximation to (2.6) wil be sought in the space VXφ , as described in the next section. 4. Collocation method 4.1. Approximate solutions. Recalling (3.8), we assume (4.1)
τ>
σ+α+1 , 2
so that VXφ ⊂ H σ+α . We shall seek (cf. Lemma 2.1) an approximate solution u e ∈ H σ+α in the form (4.2)
u e=u e0 + u e1
where u e0 ∈ ker L and u e1 ∈ VXφ .
We note that Le u1 ∈ H σ−α and recall that g ∈ H σ−α . If σ > α + 1, then Le u1 and g are continuous functions. Therefore, we can find the
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component u e1 ∈ VXφ by solving the equations (4.3)
Le u1 (xj ) = g(xj ),
By writing u e1 =
PN
i=1
j = 1, . . . , N.
αi Φi we obtain from (4.3) a matrix equation
with an N × N matrix A having entries given by Ai,j = LΦi (xj ) =
∞ X ` X
b Y`,m (xi ) Y`,m (xj ) b φ(`) L(`)
`=0 m=−`
=
1 4π
∞ X
b b φ(`)P (2` + 1)L(`) ` (xi · xj ),
`=0
where in the final step we used (3.2). The matrix A is positive-definite due to the strict positive-definiteness of the kernel Ψ defined by Ψ(x, y) :=
∞ X
` X
b b φ(`) L(`)
`=0
Y`,m (x) Y`,m (y);
m=−`
see Remark 3.1. Therefore, there exists a unique solution u e1 to (4.3). Having found u e1 , we find u e0 ∈ ker L by solving the equations (cf. (2.6)) (4.4)
µi (e u0 ) = ai − µi (e u1 ),
i = 1, . . . , M,
so that µi (e u) = µi (u) for i = 1, . . . , M . The unisolvency assumption assures the unique existence of u e0 . In the remainder, we shall convert the collocation equation (4.3) into a Galerkin equation, and use the known error analysis for Galerkin methods to estimate the error u − u e. 4.2. Error analysis. Recalling (3.9) we can rewrite (4.3) as (4.5)
hLe u1 , Φj iφ = hg, Φj iφ ,
j = 1, . . . , N.
In order to see that the above equation is a Galerkin equation, we introduce a new pseudodifferential operator Lφ and a new function
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T. D. PHAM AND T. TRAN
gφ ∈ H σ−α−2τ :
Lφ v :=
∞ X ` X b L(`) vb`,m Y`,m b φ(`)
and gφ :=
`=0 m=−`
∞ X ` X gb`,m Y`,m . b φ(`) `=0 m=−`
b b The operator Lφ has as symbol L(`)/ φ(`), and is of order 2(τ + α) due to (2.5) and (3.5). Therefore, Lφ : H τ +α → H −τ −α is bounded. Using Lφ and gφ , we can rewrite equation (4.5) as
(4.6)
hLφ u e1 , Φj i = hgφ , Φj i ,
j = 1, . . . , N.
The exact solution u1 defined in Lemma 2.1 satisfies a similar equation as shown in the following lemma. Lemma 4.1. Assume that g ∈ H σ−α with σ ≥ τ . Let u1 be given as in Lemma 2.1. Then
(4.7)
hLφ u1 , vi = hgφ , vi
∀ v ∈ H τ +α .
Proof. First we note that gφ ∈ H −τ −α under the assumption σ ≥ τ . On the other hand, Lφ maps H τ +α to H −τ −α . Therefore, equation (4.7) is well-defined. Since σ ≥ τ , there holds H α−τ ⊂ H α−σ , which together with (2.8) implies
(4.8)
hLu1 , wi = hg, wi
∀w ∈ H α−τ .
For any v ∈ H τ +α let ∞ X ` X vb`,m V := Y`,m . b φ(`) `=0 m=−`
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Then V ∈ H α−τ . By using (4.8), we obtain hLφ u1 , vi =
=
∞ X ` X b L(`) d (u b`,m 1 )`,m v b φ(`) `=0 m=−` ∞ X ` X
d b (u b L(`) 1 )`,m V`,m
`=0 m=−`
= hLu1 , V i = hg, V i ∞ X ` X gb`,m vb`,m = = hgφ , vi , b φ(`) `=0 m=−`
finishing the proof of the lemma.
