A domain decomposition method for solving the hypersingular integral equation on the sphere with...

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Numerische Mathematik ISSN 0029-599XVolume 120Number 1 Numer. Math. (2012) 120:117-151DOI 10.1007/s00211-011-0404-1

A domain decomposition method forsolving the hypersingular integral equationon the sphere with spherical splines

Duong Pham & Thanh Tran

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Numer. Math. (2012) 120:117–151DOI 10.1007/s00211-011-0404-1

NumerischeMathematik

A domain decomposition method for solvingthe hypersingular integral equation on the spherewith spherical splines

Duong Pham · Thanh Tran

Received: 3 December 2010 / Revised: 8 June 2011 / Published online: 19 July 2011© Springer-Verlag 2011

Abstract We present an overlapping domain decomposition technique for solvingthe hypersingular integral equation on the sphere with spherical splines. We prove thatthe condition number of the additive Schwarz operator is bounded by O(H/δ), whereH is the size of the coarse mesh and δ is the overlap size, which is chosen to be pro-portional to the size of the fine mesh. In the case that the degree of the splines is even,a better bound O(1 + log2(H/δ)) is proved. The method is illustrated by numericalexperiments on different point sets including those taken from magsat satellite data.

Mathematics Subject Classification (2010) 65N55 · 65N30

1 Introduction

Hypersingular integral equations have many applications, for example in acoustics,fluid mechanics, elasticity and fracture mechanics [8]. Numerous numerical methodsfor solving these equations have been proposed and developed during the last fewdecades; see for example [13,16,21]. In this paper, we are particularly interested inthe setting of the equation on the sphere, namely, we study the hypersingular integral

D. Pham · T. TranSchool of Mathematics and Statistics, The University of New South Wales,Sydney 2052, Australiae-mail: [email protected]

Present Address:D. Pham (B)Hausdorff Center for Mathematics - Universitat Bonn,Bonn, Germanye-mail: [email protected]

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Author's personal copy

118 D. Pham, T. Tran

Table 1 Unpreconditioned systems with uniform triangulations; κ(A) = O(hα)

N h d = 1 ω2 = 0.01 d = 2 ω2 = 0.01 d = 3 ω2 = 0.1κ(A) α κ(A) α κ(A) α

18 0.7071 12.27 469.36 19,109.90

66 0.3536 20.90 −0.77 736.53 −0.65 25,735.72 −0.42

258 0.1768 39.22 −0.91 1,422.77 −0.95 50,877.82 −0.98

1,026 0.0883 75.93 −0.95 2,798.35 −0.97 100,876.63 −0.99

equation of the form

− N u + ω2∫

S

u dσ = f on S, (1.1)

where N is the hypersingular integral operator given by

N v(x) := 1

4π

∂

∂νx

∫

S

v( y)∂

∂ν y

1

|x − y| dσ y, (1.2)

ω is some nonzero real constant, and S is the unit sphere in R3, that is, S = {x ∈ R

3 :|x| = 1}. Here ∂/∂νx is the normal derivative with respect to x, and |·| denotes theEuclidean norm. The equation arises from the boundary-integral reformulation of theNeumann problem with the Laplacian in the interior or exterior of the sphere; see e.g.[13,16,21].

Equation (1.1) can be solved by the Galerkin method with tensor products of uni-variate splines on a regular grid. However, when the data [i.e., the right-hand sidefunction f in (1.1)] is given by a satellite, this approach is not suitable.

Spherical radial basis functions seem to be a better choice [18]. However, theresulting matrix system from this approximation is very ill-conditioned. Even thoughoverlapping additive Schwarz preconditioners can be designed for this problem, thecondition number of the preconditioned system still depends on the number of subdo-mains and the angles between subspaces; see [18].

