PSEUDODIFFERENTIAL EQUATIONS ON THE SPHERE WITH

Mathematical Models and Methods in Applied Sciences Vol. 21, No. 9 (2011) 1933
1959 # .c World Scienti¯c Publishing Company DOI: 10.1142/S021820251100560X

PSEUDODIFFERENTIAL EQUATIONS ON THE SPHERE WITH SPHERICAL SPLINES

T. D. PHAM* and T. TRAN† School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia *[email protected] † [email protected] A. CHERNOV Hausdor® Center for Mathematics, Universit€ at Bonn, Endenicher Allee 60, 53155 Bonn, Germany [email protected] Received 29 January 2010 Revised 20 January 2011 Communicated by F. Brezzi Spherical splines are used to de¯ne approximate solutions to strongly elliptic pseudodi®erential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by using Galerkin method. We prove optimal convergence (in Sobolev norms) of the approximate solution by spherical splines to the exact solution. Our numerical results underlie the theoretical result. Keywords: Pseudodi®erential equation; sphere; spherical spline. AMS Subject Classi¯cation: 65N30, 65N38, 65N15

1. Introduction Pseudodi®erential operators have long been used13,16 as a modern and powerful tool to tackle linear boundary-value problems. Svensson27 introduces this approach to geodesists who study11,12 these problems on the sphere which is taken as a model of the earth. E±cient solutions to pseudodi®erential equations on the sphere become more demanding when given data are collected by satellites. In this paper, we study the use of spherical splines to ¯nd approximate solutions to these equations. For spherical data interpolation and approximation, tensor products of univariate splines are not a good choice since data locations are not usually spaced over a regular grid. Spherical radial basis functions seem to be a better choice28; however, the resulting matrix systems from this approximation are very ill-conditioned. 1933

1934

T. D. Pham, T. Tran & A. Chernov

Spherical splines1
3 appear to have many properties in common with classical polynomial splines over planar triangulations and are well suited for scattered data interpolation and approximation problems. Baramidze and Lai5 use these functions to solve the Laplace
Beltrami equation on the unit sphere in R 3 . There are advantages in using spherical splines, compared to other ¯nite element methods where polynomials de¯ned on a reference planar triangle are mapped to spherical triangles. Firstly, the use of barycentric coordinates facilitates the computation of basis functions, as in the case of classical splines (see Sec. 6). Secondly, by using spherical splines one can write the approximate solution in a simple form of a linear combination of piecewise homogeneous polynomials. Another advantage is the ability to control smoothness of a function and its derivatives across edges of the triangulation (see Ref. 1). In this paper, we solve strongly elliptic pseudodi®erential equations on the sphere by the Galerkin method using spherical splines. This class of equations includes the Laplace
Beltrami equation, Stokes equation, weakly singular integral equations and many others.11,12,27 Our main result is an optimal convergence rate of the approximation. The key of the analysis is proving the approximation property of spherical splines as a subset of Sobolev spaces. Since the pseudodi®erential operators to be studied can be of any real order, it is necessary to obtain an approximation property in Sobolev norms of real orders, negative and positive (see Theorem 4.1). In the implementation of the method, it is important to compute e±ciently integrals over spherical triangles. In general, this integral is transformed into an integral over the planar triangle with vertices ð0; 0Þ, ð1; 0Þ and ð0; 1Þ; see Sec. 6.1. When solving the weakly singular integral equation, one has to compute double integrals with integrands being singular functions of the form f ðxÞgðyÞ=jjx
yjj where f and g are smooth functions. Integrals of this type appear e.g. in the context of Boundary Element Methods. In Sec. 6.2.2 we describe techniques to deal with this type of singularities. The paper is organized as follows. In Sec. 2 we de¯ne Sobolev spaces and the pseudodi®erential equations to be studied. The de¯nition of spherical splines is recalled in Sec. 3. In Sec. 4 we prove the approximation property of spherical splines as a subset of Sobolev spaces. Section 5 presents the main result of the Galerkin scheme. Our numerical results are reported in Sec. 6. Throughout this paper, C is a generic constant which may take di®erent value at di®erent occurrence. 2. Preliminaries 2.1. Sobolev spaces Throughout this paper, we denote by S the unit sphere in R 3 , i.e. S :¼ fx 2 R 3 : jjxjj ¼ 1g where jj  jj is the Euclidean norm in R 3 . A spherical harmonic of order ‘ on S is the restriction to S of a homogeneous harmonic polynomial of degree ‘ in R 3 . The space of all spherical harmonics of order ‘ is the eigenspace of the Laplace
Beltrami

Pseudodi®erential Equations on the Sphere with Spherical Splines

1935

operator
S corresponding to the eigenvalue
‘ ¼
‘ð‘ þ 1Þ. The dimension of this space being 2‘ þ 1 (see, e.g. Ref. 20), one may choose for it an orthonormal basis fY‘;m g ‘m¼
‘ . The collection of all the spherical harmonics Y‘;m , m ¼
‘; . . . ; ‘ and ‘ ¼ 0; 1; . . ., forms an orthonormal basis for L2 ðSÞ. For s 2 R, the Sobolev space H s is de¯ned as usual by ( ) 1 X ‘ X s 0 2s 2 H :¼ v 2 D ðSÞ : ð‘ þ 1Þ j^ v ‘;m j < 1 ; ‘¼0 m¼
‘

where D 0 ðSÞ is the space of distributions on S and v^‘;m are the Fourier coe±cients of v, Z v^‘;m ¼ vðxÞY‘;m ðxÞdx : S

Here dx is the element of surface area. The space H s is equipped with the following norm and inner product: ! 1=2 1 X ‘ X 2s 2 jjvjjs :¼ ð‘ þ 1Þ j^ v ‘;m j ð2:1Þ ‘¼0 m¼
‘

and hv; wis :¼

1 X ‘ X ‘¼0 m¼
‘

^ ‘;m : ð‘ þ 1Þ 2s v^‘;m w

When s ¼ 0 we write h; i instead of h; i0 ; this is in fact the L2 -inner product. We note that jhv; wis j  jjvjjs jjwjjs

8 v; w 2 H s ; 8 s 2 R

ð2:2Þ

8 v 2 H s1 ; 8 s1 ; s2 2 R:

ð2:3Þ

and jjvjjs1 ¼ sup

w2H s2 w6¼0

hv; wi s1 þs2 2

jjwjjs2

In the case k belongs to the set of non-negative integers Z þ , the Sobolev space H k can also be de¯ned by using atlas for the unit sphere S; see Ref. 21. This construction uses a concept of homogeneous extensions. De¯nition 2.1. Given any spherical function v, its homogeneous extension of degree n 2 Z þ to R 3 nf0g is a function vn de¯ned by
 x n vn ðxÞ :¼ jjxjj v : jjxjj Let fðj ; j Þg Jj¼1 be an atlas for S, i.e. a ¯nite collection of charts ðj ; j Þ, where j are open subsets of S, covering S, and where j : j ! Bj are in¯nitely di®erentiable mappings whose inverses 
1 j are also in¯nitely di®erentiable. Here Bj , j ¼ 1; . . . ; J , 2 are open subsets in R . Also, let fj g Jj¼1 be a partition of unity subordinate to the atlas fðj ; j Þg Jj¼1 , i.e. a set of in¯nitely di®erentiable functions j on S vanishing

1936

T. D. Pham, T. Tran & A. Chernov

P outside the sets j , such that Jj¼1 j ¼ 1 on S. For any k 2 Z þ , the Sobolev space H k on the unit sphere is de¯ned as follows: k H k :¼ fv : ðj vÞ  
1 j 2 H ðBj Þ; j ¼ 1; . . . ; J g;

ð2:4Þ

which is equipped with a norm de¯ned by jjvjj k :¼

J X

jjðj vÞ  
1 j jjH k ðBj Þ :

ð2:5Þ

j¼1

This norm is equivalent to the norm de¯ned in (2.1); see Ref. 18. Let v 2 H k , k  1. For each l ¼ 1; . . . ; k, let vl
1 denote the unique homogeneous extension of v of degree l
1. Then X jvjl :¼ jjD  vl
1 jjL2 ðSÞ ð2:6Þ jj¼l

is a Sobolev-type seminorm of v in H k . Here jjD  vl
1 jjL2 ðSÞ is understood as the L2 -norm of the restriction of the trivariate function D  vl
1 to S. When l ¼ 0 we de¯ne jvj0 :¼ jjvjjL2 ðSÞ ; which can now be used together with (2.6) to de¯ne another norm in H k : jjvjj 0k :¼

k X

jvjl :