Assumptions (4.1) and σ ≥ τ imply VXφ ⊂ H σ+α ⊂ H τ +α . Due to (4.6) and (4.7), u e1 is a Galerkin solution to the equation Lφ u1 = gφ , with the energy space being H τ +α , namely, hLφ (u1 − u e1 ), vi = 0 ∀v ∈ VXφ .
(4.9)
It is noted that in general VXφ 6⊆ (ker L)⊥ e can be H τ +α . However, u rewritten in a form similar to (2.7) as follows. Let (4.10) u∗0 := u e0 +
` X X `∈K(L) m=−`
d (e u1 )`,m Y`,m
and u∗1 :=
` X X
d (e u1 )`,m Y`,m .
m=−` `∈K(L) /
Then (4.11)
u e = u∗0 + u∗1
with u∗0 ∈ ker L and u∗1 ∈ (ker L)⊥ H τ +α .
It should be noted that u∗1 may not belong to VXφ . Comparing (2.7) and (4.11) suggests that ku − u ekτ +α can be estimated by estimating ku0 − u∗0 kτ +α and ku1 − u∗1 kτ +α . It turns out that an estimate for the latter is sufficient, as given in the following two lemmas.
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T. D. PHAM AND T. TRAN
Lemma 4.2. Assume that σ ≥ τ . If the functionals µi , i = 1, . . . , M , are bounded in H τ +α , then ku0 − u∗0 kτ +α ≤ Cku1 − u∗1 kτ +α , where u0 , u1 , u∗0 and u∗1 are defined by (2.7) and (4.10). Proof. For i = 1, . . . , M , it follows from the linearity of µi and µi (u) = µi (e u), see (4.4), that µi (u0 ) + µi (u1 ) = µi (u∗0 ) + µi (u∗1 ), implying µi (u0 ) − µi (u∗0 ) = µi (u∗1 ) − µi (u1 ). Therefore, |µi (u0 − u∗0 )| = |µi (u1 − u∗1 )| ≤ kµi kku1 − u∗1 kτ +α , where kµi k is the operator norm of µi with respect to the H τ +α -norm. This result holds for all i = 1, . . . , M , implying ku0 − u∗0 kµ ≤ M ku1 − u∗1 kτ +α , where M := maxi=1,...,M kµi k, and where, for any v ∈ ker L, kvkµ := max |µi (v)| . i=1,...,M
(The unisolvency assumption assures us that the norm k·kµ is welldefined. Indeed, if kvkµ = 0 for some v ∈ ker L, then the unisolvency of µi confirms that v = 0.) The subspace ker L being finite dimensional (so that all norms are equivalent), we deduce ku0 − u∗0 kτ +α ≤ C ku1 − u∗1 kτ +α , proving the lemma.
Lemma 4.3. Under the assumptions of Lemma 4.2 there holds ku − u ekτ +α ≤ Cku1 − u∗1 kτ +α .
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Proof. The norm ku − u ekτ +α can be rewritten as, noting (2.7) and (4.11),
ku −
u ek2τ +α
` X
X
=
c (` + 1)2(τ +α) |b u`,m − (e u)`,m |2
`∈K(L) m=−`
+
X
` X
c (` + 1)2(τ +α) |b u`,m − (e u)`,m |2
m=−` `∈K(L) / ` X
X
=
2 ∗ d d (` + 1)2(τ +α) |(u 0 )`,m − (u0 )`,m |
`∈K(L) m=−`
+
X
` X
2 ∗ d d (` + 1)2(τ +α) |(u 1 )`,m − (u1 )`,m |
m=−` `∈K(L) /
= ku0 − u∗0 k2τ +α + ku1 − u∗1 k2τ +α . The required result now follows from Lemma 4.2.