Spherical splines (which were developed in [2–4] and have many properties in com-mon with classical polynomial splines over planar triangulations) are another tool wellsuited for scattered data interpolation and approximation problems [15]. However, asproved in Proposition 2.6 in the next section and evidenced by the numerical resultsin Table 1, the Galerkin method with spherical splines yields a linear system whichmay also be ill-conditioned, though the ill-conditionedness is not as bad as with radialbasis functions. The purpose of this paper is to overcome this ill-conditionedness bypreconditioning with additive Schwarz methods. Differently from the case of radialbasis functions, we prove that the condition number of the preconditioned systemarising from this method does not depend on the number of subdomains.

Preconditioners of this type have long been used in the finite element and boundaryelement literatures; see [17,20] and the references therein. As is usual for finite elementor boundary element methods, the additive Schwarz preconditioner is defined from

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A domain decomposition method 119

a subspace splitting of the finite dimensional space in which the solution is sought.This splitting is in turn defined by a decomposition of S into subdomains. In the finiteelement and boundary element literatures, these subdomains are usually defined froma two-level mesh in which one first defines a coarse mesh then re-defines this mesh toobtain the fine mesh.

In this paper, we work with scattered data points, which means that the right-handside function f in (1.1) may be given only at the location of the data points. There-fore, it is natural to use these points as vertices in the triangulation, so that f canbe interpolated by linear interpolation. (Any other triangulation may result in somespherical triangles containing no data points, which implies f = 0 in these triangles.This incurs unnecessary errors.) For this reason, we do not have the complete freedomto define the two-level mesh as mentioned above. We suggest to define the fine meshfrom a set of data points and the coarse mesh from another set of points which is notnecessarily a subset but has a smaller cardinality. A subdomain is constructed fromeach triangle in the coarse mesh by taking the union of all triangles in the fine meshwhich intersect this coarse triangle. This results in a set of overlapping subdomains.

We prove that the condition number of the preconditioned system is bounded byO(H/δ) for all parities of polynomial degrees, and by O(1+log2(H/δ)) in the case ofeven degree polynomials. Here, H is the mesh size of the coarse mesh and δ is the size ofthe overlap which is proportional to the mesh size of the fine mesh. The weaker estimatein the case of odd degree might be only a technical obstacle that we could not over-come. We note that the overlapping subdomains are, in general, not spherical triangles.In the analysis, we have to use another set of artificial subdomains which are triangles.

The structure of the paper is as follows. In Sect. 2, we will review spherical splines,introduce the Sobolev spaces on the unit sphere, a quasi-interpolation operator tobe used and present the hypersingular integral equation. The abstract framework forthe additive Schwarz preconditioner is reviewed in Sect. 3 followed by the additiveSchwarz method for the hypersingular integral equation in Sect. 4. In Sect. 5 we pres-ent the main theoretical results of the paper. Numerical results are presented in Sect. 6which is followed by two appendices which contain the proofs of some technicallemmas. In this paper C, C1, C2, and C3 denote generic constants which may takedifferent values at difference occurrences.

2 Preliminaries

In this section, we will first review spherical splines [2–4] and introduce our functionspaces on the unit sphere S ⊂ R

3. The quasi-interpolation operator and its bound-edness properties will be recalled. Then the hypersingular integral equation will bediscussed.

2.1 Spherical splines

Let {x1, x2, x3} be linearly independent vectors in R3. The trihedron T generated by

{x1, x2, x3} is defined by

T := {x ∈ R3 : x = b1x1 + b2x2 + b3x3 with bi ≥ 0, i = 1, 2, 3}.

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The intersection τ := T ∩S is called a spherical triangle. Let � = {τi : i = 1, . . . , T }be a set of spherical triangles. Then � is called a spherical triangulation of the sphereS if there hold

(i)⋃T

i=1 τi = S,(ii) each pair of distinct triangles in � are either disjoint or share a common vertex

or an edge.

Let Hd denote the space of trivariate homogeneous polynomials of degree drestricted to S. We define Sr

d(�) to be the space of homogeneous splines of degree dand smoothness r on a spherical triangulation �, that is,

Srd(�) := {s ∈ Cr (S) : s|τ ∈ Hd , τ ∈ �}.