ð2:7Þ

l¼0

This norm turns out to be equivalent to the Sobolev norm de¯ned in (2.5); see Ref. 21. 2.2. Pseudodi®erential operators ^ Let fLð‘Þg ‘0 be a sequence of real numbers. A pseudodi®erential operator L is a linear operator that assigns to any v 2 D 0 ðSÞ a distribution Lv :¼

1 X ‘ X ‘¼0 m¼
‘

^ v ‘;m Y‘;m : Lð‘Þ^

ð2:8Þ

^ The sequence fLð‘Þg ‘0 is referred to as the spherical symbol of L. Let ^ ¼ 0g. Then KðLÞ :¼ f‘ : Lð‘Þ ker L ¼ spanfY‘;m : ‘ 2 KðLÞ; m ¼
‘; . . . ; ‘g: Denoting M :¼ dim ker L, we assume that 0  M < 1. In this paper, we assume that L is a strongly elliptic pseudodi®erential operator of order 2, i.e. ^  C2 ð‘ þ 1Þ 2 C1 ð‘ þ 1Þ 2  Lð‘Þ

for all ‘ 62 KðLÞ;

ð2:9Þ

for some positive constants C1 and C2 . More general pseudodi®erential operators can be de¯ned via Fourier transforms by using local charts; see, e.g. Refs. 15 and 22. It can be easily seen that if L is a pseudodi®erential operator of order 2 then L : H sþ ! H s
 is bounded for all s 2 R.

Pseudodi®erential Equations on the Sphere with Spherical Splines

1937

The following commonly seen pseudodi®erential operators are strongly elliptic; see Ref. 27. .

The Laplace–Beltrami operator is an operator of order 2 and has as symbol ^ ¼ ‘ð‘ þ 1Þ. This operator is the restriction of the Laplacian on the sphere. Lð‘Þ . The hypersingular integral operator is an operator of order 1 and has as symbol ^ ¼ ‘ð‘ þ 1Þ=ð2‘ þ 1Þ. This operator arises from the boundary-integral reforLð‘Þ mulation of the Neumann problem with the Laplacian in the interior or exterior of the sphere. . The weakly singular integral operator is an operator of order
1 and has as symbol ^ ¼ 1=ð2‘ þ 1Þ. This operator arises from the boundary-integral reformulation Lð‘Þ 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 A. Let L be a pseudodi®erential operator of order 2. Given g 2 H


satisfying g^‘;m ¼ 0 for all ‘ 2 KðLÞ;

m ¼
‘; . . . ; ‘;

ð2:10Þ

¯nd u 2 H  satisfying Lu ¼ g; hi ; ui ¼  i ;

ð2:11Þ

i ¼ 1; . . . ; M ;

where  i 2 R and i 2 H
 are given. Here h; i denotes the duality inner product between H
 and H  , which coincides with the H 0 -inner product when i and u belong to H 0 . Problem A is uniquely solvable under the following assumption. Assumption B. The functionals 1 ; . . . ; N are assumed to be unisolvent with respect to ker L, i.e. for any v 2 ker L if hi ; vi ¼ 0 for all i ¼ 1; . . . ; M , then v ¼ 0. The following theorem is proved in Ref. 19. We include the proof here for completeness. Theorem 2.1. Under Assumption B, Problem A has a unique solution. Proof. Since ker L is a ¯nite-dimensional subspace of H  , we can represent H  as H  ¼ ker L  ðker LÞ ? H ;  where ðker LÞ ? H  is the orthogonal complement of ker L with respect to the H -inner product. Writing the solution u in the form

u ¼ u0 þ u1

where u0 2 ker L and u1 2 ðker LÞ ? H ;

and noting that Ljðker LÞ ?H  is injective, we can de¯ne u1 by u1 ¼ ker L by solving hi ; u0 i ¼  i
hi ; u1 i;

i ¼ 1; . . . ; M :

L
1 g

ð2:12Þ and ¯nd u0 2 ð2:13Þ

1938

T. D. Pham, T. Tran & A. Chernov

Since u0 2 ker L, it can be represented as ‘ X X

u0 ¼

c‘;m Y‘;m :

‘2KðLÞ m¼
‘

Substituting this into (2.13) yields ‘ X X

c‘;m hi ; Y‘;m i ¼  i
hi ; u1 i;

i ¼ 1; . . . ; M :

ð2:14Þ

‘2KðLÞ m¼
‘

Recalling that M ¼ dim ker L, we note that there are M unknowns c‘;m . The unisolvency Assumption B assures us that Eq. (2.13) with zero right-hand side has a unique solution u0 ¼ 0. Therefore, the matrix arising from (2.14) is invertible, which in turn implies unique existence of c‘;m , m ¼
‘; . . . ; ‘ and ‘ 2 KðLÞ. The theorem is proved. We de¯ne a bilinear form að; Þ : H þs  H 
s ! R, for any s 2 R, by aðw; vÞ :¼ hLw; vi

for all w 2 H þs ; v 2 H 
s :

We note that aðu1 ; vÞ ¼ hg; vi

for all v 2 H  :

ð2:15Þ

In the sequel, for any x; y 2 R, x ’ y means that there exist positive constants C1 and C2 satisfying C1 x  y  C2 x. The following simple results are often used in the next sections. Lemma 2.1. Let s be any real number. .

The bilinear form að; Þ : H þs  H 
s ! R is bounded, i.e. jaðw; vÞj  C jjwjjþs jjvjj
s

.

If w; v 2

H s,

for all w 2 H þs ; v 2 H 
s :

then jhLw; vis
 j  C jjwjjs jjvjjs :

.

ð2:16Þ

Assume that L is strongly elliptic. If v 2 ðker

LÞ ? H s,

ð2:17Þ

then

hLv; vis
 ’ jjvjj 2s :

ð2:18Þ

In particular, there holds aðv; vÞ ’ jjvjj 2 for all v 2 ðker LÞ ? H . Here C is a constant independent of v and w. Proof. Let w 2 H þs and v 2 H 
s . Noting (2.9) and using the Cauchy
Schwarz inequality, we have jaðw; vÞj 

1 X ‘ X

^ jLð‘Þjj w^ ‘;m jj^ v ‘;m j  C

‘¼0 m¼
‘ 1 X ‘ X

¼C

‘¼0 m¼
‘

1 X ‘ X ‘¼0 m¼
‘

ð‘ þ 1Þ 2 jw^ ‘;m jj^ v ‘;m j

^ ‘;m jð‘ þ 1Þ 
s j^ ð‘ þ 1Þ þs jw v ‘;m j

Pseudodi®erential Equations on the Sphere with Spherical Splines

C

1 X ‘ X

! 1=2 ð‘ þ 1Þ

2ðþsÞ

‘¼0 m¼
‘

^ ‘;m j jw

2

1 X ‘ X

1939

! 1=2 ð‘ þ 1Þ

‘¼0 m¼
‘

2ð
sÞ

j^ v ‘;m j

2

¼ C jjwjjþs jjvjj
s ; proving (2.16). The proof for (2.17) and (2.18) can be done similarly, noting the de¯nition (2.9) of strongly elliptic operators, and noting that v 2 ðker LÞ ? H s if and s only if v 2 H and v^‘;m ¼ 0 for all ‘ 2 KðLÞ and m ¼
‘; . . . ; ‘. In the next section, we shall de¯ne ¯nite-dimensional subspaces in which approximate solutions are sought for.