In the following, the notation a ' b means c1 a ≤ b ≤ c2 a, where c1 and c2 are positive constants independent of the variables in concern. The following simple results are often used in the remainder of the paper. Lemma 4.4. If v ∈ H τ +α satisfies vb`,m = 0 for all ` ∈ K(L) and m = −`, . . . , `, then (4.12)
hLφ v, vi ' kvk2τ +α .
Moreover, if v, w ∈ H τ +α , then (4.13)
| hLφ v, wi | ≤ Ckvkτ +α kwkτ +α .
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T. D. PHAM AND T. TRAN
Proof. Let v ∈ H τ +α be such that vb`,m = 0 for all ` ∈ K(L) and for all m = −` . . . `. Noting (2.5) and (3.5), we have hLφ v, vi =
X
` X b L(`) |b v`,m |2 b φ(`)
m=−` `∈K(L) /
'
X
` X
(` + 1)2(τ +α) |b v`,m |2
m=−` `∈K(L) /
= kvk2τ +α , proving (4.12). Inequality (4.13) can be proved similarly by using (2.5) and (3.5), and the Cauchy-Schwarz inequality.
The following lemma is technically C´ea’s Lemma; see e.g., [1]. Lemma 4.5. Under the assumptions of Lemma 4.2, there holds ku1 − u∗1 kτ +α ≤ Cku1 − vkτ +α
∀v ∈ VXφ .
Proof. It follows from (4.10) that (4.14)
hLφ w, u∗1 i = hLφ w, u e1 i
∀w ∈ H τ +α ,
and from (4.10) and (4.9) that (4.15)
hLφ (u1 − u∗1 ), vi = 0 ∀v ∈ VXφ .
Due to (4.12) there holds ku1 − u∗1 k2τ +α ' hLφ (u1 − u∗1 ), u1 − u∗1 i = hLφ (u1 − u∗1 ), u1 i − hLφ (u1 − u∗1 ), u∗1 i , which together with (4.14) implies ku1 − u∗1 k2τ +α ' hLφ (u1 − u∗1 ), u1 i − hLφ (u1 − u∗1 ), u e1 i .
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By noting that u e1 ∈ VXφ and using (4.15) we deduce, for any v ∈ VXφ , ku1 − u∗1 k2τ +α ' hLφ (u1 − u∗1 ), u1 i = hLφ (u1 − u∗1 ), u1 i − hLφ (u1 − u∗1 ), vi = hLφ (u1 − u∗1 ), u1 − vi . It follows from (4.13) that ku1 − u∗1 k2τ +α ≤ C ku1 − u∗1 kτ +α ku1 − vkτ +α . The required result now follows by cancelling similar terms.
By using Lemmas 4.3 and 4.5, and Proposition 3.2 we can now prove the main result of the paper. Theorem 4.6. Let (3.5) and (4.1) be satisfied. Assume that 1 < τ ≤ σ. Assume further that u ∈ H s for some s ∈ R satisfying τ + α ≤ s ≤ 2τ . If µi ∈ (H τ +α )∗ for i = 1, . . . , M , then (4.16)
ku − u ekτ +α ≤ ChνX1 kuks ,
where ν1 = min{s − τ − α, −2α, 2τ + |s|}. Moreover, if µi ∈ (H t )∗ , i = 1, . . . , M , for some t ∈ R satisfying 2α ≤ t ≤ τ + α, then (4.17)
ν1 +ν2 ku − u ekt ≤ ChX kuks ,
where ν2 = min{τ + α − t, −2α, 2τ + |2τ + 2α − t|}. The constant C is independent of hX . Proof. We first prove (4.16). Since τ + α ≤ τ and τ + α ≤ s ≤ 2τ , Proposition 3.2 assures us that there exists v ∈ VXφ satisfying ku1 − vkτ +α ≤ ChνX1 ku1 ks , where ν1 = min{s − τ − α, −2α, 2τ + |s|}. It follows from Lemma 4.5 and the above inequality that (4.18)
ku1 − u∗1 kτ +α ≤ ChνX1 ku1 ks .