Here d ≥ 3r + 2 if r ≥ 1, and d ≥ 1 if r = 0; see [2–4].Let τ be a spherical triangle with vertices x1, x2, and x3. For any x ∈ S, there exist

unique real numbers b1,τ (x), b2,τ (x), and b3,τ (x) satisfying

x = b1,τ (x)x1 + b2,τ (x)x2 + b3,τ (x)x3

as x1, x2, and x3 are linearly independent. These numbers b1,τ (x), b2,τ (x), andb3,τ (x) are called the spherical barycentric coordinates of x associated with τ . Wedefine the homogeneous Bernstein basis polynomials of degree d on τ to be the poly-nomials

Bd,τi jk (x) := d!

i ! j !k!b1,τ (x)i b2,τ (x) j b3,τ (x)k, i + j + k = d.

As was shown in [2], we can use these polynomials as a basis for Hd .A spherical cap centred at x ∈ S and having radius R is defined by

C(x, R) := { y ∈ S : cos−1(x · y) ≤ R}.

For any spherical triangle τ , let |τ | denote the diameter of the smallest spherical capcontaining τ , and ρτ denote the diameter of the largest spherical cap contained in τ .We define

|�| := max{|τ | : τ ∈ �} and ρ� := min{ρτ : τ ∈ �},

and refer to |�| as the mesh size. In this paper, we assume that our triangulations areregular, i.e., for some β > 1, there holds

|τ |≤βρτ ∀τ ∈ �,

and quasi-uniform, i.e, for some positive number γ < 1, there holds

|τ |≥γ |�| ∀τ ∈ �.

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Roughly speaking, regularity guarantees that the smallest angle in the triangulationis not too small so that there are no long and thin triangles, and quasi-uniformityguarantees that the sizes of triangles in a triangulation are not too much different.

Let X := {x1, . . . , xK } be a set of points on S. We denote by �h the sphericaltriangulation generated by X , which will be referred to as the fine mesh. Here thesubscript h is related to the mesh size |�h | of �h by

h = tan(|�h |/2).

Throughout the paper, we assume that |�h | ≤ 1.The construction of a preconditioner by additive Schwarz method, as is usual,

requires a coarse mesh �H which will be defined from a set of points Y :={ y1, . . . , yJ } on S with J < K . As with the fine mesh, here

H = tan(|�H |/2).

We assume that |�H | > |�h |. We also assume that both �H and �h are regular andquasi-uniform.

To differentiate triangles in the two different meshes, we denote a triangle in thefine mesh by τ , and a triangle in the coarse mesh by τH . For each τ ∈ �h , we denoteby Aτ its area.

To invoke results in [5,12] we also define

ς = tan(ρ�h /2), hτ = tan(|τ |/2), and ςτ = tan(ρτ /2) for τ ∈ �h

and, similarly for the coarse mesh,

ςH = tan(ρ�H /2), HτH = tan(|τH |/2), and ςτH = tan(ρτH /2) for τH ∈ �H .

In the sequel, for any x, y ∈ R, x � y means that there exists a constant C1 satisfying

x ≤ C1 y,

and x y means there exists a constant C2 such that

x ≥ C2 y.

When x and y satisfy x � y and x y we write x y. Using this notation, it isstraightforward to see that since �h and �H are regular and quasi-uniform, there hold

ρτ |τ | |�h | h hτ ςτ ς

ρτH |τH | |�H | H HτH ςτH ςH .(2.1)

Denoting by star1(x) the union of all triangles in �h that share the vertex x, we define

stark(x) := ∪{star1(w) : w is a vertex of stark−1(x)}, k > 1,

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and

stark(τ ) := ∪{stark(x) : x is a vertex of τ }, k ≥ 1.

We make use of the radial projection defined in [12]. Let � be a subset of S. We denoteby r� the centre of a spherical cap of smallest possible radius containing �, and by�� the tangential plane touching S at r�. For each point x ∈ �, the intersection of�� and the ray passing through the origin and x is denoted by x. We define

R(�) := {x ∈ �� : x ∈ �}. (2.2)

The radial projection R� is defined by

R� : R(�) → �

x → x := x/|x|. (2.3)

It is clear that R� is invertible. The following result is proved in [5,12].