3. Approximation Subspaces The ¯nite-dimensional subspace to be used in the approximation will be de¯ned from spherical splines. In this section we review several basic concepts and present some properties of spherical splines. More details can be found in Refs. 1
3. Let fv1 ; v2 ; v3 g be linearly independent vectors in R 3 . The triheron T generated by fv1 ; v2 ; v3 g is de¯ned by T :¼ fv 2 R 3 : v ¼ b1 v1 þ b2 v2 þ b3 v3 with bi  0; i ¼ 1; 2; 3g: The intersection  :¼ T \ S is called a spherical triangle. For each v 2 , where  is a spherical triangle having vertices v1 , v2 , v3 , there exist uniquely b1 ðvÞ, b2 ðvÞ, b3 ðvÞ satisfying v ¼ b1 ðvÞv1 þ b2 ðvÞv2 þ b3 ðvÞv3 ;

ð3:1Þ

which are called the spherical barycentric coordinates of v with respect to . Let
¼ f i : i ¼ 1; . . . ; T g be a set of spherical triangles. If
satis¯es ST . i¼1  i ¼ S, . each pair of distinct triangles in
is either disjoint or share a common vertex or an edge, then
is called a spherical triangulation of the sphere S. Given X ¼ fx1 ; . . . ; xN g a set of points on S, we can form a spherical triangulation
which contains triangles whose vertices are elements of X. In the experiments (Sec. 6), we used the FORTRAN package STRIPACK (http://www.netlib.org/ toms/77) to obtain the spherical triangulation
from the set X. Given non-negative integers r and d, the set of spherical splines of degree d and smoothness r associated with
is de¯ned by S dr ð
Þ :¼ fs 2 C r ðSÞ : sj 2 P d ;  2
g:

1940

T. D. Pham, T. Tran & A. Chernov

Here, P d is the space of restrictions to S of homogeneous polynomials of degree d in R 3 . If  has vertices v1 , v2 , v3 , then sj can be written as X d;  sj ðvÞ ¼ c ijk B ijk ðvÞ; v 2 ; iþjþk¼d  where the coe±cients c ijk are real numbers and the functions d; B ijk ðvÞ :¼

d! i b ðvÞb j2 ðvÞb k3 ðvÞ; i!j!k! 1

i þ j þ k ¼ d;

are called the spherical Bernstein
Bezier basis polynomials of degree d relative to . Here, bi ðvÞ, i ¼ 1; 2; 3, are given by (3.1). For more details, see Ref. 1. In this paper, we assume that the integers d and r de¯ning S dr ð
Þ satisfy  d  3r þ 2 if r  1; ð3:2Þ d1 if r ¼ 0: We now use the de¯nition of Sobolev spaces by local charts (see (2.4)) to show the following necessary result. Proposition 3.1. Let
be a spherical triangulation on the unit sphere S. Assume that (3.2) holds. There holds S dr ð
Þ H rþ1 : Proof. We prove the result for r ¼ 0 and use induction to obtain the result for r > 0. Here we use a speci¯c atlas of charts to de¯ne the Sobolev space H 1 . We denote by n ¼ ð0; 0; 1Þ and s ¼ ð0; 0;
1Þ the north and south poles of S, respectively. For any point x 2 S and R > 0, we denote the spherical cap centered at x and having radius R by C ðx; RÞ, i.e. C ðx; RÞ :¼ fy 2 S : cos
1 ðx  yÞ  Rg: The interior of C ðx; RÞ is denoted by

C o ðx; RÞ.

U1 ¼ C o ðn; 0 Þ and

ð3:3Þ

Let

U2 ¼ C o ðs; 0 Þ;

with 0 2 ð =2; 2 =3Þ. Assume that
is ¯ne enough such that S admits a simple cover S ¼ 1 [ 2 , where [ j ¼ ; j ¼ 1; 2:  Uj

The stereographic projection 1 from the punctured sphere Snfng onto R 2 is de¯ned as a mapping that maps x 2 Snfng to the intersection of the equatorial hyperplane fz ¼ 0g and the extended line that passes through x and n. The stereographic projection 2 based on s can be de¯ned similarly. The scaled projections 1 :¼

1 tanð0 =2Þ

2 j 1

and

2 :¼

1 tanð0 =2Þ

1 j 2

Pseudodi®erential Equations on the Sphere with Spherical Splines

1941

map 1 and 2 , respectively, onto the unit ball of R 2 . It is clear that fj ; j g 2j¼1 is a C 1 atlas for S. Let Bj ¼ j ðj Þ, j ¼ 1; 2, and let fj : S ! Rg 2j¼1 be the partition of unity subordinate to fðj ; j Þg 2j¼1 ; see Sec. 2. For any v 2 S d0 ð
Þ, it su±ces to show that (see (2.4)) 1 wj :¼ ðj vÞ  
1 j 2 H ðBj Þ for j ¼ 1; 2:

This holds because wj is continuous on Bj and wj j belongs to H 1 ðÞ, where f :¼ j ðÞ :  j g forms a partition of Bj ; see p. 38 of Ref. 8. 4. Approximation Property Before studying the approximation property of spherical splines, we introduce the concept of quasiuniform triangulation on S. For any spherical triangle , we denote by jj the diameter of the smallest spherical cap containing , and by  the diameter of the largest spherical cap inside . Here the diameter of a cap is, as usual, twice its radius; see (3.3). We de¯ne j
j :¼ maxfjj;  2
g;


:¼ minf  ;  2
g and

h
:¼ tan

j
j : 2

ð4:1Þ

De¯nition 4.1. Let  be a positive real number. A triangulation
is said to be -quasiuniform provided that j
j  :


In the following we brie°y introduce the construction of a quasi-interpolation operator Q : L2 ðSÞ ! S dr ð
Þ which is introduced in Ref. 21. First we introduce the set of domain points of
to be   [ iv þ jv2 þ kv3 D :¼ ijk ¼ 1 : d iþjþk¼d  ¼hv ;v ;v i2
1

2

3

Here,  ¼ hv1 ; v2 ; v3 i denotes the spherical triangle whose vertices are v1 ; v2 ; v3 . We denote the domain points by 1 ; . . . ; D , where D ¼ dim S d0 ð
Þ. Let fBl : l ¼ 1; . . . ; Dg be a basis for S d0 ð
Þ such that the restriction of Bl on the triangle containing l is the Bernstein
Bezier polynomial of degree d associated with this point, and that Bl vanishes on other triangles. A set M :¼ f l g M set for S dr ð
Þ if, for every l¼1 D is called a minimal determiningP r s 2 S d ð
Þ, all the coe±cients  l ðsÞ in the expression s ¼ D l¼1  l ðsÞBl are uniquely determined by the coe±cients corresponding to the basis functions which are associated with points in M. Given a minimal determining set, we construct a basis r fB l g M l¼1 for S d ð
Þ by requiring  l 0 ðB l Þ ¼ l;l 0 ;

1  l; l 0  M :

1942

T. D. Pham, T. Tran & A. Chernov

By using Hahn
Banach theorem we extend the linear functionals  l , l ¼ 1; . . . ; M , to all of L2 ðSÞ. We continue to use the same symbol for these extensions. The quasi-interpolation operator Q : L2 ðSÞ ! S dr ð
Þ is now de¯ned by Qv :¼

M X

 l ðvÞB l ;

v 2 L2 ðSÞ:

l¼1

The following proposition is a special case of a result established in Theorem 2 in Ref. 4. Proposition 4.1. Assume that
is a -quasiuniform spherical triangulation with j
j  1, and that (3.2) holds. Then for any v 2 H m there holds m
k jv
Qvjk  Ch
jvjm ;

for all k ¼ 0; . . . ; minfm
1; r þ 1g, and  1; 3; . . . ; d þ 1 m¼ 2; 4; . . . ; d þ 1

if d is even; if d is odd:

Here C is a positive constant depending only on d and the smallest angle in
. Remark 4.1. (i) The condition k  r þ 1 is to ensure that Qv 2 H k ; see Proposition 3.1. (ii) Theorem 2 in Ref. 4 proves the result for r  0 and d  3r þ 2. In fact, the result can also be proved when r ¼ 0 and d ¼ 1 by using the same argument. The following proposition uses the above result to show the approximation property of S dr ð
Þ in positive order Sobolev spaces. Proposition 4.2. Assume that
is a -quasiuniform spherical triangulation with j
j  1, and that (3.2) holds. Then for any v 2 H m there exists  2 S dr ð
Þ satisfying m
n jjv
jjn  Ch
jjvjjm ;