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T. D. PHAM AND T. TRAN
This inequality, Lemma 4.3, and the inequality ku1 ks ≤ kuks imply (4.16). Estimate (4.17) can be obtained by using the well-known duality argument. We include the proof here for completeness. Under the assumption µi ∈ (H t )∗ for i = 1, . . . , M , we can use the same arguments as in the proofs of Lemmas 4.2 and 4.3 to obtain ku − u ekt ≤ C ku1 − u∗1 kt . By using (2.3), and noting that the order of Lφ is 2(τ + α), we deduce from the above inequality ku − u ekt ≤ C
sup v∈H 2τ +2α−t v6=0
≤C
sup v∈H 2τ +2α−t v6=0
hu1 − u∗1 , viτ +α kvk2τ +2α−t hLφ (u1 − u∗1 ), vi . kvk2τ +2α−t
It follows from (4.15) and (4.13) that, for any η ∈ VXφ , ku − u ekt ≤ C
sup v∈H 2τ +2α−t v6=0
hLφ (u1 − u∗1 ), v − ηi kvk2τ +2α−t
≤ Cku1 − u∗1 kα+τ
sup v∈H 2τ +2α−t v6=0
kv − ηkα+τ . kvk2τ +2α−t
By using Proposition 3.2, noting that 2τ + 2α − t ≤ 2τ due to 2α ≤ t and τ + α ≤ 2τ + 2α − t due to t ≤ τ + α, we can choose η ∈ VXφ such that sup v∈H 2τ +2α−t v6=0
kv − ηkα+τ ≤ ChνX2 . kvk2τ +2α−t
The desired result now follows from (4.18) and the inequality ku1 ks ≤ kuks .
5. Conclusions
We presented a collocation solution to pseudodifferential equations of negative orders on the sphere, using spherical radial basis functions.
PSEUDODIFFERENTIAL EQUATIONS WITH RBFS
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The analysis follows that of the Galerkin method thanks to an observation that the collocation equation can be rewritten as a Galerkin equation. Numerical results for the case of an operator of order −1 can be found in a forthcoming paper [8]. Acknowledgements The first author is supported by the University International Postgraduate Award offered by the University of New South Wales. References [1] S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer, Berlin, 2002. [2] D. Chen, V. A. Menegatto, and X. Sun. A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 131 (2003), 2733–2740. [3] W. Freeden, T. Gervens, and M. Schreiner. Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford, 1998. [4] T. M. Morton. Improved error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels. In Trends in approximation theory (Nashville, TN, 2000), Innov. Appl. Math., pages 317–326. Vanderbilt Univ. Press, Nashville, TN, 2001. [5] T. M. Morton and M. Neamtu. Error bounds for solving pseudodifferential equations on spheres. J. Approx. Theory, 114 (2002), 242–268. [6] C. M¨ uller. Spherical Harmonics, volume 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1966. [7] J.-C. N´ed´elec. Acoustic and Electromagnetic Equations. Springer-Verlag, New York, 2000. [8] T. D. Pham, T. Tran, and Q. T. Le Gia. Numerical solutions to a boundaryintegral equation with spherical radial basis functions. In Proceedings of the 14th Biennial Computational Techniques and Applications Conference, CTAC2008, G. N. Mercer and A. J. Roberts, editors, ANZIAM J., 2008. In review. [9] I. J. Schoenberg. Positive definite function on spheres. Duke Math. J., 9 (1942), 96–108.
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[10] S. Svensson. Pseudodifferential operators – a new approach to the boundary problems of physical geodesy. Manuscr. Geod., 8 (1983), 1–40. [11] T. Tran, Q. T. Le Gia, I. H. Sloan, and E. P. Stephan. Boundary integral equations on the sphere with radial basis functions: Error analysis. Technical Report AMR07/11, School of Mathematics and Statistics, The University of New South Wales, 2007, (to appear in Appl. Numer. Math.) http://www.maths.unsw.edu.au/applied/pubs/apppreprints2007.html. [12] H. Wendland. Meshless Galerkin methods using radial basis functions. Math. Comp., 68 (1999), 1521–1531. [13] H. Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2005. [14] Y. Xu and E. W. Cheney. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116 (1992), 977–981.
School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia. mailto:
[email protected] School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia. mailto:
[email protected] 