Lemma 2.1 Under the assumptions that �h is a regular, quasi-uniform triangulationsatisfying |�h | ≤ 1 we have, for any τ ∈ �h,

(i) Aτ h2,(ii) νk(τ ) � (2k + 1)2

where Aτ denotes the area of τ , and νk(τ ) denotes the number of triangles in stark(τ ).Here, the constants depend only on the smallest angle of the triangulation.

2.2 Sobolev spaces

Let � ⊂ S be a Lipschitz domain. We use the Sobolev space H1(�) := {v ∈ L2(�) :‖v‖H1(�) < ∞}, which is equipped with a seminorm

|v|2H1(�):=

∫

�

|∇∗v|2 dσ

and a norm

‖v‖2H1(�)

:=∫

�

|∇∗v|2 dσ +∫

�

|v|2 dσ. (2.4)

Here ∇∗ is the surface gradient, and dσ is the Lebesgue measure on S. The spaceH1/2(�) is defined by Hilbert space interpolation [6] so that

H1/2(�) := [L2(�), H1(�)]1/2 (2.5)

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with the norm

‖v‖2H1/2(�)

=∞∫

0

K (t, v)2 dt

t2 , (2.6)

where the K -functional is defined, for v ∈ L2(�) + H1(�), by

K (t, v)2 = infv=v0+v1

(‖v0‖2L2(�) + t2‖v1‖2

H1(�)).

Similarly, we define the subspace H1/2(�) ⊂ H1/2(�) by

H1/2(�) := [L2(�), H10 (�)]1/2,

where H10 (�) = {v ∈ H1(�) : v = 0 on ∂�} with ∂� being the boundary of �.

The spaces H−1/2(�) and H−1/2(�) are defined as the dual spaces of H1/2(�) andH1/2(�), respectively, with respect to the L2 duality which is the usual extension ofthe L2 inner product on �.

For the analysis in this paper we also define the following norms:

�v�2H1/2(�)

:= 1

diam(�)‖v‖2

L2(�) + |v|2H1/2(�)(2.7)

and

�v�2H1/2(�)

:= |v|2H1/2(�)+

∫

�

v2(x)

dist(x, ∂�)dσx, (2.8)

where

|v|2H1/2(�):=

∫

�

∫

�

|v(x) − v( y)|2|x − y|3 dσx dσ y. (2.9)

We will occasionally use the Sobolev space Hk(�) := {v ∈ L2(�) : ‖v‖′Hk (�)

< ∞}for nonnegative integer k, where the norm ‖·‖′

Hk (�)is defined by using homogeneous

extensions:

‖v‖′Hk (�)

:=k∑

l=0

|v|′Hl (�), (2.10)

with

|v|′Hl (�):=

{ ‖v‖L2(�), l = 0,∑|α|=l ‖Dαvl−1‖L2(�), l = 1, . . . , k.

(2.11)

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Here, vl−1 is the homogeneous extension (of degree l − 1) of v to R3\{0}, defined by

vl−1(x) := |x|l−1v

(x|x|

), x ∈ R

3\{0}, (2.12)

and Dα denotes the derivative of order |α| where α = (α1, α2, α3) ∈ N3 and |α| =

α1 + α2 + α3. The norm ‖·‖′Hk (�)

defined by (2.10), with k = 1, turns out to beequivalent to the norm ‖ · ‖H1(�); see [10].