ð4:2Þ

where m¼

 1; 2; . . . ; d þ 1 2; 3; . . . ; d þ 1

if d is even; if d is odd;

and where n ¼ 0; 1; . . . ; minfm
1; r þ 1g. Here the positive constant C depends only on d and the smallest angle in
. Proof. Since the three norms de¯ned in (2.1), (2.5) and (2.7) are equivalent, it su±ces to prove the result with the norm jj  jj 0 . Case 1. Consider ¯rst the case when m and d are of di®erent parity, i.e.  1; 3; . . . ; d þ 1 if d is even; m¼ 2; 4; . . . ; d þ 1 if d is odd:

Pseudodi®erential Equations on the Sphere with Spherical Splines

1943

It follows from Proposition 4.1 that, for any v 2 H m , its quasi-interpolant satis¯es m
k jv
Qvjk  Ch
jvjm ;

k ¼ 0; . . . ; n:

By summing over all k ¼ 0; . . . ; n and noting that h
< 1, we obtain m
n jjv
Qvjj 0n  Ch
jvjm :

Since jvjm  jjvjj 0m , we deduce the required inequality. Case 2. Next, we consider the case when m and d have the same parity, i.e.  2; 4; . . . ; d if d is even; m¼ 3; 5; . . . ; d if d is odd: Note that here we can use the result proved in Case 1 for m
1 and m þ 1, but not for m, due to the parity of m when compared to d. Hence, we further consider di®erent sub-cases. Case 2.1. When n  m
2, we have by using the result in Case 1 with n and m
1 m
1
n jjv
Qvjj 0n  Ch
jjvjj 0m
1

8 v 2 H m
1 ;

and then with n and m þ 1 mþ1
n jjv
Qvjj 0n  Ch
jjvjj 0mþ1

8 v 2 H mþ1 :

By using interpolation (see, e.g. Theorem 8.12 in Ref. 17) we obtain the required inequality for v 2 H m . Case 2.2. Next, consider the case n ¼ m
1  r. The result in Case 1 gives (noting that m  2 in this case) jjv
Qvjj 0m
2  Ch
jjvjj 0m
1

8 v 2 H m
1 ;

and jjv
Qvjj 0m  Ch
jjvjj 0mþ1

8 v 2 H mþ1 :

By using interpolation again, we obtain jjv
Qvjj 0m
1  Ch
jjvjj 0m

8 v 2 H m:

Case 2.3. Finally, consider the case n ¼ m
1 ¼ r þ 1. Let Prþ1 be the projection from H rþ1 onto S dr ð
Þ de¯ned by hPrþ1 v; wirþ1 ¼ hv; wirþ1

8 v 2 H rþ1 and w 2 S dr ð
Þ:

It is well known that jjPrþ1 v
vjj 0rþ1 

inf jjw
vjj 0rþ1 ;

w2S dr ð
Þ

so that jjPrþ1 v
vjj 0rþ1  jjvjj 0rþ1

ð4:3Þ

1944

T. D. Pham, T. Tran & A. Chernov

and jjPrþ1 v
vjj 0rþ1  jjQv
vjj 0rþ1 : Using the result proved in Case 2.1 with r þ 1 and r þ 3 we have 2 jjQv
vjj 0rþ1  Ch
jjvjj 0rþ3 ;

implying 2 jjPrþ1 v
vjj 0rþ1  Ch
jjvjj 0rþ3 :

The above inequality and (4.3) yield jjPrþ1 v
vjj 0rþ1  Ch
jjvjj 0rþ2 ; proving the proposition. Remark 4.2. The approximant  to v in the above proposition is the quasiinterpolant Qv (Proposition 4.2), except when r and d have the same parity and n ¼ r þ 1 and m ¼ r þ 2. We believe that this restriction is only a technical di±culty. When solving pseudodi®erential equations of order 2 by the Galerkin method, it is natural to carry out error analysis in the energy space H  . Since the order 2 may be negative (as in the case of the weakly singular integral equation discussed after De¯nition 2.9) it is necessary to show an approximation property of the form (4.2) for a wider range of n and m, including negative real values. The following proposition extends Proposition 4.2 to the case when n and m are non-negative real numbers. It will also relax the condition that n  m
1. Proposition 4.3. Assume that
is a -quasiuniform spherical triangulation with j
j  1, and that (3.2) holds. Then for any v 2 H s , there exists  2 S dr ð
Þ satisfying s
t jjv
jjt  Ch
jjvjjs ;

where 0  t  r þ 1 and t  s  d þ 1. Here C is a positive constant depending only on d and the smallest angle in
. Proof. The proposition will be proved by considering di®erent cases. The main idea of the proof is to use Proposition 4.2 and interpolation. Case 1. t  r þ 1 and s ¼ d þ 1. By using Proposition 4.2 we have for any v 2 H dþ1 dþ1
btc

jjvjjdþ1 ;

dþ1
dte

jjvjjdþ1 :

jjv
Qvjjbtc  Ch
and jjv
Qvjjdte  Ch
By using interpolation we derive

dþ1
t jjv
Qvjjt  Ch
jjvjjdþ1 :

ð4:4Þ

Pseudodi®erential Equations on the Sphere with Spherical Splines

1945

We note that the only case that this argument fails is when dte ¼ r þ 1 and d þ 1 ¼ r þ 2, with d and r having the same parity; see Remark 4.2. This case never happens. Case 2. t  r þ 1 and t  s < d þ 1. Let Pt : H t ! S dr ð
Þ be the projection de¯ned by hPt v; wit ¼ hv; wit

8 v 2 H t and w 2 S dr ð
Þ:

Then jjPt v
vjjt 

inf jjw
vjjt ;

w2S dr ð
Þ

so that jjPt v
vjjt  jjvjjt ;

ð4:5Þ

and jjPt v
vjjt  jjQv
vjjt : Due to (4.4) there holds dþ1
t jjPt v
vjjt  Ch
jjvjjdþ1 :

This inequality and (4.5) yield s
t jjPt v
vjjt  Ch
jjvjjs ;

proving the proposition. The following theorem is our most general result on the approximation property of which includes negative values of t and s.

S dr ð
Þ,

Theorem 4.1. Assume that
is a -quasiuniform spherical triangulation with j
j  1, and that (3.2) holds. Then for any v 2 H s , there exists  2 S dr ð
Þ satisfying s
t jjv
jjt  Ch
jjvjjs ;

where t  r þ 1 and t  s  d þ 1. Here C is a positive constant depending only on d and the smallest angle in
. Proof. When both t and s are non-negative, the result is proved in Proposition 4.3. The remaining cases are considered separately below. Step 1.
d
1  t  0  s  d þ 1. We de¯ne P0 : H 0 ! S dr ð
Þ by hP0 v; wi ¼ hv; wi

8 v 2 H 0 and w 2 S dr ð
Þ:

It is well known that jjP0 v
vjj0 

inf jjv
wjj0 ;

w2S dr ð
Þ

which implies, together with Proposition 4.3, that  jjP0 v
vjj0  h
jjvjj

8 v 2 H ;

0    d þ 1:

1946

T. D. Pham, T. Tran & A. Chernov

Noting (2.3) and using the above estimate with  ¼ s and  ¼
t we obtain jjP0 v
vjjt ¼ sup

w2H
t w6¼0

hP0 v
v; wi hP0 v
v; w
P0 wi ¼ sup jjwjj
t jjwjj
t
t w2H w6¼0

jjw
P0 wjj0  jjP0 v
vjj0 sup jjwjj
t w2H
t w6¼0

s
t  Ch
jjvjjs :

Step 2. 2s
d
1  t  s  0 and
d
1  s. Let Ps : H s ! S dr ð
Þ be de¯ned by hPs v; wis ¼ hv; wis

8 v 2 H s and w 2 S dr ð
Þ:

Then it is obvious that jjPs v
vjjs  jjvjjs

8 v 2 H s:

ð4:6Þ

If 2s
d
1  t  2s, so that 0  2s
t  d þ 1, then the result proved in Step 1 yields s
t jjP0 w
wjjs  Ch
jjwjj2s
t

8 w 2 H 2s
t :

ð4:7Þ

Combining (2.3), (4.6) and (4.7), we have jjPs v
vjjt ¼ sup

2s
t

w2H w6¼0

hPs v
v; wis hPs v
v; w
P0 wis ¼ sup jjwjj2s
t jjwjj2s
t 2s
t w2H w6¼0

jjw
P0 wjjs  jjPs v
vjjs sup jjwjj2s
t 2s
t w2H w6¼0

s
t  Ch
jjPs v
vjjs s
t  Ch
jjvjjs :