For a subset R of R2, the Sobolev spaces H1/2(R) and H1/2(R) can be defined

similarly to the case of � ⊂ S, with norms and seminorms given by (2.7)–(2.9)accordingly. In particular, when R = I × J , where I, J are intervals in R, there hold

‖v‖H1/2(R) �v�H1/2(R) ∀v ∈ H1/2(R) (2.13)

and

|v|2H1/2(R)

∫

I

∫

I

‖v(x, ·) − v(x ′, ·)‖2L2(J )

|x − x ′|2 dx dx ′

+∫

J

∫

J

‖v(·, y) − v(·, y′)‖2L2(I )

|y − y′|2 dy dy′. (2.14)

Here, the constants in the equivalences are independent of the sizes of I and J . Theresult (2.13) is proved in [1, Lemma 2] and (2.14) in [14, Lemma 5.3] (see alsoExercise 5.1 following that lemma).

2.3 Quasi-interpolation

In this subsection we briefly discuss the construction of a quasi-interpolation operatorI h : L2(S) → Sr

d(�h) which is defined in [12]. This operator will be used frequentlyin the rest of the paper. First we introduce the set of domain points of �h to be

D :=⋃

τ=〈x1,x2,x3〉∈�h

{ξτ

i jk = i x1 + j x2 + kx3

d: i, j, k ∈ N and i + j + k = d

}.

Here, τ = 〈x1, x2, x3〉 denotes the spherical triangle whose vertices are x1, x2, x3.We denote the domain points by ξ1, . . . , ξD , where D = dim S0

d (�). Let {Bl : l =1, . . . , D} be a basis for S0

d (�h) such that the restriction of Bl on a triangle containingξl is the Bernstein-Bézier polynomial of degree d associated with this point, and thatBl vanishes on other triangles.

A set M := {ζl}Nl=1 ⊂ D is called a minimal determining set for Sr

d(�h) if, for

every s ∈ Srd(�h), all the coefficients νl(s) in the expression s = ∑D

l=1 νl(s)Bl areuniquely determined by the coefficients corresponding to the basis functions which

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are associated with points in M. Given a minimal determining set, we construct abasis {B∗

l }Nl=1 for Sr

d(�h) by requiring

νl ′(B∗l ) = δl,l ′ , 1 ≤ l, l ′ ≤ N .

The use of Hahn–Banach Theorem extends the linear functionals νl , l = 1, . . . , N ,to be defined for all functions in L2(S). We continue to use the same symbol for theseextensions.

The quasi-interpolation operator I h : L2(S) → Srd(�h) is now defined by

I hv :=N∑

l=1

νl(v)B∗l , v ∈ L2(S). (2.15)

The following lemma states the stability for these quasi-interpolants.

Lemma 2.2 For any τ ∈ �h, let ωτ := ⋃i∈Iτ ωi , where ωi := supp(B∗

i ) and Iτ :={i ∈ {1, . . . , N } : τ ⊂ ωi }.

(i) For v ∈ L2(S) and k = 0, 1, there holds

| I hv|Hk (τ ) � h−k‖v‖L2(ωτ ).

(ii) For v ∈ L2(S) and k = 0, 1/2, 1, there holds

‖ I hv‖Hk (S) ≤ Ch−k‖v‖L2(S). (2.16)

Here, the constants depend only on the smallest angle ��h of �h and the polynomialdegree d.

Proof The proof for (i) can be found in [12, Proposition 5.2]. To prove (ii) we firstnote that by (i) we have

‖ I hv‖L2(τ ) � ‖v‖L2(ωτ ).

It follows that

‖ I hv‖2L2(S) =

∑τ∈�h

‖ I hv‖2L2(τ ) �

∑τ∈�h

‖v‖2L2(ωτ )

=∑τ∈�h

∑τ ′⊂ωτ

τ ′∈�h

‖v‖2L2(τ ′) =

∑τ ′∈�h

#{τ : τ ′ ⊂ ωτ }‖v‖2L2(τ ′),

where # denotes the cardinality of a set. We deduce

‖ I hv‖2L2(S) � max

τ ′∈�h

{#{τ : τ ′ ⊂ ωτ }}‖v‖2L2(S) ≤ M‖v‖2

L2(S),

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where M := maxτ ′∈�h {#{τ : τ ′ ⊂ ωτ }} depends only on the smallest angle of thetriangulation (see [12]). This proves (ii) for the case k = 0.