In particular, we have
s jjPs v
vjj2s  Ch
jjvjjs :

The result for 2s < t  s is obtained by interpolation, noting (4.6). Step 3.
2ðd þ 1Þ  t 
d
1 and 0  s  d þ 1. (This step extends the result of Step 1.) Let P
d
1 : H
d
1 ! S dr ð
Þ be the projection de¯ned by hP
d
1 v; wi
d
1 ¼ hv; wi
d
1

8 w 2 S dr ð
Þ:

Then P
d
1 v is the best approximation of v from S dr ð
Þ in the H
d
1 -norm. The result in Step 1 gives sþdþ1 jjP
d
1 v
vjj
d
1  Ch
jjvjjs

8 v 2 H s:

ð4:8Þ

Pseudodi®erential Equations on the Sphere with Spherical Splines

1947

Since hP
d
1 v
v; i
d
1 ¼ 0 for all  2 S dr ð
Þ, we have jjP
d
1 v
vjjt ¼

sup w2H
t
2ðdþ1Þ w6¼0

¼

sup w2H
t
2ðdþ1Þ

hP
d
1 v
v; wi
d
1 jjwjj
t
2ðdþ1Þ hP
d
1 v
v; w
i
d
1 jjwjj
t
2ðdþ1Þ

w6¼0

 jjP
d
1 v
vjj
d
1

sup w2H
t
2ðdþ1Þ w6¼0

jjw
jj
d
1 : jjwjj
t
2ðdþ1Þ

ð4:9Þ

Noting that
d
1 
t
2ðd þ 1Þ  0, we can use the result in Step 2 to choose, for any w 2 H
t
2ðdþ1Þ , a function  2 S dr ð
Þ satisfying
t
d
1 jjwjj
t
2ðdþ1Þ : jjw
jj
d
1  Ch


ð4:10Þ

Substituting (4.8) and (4.10) into (4.9), we infer s
t jjP
d
1 v
vjjt  Ch
jjvjjs :

Step 4.
d
1  s  0 and 4s
2ðd þ 1Þ  t  2s
d
1. (This step extends the result of Steps 2 and 3.) De¯ning the projection P2s
d
1 : H 2s
d
1 ! S dr ð
Þ in the same manner as P
d
1 , using the same argument as in Step 3, and invoking the result in Step 2, we obtain the required estimate. The results for the cases t <
2ðd þ 1Þ or s <
d
1 can be obtained similarly by repeating the above arguments. The theorem is proved. In the next section we will use the result developed in this section to estimate the error of the Galerkin approximation. 5. Galerkin Method 5.1. Approximate solution Noting (2.12), we shall seek an approximate solution u~ 2 H þ in the form u~ ¼ u~0 þ u~1

where u~0 2 ker L and u~1 2 S dr ð
Þ:

ð5:1Þ

The solution u~1 will be found by solving the Galerkin equation hL  u~1 ; vi ¼ hf ; vi

8 v 2 S dr ð
Þ;

ð5:2Þ

in which L  is a strongly elliptic pseudodi®erential operator of order 2 whose symbol is given by ( ^ Lð‘Þ if ‘ 62 KðLÞ;  c L ð‘Þ ¼ 2 ð1 þ ‘Þ if ‘ 2 KðLÞ:

1948

T. D. Pham, T. Tran & A. Chernov

Noting (2.9), we have c ð‘Þ ’ ð1 þ ‘Þ 2 L This con¯rms that the bilinear form

a

:

H

8 ‘ 2 N:

 H  ! R de¯ned by

a  ðv; wÞ :¼ hL  v; wi;

v; w 2 H  ;

is continuous and coercive in H  . The well-known Lax
Milgram theorem con¯rms the unique existence of u~1 2 S dr ð
Þ. Having found u~1 , we will ¯nd u~0 2 ker L by solving the equations (cf. (2.11)) hi ; u~0 i ¼  i
hi ; u~1 i;

i ¼ 1; . . . ; M ;

so that hi ; u~i ¼ hi ; ui;

i ¼ 1; . . . ; M :

ð5:3Þ

The unique existence of u~0 follows from Assumption B in exactly the same way as that of u0 ; see Theorem 2.1. ~ can be rewritten in a It is noted that in general S dr ð
Þ µ 6 ðker LÞ ? H  . However, u form similar to (2.12) as follows. Let u 0 :¼ u~0 þ

‘ X X

~1 Þ‘;m Y‘;m ðuc

ð5:4Þ

‘2KðLÞ m¼
‘

and u 1 ¼

‘ X X

~1 Þ‘;m Y‘;m : ðuc

ð5:5Þ

‘62KðLÞ m¼
‘

Then u~ ¼ u 0 þ u 1

with u 0 2 ker L and u 1 2 ðker LÞ ? H : u 1

ð5:6Þ

S dr ð
Þ,

does not belong to and that this It should be noted that, in general, function is introduced purely for analysis purposes. We do not explicitly compute u 1 , nor u 0 . 5.2. Error analysis Assume that the exact solution u and the approximate solution u~ of Problem A belong to H t for some t 2 R, and assume that i 2 H
t for i ¼ 1; . . . ; M . Comparing (2.12) and (5.6) suggests that jju
u~jjt can be estimated by estimating jju0
u 0 jjt and jju1
u 1 jjt . It turns out that an estimate for the latter is su±cient, as shown in the following two lemmas. Lemma 5.1. Let u0 , u1 , u 0 and u 1 be de¯ned by (2.12), (5.4) and (5.5). For i ¼ 1; . . . ; M , if i 2 H
t for some t 2 R, then jju0
u 0 jjt  C jju1
u 1 jjt ; where C is independent of u.

Pseudodi®erential Equations on the Sphere with Spherical Splines

1949

Proof. For i ¼ 1; . . . ; M , it follows from (5.3) that hi ; u0 i þ hi ; u1 i ¼ hi ; u 0 i þ hi ; u 1 i; implying hi ; u0
u 0 i ¼ hi ; u 1
u1 i. Inequality (2.3) yields jhi ; u0
u 0 ij ¼ jhi ; u1
u 1 ij  jji jj
t jju1
u 1 jjt : This result holds for all i ¼ 1; . . . ; M , implying jju0
u 0 jj  Mjju1
u 1 jjt ; where M :¼ maxi¼1;...;M jji jj
t , and jjvjj :¼ maxi¼1;...;M jhi ; vij for all v 2 ker L. (The unisolvency assumption assures us that the above norm is well-de¯ned.) The subspace ker L being ¯nite-dimensional, we deduce jju0
u 0 jjt  C jju1
u 1 jjt ; proving the lemma. Lemma 5.2. Under the assumptions of Lemma 5.1, there holds jju
u~jjt  C jju1
u~1 jjt : Proof. Noting (2.12) and (5.6), the norm jju
u~jjt can be rewritten as jju
u~jj 2t ¼

X

‘ X

d ð‘ þ 1Þ 2t j^ u ‘;m
ð~ uÞ‘;m j 2

‘2KðLÞ m¼
‘ ‘ X X

þ ¼

‘62KðLÞ m¼
‘ ‘ X

X

2  dÞ
ðu d ð‘ þ 1Þ 2t jðu 0 ‘;m 0 Þ‘;m j

‘2KðLÞ m¼
‘ ‘ X X

¼

d ð‘ þ 1Þ 2t j^ u ‘;m
ð~ u Þ‘;m j 2

þ

2  dÞ
ðu d ð‘ þ 1Þ 2t jðu 1 ‘;m 1 Þ‘;m j

‘62KðLÞ m¼
‘ jju0
u 0 jj 2t þ

jju1
u 1 jj 2t :

This, together with the result in Lemma 5.1, implies jju
u~jjt  C jju1
u 1 jjt : Noting that jju1
u 1 jjt  jju1
u~1 jjt , we deduce the required result. We now prove the main theorem of the paper. Theorem 5.1. Assume that
is a -quasiuniform spherical triangulation with j
j  1 and that (3.2) hold. If the order 2 of the pseudodi®erential operator L satis¯es   r þ 1, and if u and u~ satisfy, respectively, (2.15) and (5.1), then s
t jju
u~jjt  Ch
jjujjs ;