A similar argument can be used to obtain the result for k = 1. The interpolationinequality (see e.g. [10, Proposition 2.3])

‖w‖H1/2(S) � ‖w‖1/2H1(S)

‖w‖1/2L2(S)

∀w ∈ H1(S)

yields

‖ I hv‖H1/2(S) � h−1/2‖v‖L2(S),

completing the proof of the lemma. ��In the next lemma we will prove the boundedness of the quasi-interpolation opera-

tor I h in H1/2(S) when d is even. The requirement that d is even is to ensure that thespace Sr

d(�h) contains constant functions. This inclusion of constants in the space iscrucial in the proof of this lemma which uses the spherical Poincaré and Friedrichstype inequalities (see Appendix A). The fact that Sr

d(�h) contains constant functionsif and only if d is even can be easily seen. Indeed, if d is even then

v(x) = (x2 + y2 + z2)d/2 ∀ x = (x, y, z) ∈ S

belongs to the space Srd(�h) and v(x) ≡ 1. On the other hand, if d is odd and if there

exists v ∈ Srd(�h) such that v(x) ≡ 1 then v ∈ Hd , and by homogeneity we have

1 = v(−x) = (−1)dv(x) = −1.

Lemma 2.3 Let �h be a regular and quasi-uniform spherical triangulation on S andlet I h : L2(S) → Sr

d(�h) be the quasi-interpolation operator defined by (2.15) withd even. Then for any v ∈ H1/2(S), there holds

‖ I hv‖H1/2(S) ≤ C‖v‖H1/2(S). (2.17)

Proof For any τ ∈ �h , let ωτ be defined as in Lemma 2.2. We define αv :=|ωτ |−1

∫ωτ

v dσ and v := v − αv . Since Srd(�h) contains constant functions when

d is even, the quasi-interpolant I h reproduces these functions. Therefore, there holds

| I hv|H1(τ ) = | I hv − αv|H1(τ ) = | I hv − I hαv|H1(τ ) = | I h v|H1(τ ).

Noting that∫ωτ

v dσ = 0 and applying the results in Lemmas 2.2 and 7.2, we obtain

| I hv|H1(τ ) � h−1 |v|L2(ωτ ) � |v|H1(ωτ ) = |v|H1(ωτ ).

Repeating the argument used in the proof of Lemma 2.2, we obtain

| I hv|H1(S) � |v|H1(S).

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This together with the result in Lemma 2.2(ii) for k = 0 implies

‖ I hv‖H1(S) ≤ C‖v‖H1(S). (2.18)

Inequality (2.17) is then obtained by using the interpolation inequality (see e.g. [11,Theorem B2]), noting (2.18) and the results in Lemma 2.2(ii) for k = 0. ��

2.4 The hypersingular integral equation

Recall the hypersingular integral equation (1.1) where f is some given smooth func-tion and the hypersingular integral operator N is given by (1.2). To set up a weakformulation, we introduce the bilinear form

a(u, v) := − 〈N u, v〉 + ω2 〈u, 1〉〈v, 1〉 , u, v ∈ H1/2(S),

where 〈u, v〉 = ∫S

uv dσ . We note that (see [13])

a(v, v) ‖v‖2H1/2(S)

∀v ∈ H1/2(S). (2.19)

A natural weak formulation of equation (1.1) is: Find u ∈ H1/2(S) satisfying

a(u, v) = 〈 f, v〉 ∀v ∈ H1/2(S).

This bilinear form is clearly bounded and coercive (cf. [7]). This guarantees the uniquesolvability of the equation. The Ritz-Galerkin approximation problem is: Find uh ∈Sr

d(�h) satisfying

a(uh, vh) = 〈 f, vh〉 ∀vh ∈ Srd(�h). (2.20)

Denoting {φi : i = 1, . . . , N } a basis for Srd(�h), and writing uh = ∑N

i=1 ciφi andc = (ci )

Ni=1, Eq. (2.20) reduces to the following linear system

Ac = f, (2.21)

where for i, j = 1, . . . , N , the entries Ai j of the matrix A and fi of the right-handside vector f are, respectively, Ai j = a(φi , φ j ) and fi = 〈 f, φi 〉.