1950

T. D. Pham, T. Tran & A. Chernov

where s  d þ 1 and 2
d
1  t  minfs; g. Here C is a positive constant depending only on d and the smallest angle in
. Proof. The proof is standard using Cea's lemma and Nitsche's trick. We include it here for completeness. Noting that u1 2 ðker LÞ ? H  and (2.15), we have a  ðu1 ; vÞ ¼ hg; vi 8 v 2 H  : Cea's lemma gives jju1
vjj ; jju1
u~1 jj  C min r v2S d ð
Þ

which, together with Theorem 4.1, implies that there exists a constant C satisfying s
 jju1
u~1 jj  Ch
jju1 jjs :

Noting that jju1 jjs  jjujjs and the result in Lemma 5.2 we have s
 jjujjs : jju
u~jj  Ch


ð5:7Þ

Now consider t < . It follows from (2.3) and (2.9) that jju1
u~1 jjt  C sup

v2H 2
t v6¼0

hu1
u~1 ; vi a  ðu1
u~1 ; vÞ  C sup : jjvjj2
t jjvjj2
t v2H 2
t v6¼0

By using successively (2.15), (5.2) we deduce that for arbitrary  2 S dr ð
Þ jju1
u~1 jjt  C sup

v2H 2
t v6¼0

a  ðu1
u~1 ; v
Þ jjv
jj  C jju1
u~1 jj sup : jjvjj2
t 2
t jjvjj2
t v2H v6¼0

Since 2
d
1  t < , there holds  < 2
t  d þ 1. By using Theorem 4.1 again, we can choose  2 S dr ð
Þ satisfying 
t jjv
jj  Ch
jjvjj2
t :

Hence, 
t s
t jju1
u~1 jjt  Ch
jju1
u~1 jj  Ch
jjujjs :

Using Lemma 5.2, we obtain s
t jju
u~jjt  Ch
jjujjs :

6. Numerical Experiments In this section, we present the numerical results obtained from our experiments with di®erent sets of points X ¼ fx1 ; x2 ; . . . ; xN g, where xi ¼ ðxi;1 ; xi;2 ; xi;3 Þ, i ¼ 1; . . . ; N , are points on the sphere generated by a simple algorithm23 which partitions the sphere into equal areas. From these sets of data points, we used the FORTRAN package STRIPACK to obtain the spherical triangulations
.

Pseudodi®erential Equations on the Sphere with Spherical Splines

1951

We solved pseudodi®erential equations by using the space of spherical splines S 10 ð
Þ, which is the space of continuous and piecewise homogeneous polynomial of degree 1. A set of basis functions for S 10 ð
Þ is fBi : i ¼ 1; . . . ; N g; where, for each i ¼ 1; . . . ; N , Bi is the nodal basis function corresponding to the vertex xi de¯ned as follows. For any x ¼ ðx1 ; x2 ; x3 Þ 2 S, there exists a spherical triangle  2
containing x. If xi is not a vertex of  then Bi ðxÞ ¼ 0. Otherwise, assume that  is formed by three vertices xi , xj and xk . Then 2 3 0 2 31
1 xi;1 xj;1 xk;1 x1 xj;1 xk;1 Bi ðxÞ ¼ det4 x2 xj;2 xk;2 5  @det4 xi;2 xj;2 xk;2 5A ; ð6:1Þ x3 xj;3 xk;3 xi;3 xj;3 xk;3 see Ref. 1. 6.1. The Laplace
Beltrami equation We solved the equation on the unit sphere of the form Lu ¼ g

on S;

ð6:2Þ

where Lu ¼

S u þ u. This equation arises, for example, when one discretizes in time the di®usion equation on the sphere. We solved (6.2) with gðxÞ ¼ 7x 33
6x3 ðx 12 þ x 22 Þ; so that the exact solution is uðxÞ ¼ x 33 : We carried out the experiment with various numbers of points N , namely N ¼ 101, 200, 500, 1001, 2000, 4000 and 8001. The entry Aij of the sti®ness matrix A is computed by Z Aij ¼ ðrBi ðxÞ  rBj ðxÞ þ Bi ðxÞBj ðxÞÞdx S XZ ¼ ðrBi ðxÞ  rBj ðxÞ þ Bi ðxÞBj ðxÞÞdx ; 2




where, for any i ¼ 1; . . . ; N , Bi ðxÞ is computed by using (6.1) and rBi is the surface gradient of Bi . The right-hand side of the linear system has entries given by Z XZ bi ¼ Bi ðxÞgðxÞdx ¼ Bi ðxÞgðxÞdx ; i ¼ 1; . . . ; N : ð6:3Þ S

2




The integral of a smooth function f over a spherical triangle  with vertices xi ¼ ðxi;1 ; xi;2 ; xi;3 Þ, xj ¼ ðxj;1 ; xj;2 ; xj;3 Þ and xk ¼ ðxk;1 ; xk;2 ; xk;3 Þ is computed as

1952

T. D. Pham, T. Tran & A. Chernov

follows. Let T be the planar triangle (in R 3 ) with the same vertices as , and let T^ be the planar triangle of vertices ð1; 0; 0Þ, ð0; 1; 0Þ and ð0; 0; 1Þ. A point on T^ can be represented as u ¼ ðu1 ; u2 ; 1
u1
u2 Þ where ðu1 ; u2 Þ satis¯es 0  u2  1
u1 and 0  u1  1. It can be shown that  Z Z 1 Z 1
u1
F ðuÞ du2 du1 f ðxÞdx ¼ jdet F j f ; ð6:4Þ jjF ðuÞjj jjF ðuÞjj 3  0 0 where F : T^ ! T is de¯ned by 2 32 3 u1 xi;1 xj;1 xk;1 5: u2 F ðuÞ ¼ 4 xi;2 xj;2 xk;2 54 ð6:5Þ xi;3 xj;3 xk;3 1
u1
u2 Having solved the linear system, we computed the L2 -norm of the error uN
u, where in this section we denoted the approximate solution by uN instead of u~. It is expected 2 from our theoretical result (Theorem 5.1) that jjuN
ujj0 ¼ Oðh
Þ. The estimated orders of convergence (EOC) shown in Table 1 agree with our theoretical result. 6.2. Weakly singular integral equation 6.2.1. The equation We also solved the weakly singular integral equation Lu ¼ g where L is the operator given by LvðxÞ ¼

1 4

on S;

Z vðyÞ S

ð6:6Þ

1 d ; jjx
yjj y

for any v 2 D 0 ðSÞ. This equation is the boundary integral reformulation of the Dirichlet problem for the Laplacian in the exterior or interior of the sphere.11 Note that the weakly singular integral operator L is a pseudodi®erential operator of order
1 with symbol 1=ð2‘ þ 1Þ. Equation (6.6) was solved with gðxÞ ¼ x1 x2 =5; so that the exact solution is uðxÞ ¼ x1 x2 . We solved this equation with various values of N , namely N ¼ 10, 20, 30, 40, 51, 80, 101, 125, 150, 175 and 200. Each entry Aij of the sti®ness matrix A is Table 1. Errors in the L2 -norm for the Laplace
Beltrami equation. N

h


jjuN
ujj0

EOC

101 200 500 1001 2000 4000 8001

0.2618 0.1819 0.1136 0.0798 0.0570 0.0397 0.0283

5.853E
2 2.713E
2 1.093E
2 5.230E
3 2.675E
3 1.316E
3 6.627E
4




2.11 1.93 2.09 1.99 1.96 2.02

Pseudodi®erential Equations on the Sphere with Spherical Splines

given by Aij ¼

1 4

Z Z S

S

Bi ðxÞBj ðyÞ dy dx ; jjx
yjj

i; j 2 f1; . . . ; N g:

1953

ð6:7Þ

Computation of Aij is explained in Sec. 6.2.2. Computation of the right-hand side is analogous to computation of (6.3) and is explained in Sec. 6.1. To check the accuracy of the approximation, we computed the H
1=2 -norm of the error u
uN , which was calculated as follows: jju
uN jj 2
1=2 ’ hLðu
uN Þ; u
uN i ¼ hLðu
uN Þ; ui Z Z ¼ hLu; ui
huN ; Lui ¼ gud
guN d: S

S

The estimated orders of convergence shown in Table 2 do not quite agree with the 5=2 theoretical result, which states that jju
uN jj
1=2 ¼ Oðh
Þ. A reason may be the fact that sets of Sa® points used in this experiment are not nested. Indeed, the numbers shown in Table 3, which were obtained by using nested sets of points, seem to agree with the theoretical result. The starting mesh contains eight equal spherical triangles with six nodes (four on the equator and two at the poles). Every further re¯nement consists of partitioning every spherical triangle into four smaller spherical triangles by joining the midpoints of the edges (red re¯nement). 6.2.2. Computation of the sti®ness matrix with numerical quadrature Computation of elements of the sti®ness matrix (6.7) requires evaluation of integrals of the type Z Z Bi ðxÞBj ðyÞ 1 I ¼ dy dx ; i; j 2 f1; . . . ; N g; ð6:8Þ 4  ð1Þ  ð2Þ jjx
yjj where  ð1Þ ;  ð2Þ are spherical triangles from
and the basis functions Bi ðxÞ and Bj ðyÞ are analytic for all x 2  ð1Þ and y 2  ð2Þ (cf. (6.1)). Due to the nonlocal integral kernel Table 2. Errors in the H
1=2 -norm for the weakly singular integral equation. N

h


jju
uN jj
1=2

EOC

20 30 40 51 80 101 125 150 175 200 226 250 275

0.7432 0.5153 0.4401 0.3757 0.2958 0.2617 0.2316 0.2104 0.1987 0.1819 0.1692 0.1625 0.1538

0.4609E
01 0.1349E
01 0.1110E
01 0.8246E
02 0.4337E
02 0.3072E
02 0.2371E
02 0.1894E
02 0.1602E
02 0.1350E
02 0.1179E
02 0.1046E
02 0.8856E
03




3.36 1.24 1.87 2.69 2.82 2.11 2.35 2.90 1.95 1.87 2.96 3.00

1954

T. D. Pham, T. Tran & A. Chernov Table 3. Errors in the H
1=2 -norm for the weakly singular integral equation with nested triangulations. N

h


jju
uN jj
1=2

EOC

6 18 66 258

1.5707 0.7853 0.3926 0.1963

4.0933E
01 6.2962E
02 7.6483E
03 1.2607E
03

0.00 2.70 3.04 2.60

jjx
yjj
1 the integral value I is in general di®erent from zero, even if Bi and Bj have disjoint supports. As a consequence, the sti®ness matrix (6.7) is densely populated. We computed (6.8) approximately by a numerical quadrature. Note that an accurate numerical approximation of (6.8) becomes a challenging task if  ð1Þ and  ð2Þ share at least one point and hence jjx
yjj
1 may become singular. For ease of presentation, in this subsection we introduce the following technical notations. Suppose that the spherical triangle  ðqÞ , q ¼ 1; 2, has vertices ðqÞ ðqÞ ðqÞ fv 1 ; v 2 ; v 3 g. Extending the notations of Sec. 6.1 we denote by T ðqÞ the planar triangle sharing its vertices with the spherical triangle  ðqÞ . Let T^ be the planar triangle in R 3 with vertices ð1; 0; 0Þ, ð0; 1; 0Þ and ð0; 0; 1Þ and K be the planar triangle in R 2 with vertices ð0; 0Þ, ð1; 0Þ and ð0; 1Þ. Any two points u; v 2 T^ can be uniquely represented by two pairs ðu1 ; u2 Þ; ðv1 ; v2 Þ 2 K by u ¼ ðu1 ; u2 ; 1
u1
u2 Þ

and

v ¼ ðv1 ; v2 ; 1
v1
v2 Þ:

Firstly, we describe a numerical approximation for (6.8) for two disjoint spherical triangles  ð1Þ and  ð2Þ (a so-called far-¯eld computation). In this case the kernel jjx
yjj
1 and the integrand f ðx; yÞ :¼

Bi ðxÞBj ðyÞ 4 jjx
yjj

are analytic on  ð1Þ   ð2Þ . Similarly to (6.4), I can be represented as an integral over K K Z Z I ¼ f ðx; yÞdx dy  ð1Þ

 ð2Þ

¼ jdet F ð1Þ jjdet F ð2Þ j  Z 1Z 1
u1Z 1Z 1
v1
ð1Þ F ðuÞ F ð2Þ ðvÞ dv2 dv1 du2 du1  f ; ; 3 ð1Þ ð2Þ ð2Þ jjF ðuÞjj jjF ðvÞjj jjF ðvÞjj jjF ð1Þ ðuÞjj 3 0 0 0 0 ð6:9Þ where F ðqÞ : T^ ! T , q ¼ 1; 2 is de¯ned by 2 ðqÞ ðqÞ ðqÞ 3 3 v 1;1 v 2;1 v 3;1 2 u1 6 7 ðqÞ ðqÞ ðqÞ 74 5: u2 F ðqÞ ðuÞ ¼ 6 4 v 1;2 v 2;2 v 3;2 5 1
u1
u2 ðqÞ ðqÞ ðqÞ v 1;3 v 2;3 v 3;3

ð6:10Þ

Pseudodi®erential Equations on the Sphere with Spherical Splines

1955

The mappings ( K ! T ðqÞ ; ðu1 ; u2 Þ 7! F ðqÞ ðuÞ;

q ¼ 1; 2

are a±ne, hence the mappings 8 ðqÞ > :ðu1 ; u2 Þ 7!

F ðqÞ ðuÞ ; jjF ðqÞ ðuÞjj

q ¼ 1; 2

are analytic if and only if T ðqÞ does not contain the origin, i.e. jjF ðqÞ ðuÞjj 6¼ 0. This requirement is always satis¯ed if the underlying spherical triangulation
is su±ciently re¯ned. Thus the integrand in (6.9) is a composition of analytic functions and hence is analytic. Therefore the integral I in (6.8) can be computed numerically by standard quadrature rules in case of disjoint spherical triangles  ð1Þ and  ð2Þ . Secondly, we consider numerical approximation of (6.8) for  ð1Þ and  ð2Þ sharing at least one point (a so-called near-¯eld computation). Depending on the mutual location of  ð1Þ and  ð2Þ we introduce an auxiliary parameter k as follows: 8 ð1Þ and  ð2Þ share a vertex; > : 2 if  ð1Þ and  ð2Þ are identical: E±cient and accurate quadratures for integrals of this type have been intensively developed over the last three decades mostly in the context of boundary element methods.9,14,25,26 It is known that there exists a sequence of regularizing coordinate transformations which removes the singularity in (6.8) completely. Note that this coordinate transformation is not unique. In this paper we employ the approach in Ref. 7 which also extends to arbitrary dimension and noninteger singularity orders, Ref. 6 for computational aspects in this approach and Refs. 10 and 24 for alternative transformations. In the remainder of this section, we describe details of the computational process. Without loss of generality we assume that  ð1Þ and  ð2Þ intersect at their ¯rst k þ 1 vertices, i.e. ð1Þ

vj

ð2Þ

¼ vj ;

j ¼ 1; . . . ; k þ 1;

ð6:11Þ

and F ðqÞ is de¯ned by (6.10) according to this order. The integrand in the right-hand side of (6.9) has a singularity on K  K . However, location of singularity is easily described by agreement (6.11): for ððu1 ; u2 Þ; ðv1 ; v2 ÞÞ 2 K  K , the integrand in the right-hand side of (6.9) is singular if 8 0 independent of Nk ; see Ref. 7. Based on this result and in view of the transformation (6.9) we approximate the integral I in (6.8) by the quadrature rule  Nk
X F ð1Þ ðul Þ F ð2Þ ðvl Þ wl QNk ¼ jdet F ð1Þ jjdet F ð2Þ j f : ; ð1Þ ðu Þjj jjF ð2Þ ðv Þjj ð2Þ ðv Þjj 3 jjF ð1Þ ðu Þjj 3 jjF jjF l l l l l¼1

Here we adopt the notation ul ¼ ðu1;l ; u2;l ; 1
u1;l
u2;l Þ

and

vl ¼ ðv1;l ; v2;l ; 1
v1;l
v2;l Þ

for l ¼ 1; . . . ; Nk :