It is well known that the matrix A is ill-conditioned, namely, the condition numberof A, defined by κ(A) := λmax(A)/λmin(A), potentially grows like h−1 as h → 0 (i.e.|�h | → 0). Since we cannot find a reference for this seemingly well-known result,we include the proof here for completeness.

We will use the following two results from [5].

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Lemma 2.4 [5, Lemma 5] Let p be a homogeneous polynomial of degree d on aspherical triangle τ . If p is written in Bernstein–Bézier form as

p(x) =∑

i+ j+k=d

c(τ )i jk Bd,τ

i jk (x), x ∈ τ,

then

‖p‖L2(τ ) A1/2τ |cτ |,

Here Aτ is the area of τ, cτ is the vector of components c(τ )i jk , i + j + k = d, and |cτ |

is its Euclidean norm.

Lemma 2.5 [5, Lemma 6] Let p be a homogeneous polynomial of degree d on aspherical triangle τ . Then there holds

|p|H1(τ ) � ς−1τ ‖p‖L2(τ ).

We can now prove an upper bound for the condition number of A.

Proposition 2.6 The condition number of the stiffness matrix A is bounded by

κ(A) � h−1.

Proof Recall that {φi }Ni=1 is a basis for Sr

d(�h). Let c = (ci )Ni=1 ∈ R

N . We define

u := ∑Ni=1 ciφi ∈ Sr

d(�h). Noting (2.19), we have

cT Ac = a(u, u) ‖u‖2H1/2(S)

. (2.22)

We note that I hu = u for all u ∈ Srd(�h), where I h is the quasi-interpolation operator

from L2(S) to Srd(�h) defined in Sect. 2.3. It follows from (2.22) and Lemma 2.2 that

cT Ac ‖ I hu‖2H1/2(S)

� h−1‖u‖2L2(S) � h−1

∑τ∈�h

‖u‖2L2(τ ).

By using Lemma 2.4 and noting Aτ |τ |2 h2 we obtain

cT Ac � h−1∑τ∈�h

Aτ |cτ |2 � h∑τ∈�h

|cτ |2. (2.23)

Here, cτ is a vector whose components are those of the vector c corresponding to thebasis functions φi whose supports contain τ . Proposition 5.1 in [12] ensures that thesupport of φi lies inside star3(τ ) if the domain point corresponding to φi belongs to τ .

123



Therefore, if μ = max{ν3(τ ) : τ ∈ �h} then each component of c appears at most μ

times in the sum on the right-hand side of (2.23). This together with Lemma 2.1 yields

cT Ac � hμ|c|2 � h|c|2 = h cT c,

which implies

λmax(A) � h.

Using (2.22) and a similar argument as above, we have

cT Ac ‖u‖2H1/2(S)

‖u‖2L2(S) =

∑τ∈�h

‖u‖2L2(τ )

∑τ∈�h

Aτ |cτ |2 h2cT c,

implying

λmin(A) h2.

An upper bound for κ(A) can now be derived. ��This behaviour of κ(A) subjected to the change of h is corroborated by the numer-

ical results in Table 1.

3 Abstract framework of additive Schwarz methods

Additive Schwarz methods provide fast solutions to (2.20) by solving, at the sametime, problems of smaller size. Let the space V = Sr

d(�h) be decomposed as

V = V0 + · · · + VJ , (3.1)

where Vi , i = 0, . . . , J, are subspaces of V , and let Pi : V → Vi , i = 0, . . . , J , beprojections defined by

a(Piv,w) = a(v,w) ∀v ∈ V, ∀w ∈ Vi . (3.2)

If we define

P := P0 + · · · + PJ , (3.3)

then the additive Schwarz method for (2.20) consists of solving, by an iterative method,the equation

Pu = g, (3.4)