Using again transformation (6.9) we express QNk in a simpler form as a quadrature rule over  ð1Þ   ð2Þ : Q Nk ¼

Nk X Bi ðxl ÞBj ðyl Þ l¼1

jjxl
yl jj

Wl ;

ð6:15Þ

where xl :¼

F ð1Þ ðul Þ ; jjF ð1Þ ðul Þjj

yl :¼

F ð2Þ ðvl Þ ; jjF ð2Þ ðvl Þjj

Wl :¼

jdet F ð1Þ jjdet F ð2Þ jwl : 4 jjF ð1Þ ul jj 3 jjF ð2Þ vl jj 3

ð6:16Þ

According to Refs. 7 and 24 the quadrature error is bounded by 1=4

jI
QNk j  C expð
bN k Þ

ð6:17Þ

with b; C > 0 independent on Nk . This quadrature rule is used in our computations. We illustrate the performance of QNk on the simpli¯ed integral Z Z 1 d d ð6:18Þ jjx
yjj x y ð1Þ ð2Þ   with  ð1Þ ¼  m , m ¼
1; 0; 1; 2,  ð2Þ ¼  2 , where  m is de¯ned by 

 1 1 1 1 
1 ¼ ð0; 1; 0Þ;
pffiffiffi ; pffiffiffi ; 0 ; 0; pffiffiffi ; pffiffiffi ; 2 2 2 2 

 1 1 1 1  1 ¼ ð1; 0; 0Þ; pffiffiffi ; pffiffiffi ; 0 ; pffiffiffi ; 0;
pffiffiffi ; 2 2 2 2

Pseudodi®erential Equations on the Sphere with Spherical Splines



1957





 1 1 1 1  0 ¼ ð1; 0; 0Þ; pffiffiffi ;
pffiffiffi ; 0 ; pffiffiffi ; 0;
pffiffiffi ; 2 2 2 2 

 1 1 1 1 ;  2 ¼ ð1; 0; 0Þ; pffiffiffi ; pffiffiffi ; 0 ; pffiffiffi ; 0; pffiffiffi 2 2 2 2 as shown in Fig. 1(b). Thus,  ð1Þ and  ð2Þ are disjoint spherical triangles if m ¼
1, share a node if m ¼ 0, share an edge if m ¼ 1 and are identical triangles if m ¼ 2. In case m ¼
1 we use a quadrature rule on K  K obtained from the Gauss
Legendre quadrature rule on ½0; 1 2  ½0; 1 2 by the Du®y transformation 8 2 > > ³ ! 7 : > :

2 In Fig. 1(a) we give the relative error of the constructed quadrature approximation. In the numerical experiments whose results were presented in Tables 2 and 3 we choose the quadrature rules with the following total number of points: N
1 ¼ 10 4 , N0 ¼ 2  11 4 , N1 ¼ 6  11 4 and N2 ¼ 6  12 4 . The corresponding relative errors are marked by solid symbols in Fig. 1(a). Remark 6.1. The most time-consuming part in the numerical simulation is the computation and assembly of the sti®ness matrix, which is fully populated due to nonlocal nature of L in (6.6). It is possible to reduce the order of the quadrature rule if the distance between  ð1Þ and  ð2Þ is large, see Sec. 5 in Ref. 24. This reduces the number of function evaluations, but requires quadratures of di®erent order in the far¯eld computation. 0

10

distant common vertex common edge identical

1

−5

10

0.5

2

−10

10

τ

τ

−0.5

1

0

−1 −1 −15

10

2

4

6

8 10 12 14 1/4 (# quadrature points)

(a)

16

18

20

τ1

τ

0

−1 −0.5

0

0 0.5 1

1

(b)

Fig. 1. (a) Relative error of the quadrature rule for approximation of (6.18) and (b) location of the spherical triangles  m , m ¼
1; 0; 1; 2.

1958

T. D. Pham, T. Tran & A. Chernov

Acknowledgments The ¯rst author was supported by the University International Postgraduate Award o®ered by the University of New South Wales. The second author was partially supported by the grant FRG PS17166. References 1. P. Alfeld, M. Neamtu and L. L. Schumaker, Bernstein
Bezier polynomials on spheres and sphere-like surfaces, Comput. Aided Geom. Design 13 (1996) 333
349. 2. P. Alfeld, M. Neamtu and L. L. Schumaker, Dimension and local bases of homogeneous spline spaces, SIAM J. Math. Anal. 27 (1996) 1482
1501. 3. P. Alfeld, M. Neamtu and L. L. Schumaker, Fitting scattered data on sphere-like surfaces using spherical splines, J. Comput. Appl. Math. 73 (1996) 5
43. 4. V. Baramidze and M.-J. Lai, Error bounds for minimal energy interpolatory spherical splines, in Approximation Theory XI : Gatlinburg 2004, Mod. Methods Math. (Nashboro Press, 2005), pp. 25
50. 5. V. Baramidze and M.-J. Lai, Spherical spline solution to a PDE on the sphere, in Wavelets and Splines : Athens 2005, Mod. Methods Math. (Nashboro Press, 2006), pp. 75
92. 6. A. Chernov, T. von Petersdor® and C. Schwab, Quadrature algorithms for highdimensional singular integrands on simplices, in preparation. 7. A. Chernov, T. von Petersdor® and C. Schwab, Exponential convergence of hp quadrature for integral operators with Gevrey kernels, M2AN, Math. Model. Numer. Anal. 45 (2011) 387
422. 8. P. G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, 1978). 9. M. G. Du®y, Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. Anal. 19 (1982) 1260
1262. 10. S. Erichsen and S. Sauter, E±cient automatic quadrature in 3-D Galerkin BEM, Comp. Methods Appl. Mech. Engrg. 157 (1998) 215
224. 11. W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere with Applications to Geomathematics (Oxford Univ. Press, 1998). 12. E. W. Grafarend, F. W. Krumm and V. S. Schwarze, eds., Geodesy : The Challenge of the 3rd Millennium (Springer, 2003). 13. L. H€ormander, Pseudodi®erential operators, Commun. Pure Appl. Math. 18 (1965) 501
517. 14. G. C. Hsiao, P. Kopp and W. L. Wendland, A Galerkin collocation method for some integral equations of the ¯rst kind, Computing 25 (1980) 89
130. 15. G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences, Vol. 164 (Springer-Verlag, 2008). 16. J. Kohn and L. Nirenberg, On the algebra of pseudodi®erential operators, Commun. Pure Appl. Math. 18 (1965) 269
305. 17. R. Kress, Linear Integral Equations (Springer-Verlag, 1989). 18. J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I (Springer-Verlag, 1972). 19. T. M. Morton and M. Neamtu, Error bounds for solving pseudodi®erential equations on spheres, J. Approx. Theory 114 (2002) 242
268. 20. C. M€ uller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17 (Springer-Verlag, 1966). 21. M. Neamtu and L. L. Schumaker, On the approximation order of splines on spherical triangulations, Adv. Comput. Math. 21 (2004) 3
20.

Pseudodi®erential Equations on the Sphere with Spherical Splines

1959

22. B. E. Petersen, Introduction to the Fourier Transform & Pseudodi®erential Operators, Monographs and Studies in Mathematics, Vol. 19 (Pitman, 1983). 23. E. B. Sa® and A. B. J. Kuijlaars, Distributing many points on a sphere, Math. Intelligencer 19 (1997) 5
11. 24. S. Sauter and C. Schwab, Randelementmethoden (B. G. Teubner, 2004). 25. S. A. Sauter and C. Schwab, Quadrature for hp-Galerkin BEM in R 3 , Numer. Math. 78 (1997) 211
258. 26. C. Schwab and W. L. Wendland, On numerical cubatures of singular surface integrals in boundary element methods, Numer. Math. 62 (1992) 343
369. 27. S. Svensson, Pseudodi®erential operators


A new approach to the boundary problems of physical geodesy, Manuscr. Geod. 8 (1983) 1
40. 28. T. Tran and T. D. Pham, Pseudodi®erential equations on the sphere with radial basis functions: Error analysis, Preprint, 2008, http://www.maths.unsw.edu.au/applied/pubs/ apppreprints2008.html.