123



where g = ∑Ji=0 gi , with gi ∈ Vi being solutions of

a(gi , w) = 〈 f, w〉 ∀w ∈ Vi . (3.5)

The equivalence of (2.20) and (3.4) was discussed in [19]. For completeness, we brieflyexplain that equivalence here. Let uh be a solution of (2.20). From the definition ofPi and gi we deduce

a(Pi uh, v) = a(uh, v) = 〈 f, v〉 = a(gi , v) ∀v ∈ V,

i.e. Pi uh = gi . Hence Puh = g. On the other hand, if P : V → V is invertible and uh

is a solution of (3.4), then by using successively the symmetry of P , (3.2) and (3.5),we obtain

a(uh, v) = a(P−1g, v) = a(g, P−1v)

=J∑

i=1

a(gi , P−1v) =J∑

i=1

a(gi , Pi P−1v)

=J∑

i=1

⟨f, Pi P−1v

⟩= 〈 f, v〉 for any v ∈ V .

A practical method to solve (3.4) is the conjugate gradient method; the additiveSchwarz method can be viewed as a preconditioned conjugate gradient method.

Bounds for eigenvalues the condition number of the additive Schwarz operator P ,can be obtained by using the following lemma; see [17].

Lemma 3.1 Assume that for any u ∈ V satisfying u = ∑Ji=0 ui with ui ∈ Vi for

i = 0, . . . , J there holds

a(u, u) ≤ C1

J∑i=0

a(ui , ui ). (3.6)

Assume further that for any u ∈ V , there exists a decomposition u = ∑Ji=0 u′

i withu′

i ∈ Vi for i = 0, . . . , J satisfying

J∑i=0

a(u′i , u′

i ) ≤ C2a(u, u). (3.7)

Then the extremal eigenvalues of the additive Schwarz operator P are bounded by

C−12 ≤ λmin(P) ≤ λmax(P) ≤ C1, (3.8)

and thus the condition number κ(P) = λmax(P)/λmin(P) is bounded by

κ(P) ≤ C1C2.

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4 Additive Schwarz method for the hypersingular integral equation on the unitsphere

In this section we will define a subspace decomposition of the form (3.1), and in thisway define the additive Schwarz operator for problem (2.20). Let V ′

0 = Srd(�H ) and

Vj = V ∩ H10 (� j ) for j = 1, . . . , J , where � j are overlapping subdomains (regarded

as open) which will be defined below. Since V ′0 is not a subspace of V we define the

subspace V0 = I h V ′0. The use of a quasi-interpolation as opposed to a “regular” inter-

polation is to allow the use of the results in [12]. We have now decomposed V as in(3.1) and can hence define the Schwarz operator P by (3.2) and (3.3).

We construct one subdomain from each triangle in �H , hence the number of sub-domains is J , the number of triangles in �H . Consider a triangle τ i

H ∈ �H , i =1, . . . , J . The subdomain �i corresponding to τ i

H is given by

�i = ∪{τ ∈ �h : τ ∩ τ iH �= φ}, i = 1, . . . , J.

Since a triangle in the fine mesh can intersect more than one triangle in the coarsemesh, the subdomains are overlapping. We denote the size of the overlapping by δ,which is proportional to h by our construction. It will be assumed that the subdomainscan be coloured using at most M colours in such a way that subdomains with the samecolour are disjoint.

In the succeeding sections a lower bound for the minimum eigenvalue of P will beobtained by using the quasi-interpolation operator and a family of functions associatedwith our set of overlapping subdomains that form a partition of unity.

5 Main results

In this section we prove a bound on the condition number of P by using the abstractresult in Lemma 3.1. We first prove (3.6).

Lemma 5.1 There exists a positive constant C independent of �h such that for anyu ∈ V satisfying u = ∑J

i=0 ui with ui ∈ Vi for i = 0, . . . , J,

a(u, u) ≤ CJ∑

i=0

a(ui , ui ),

where the constant C depends on the smallest angle of the triangulation.

Proof By the assumption on the colouring of the subdomains there are at most Msubdomains to which any x ∈ S can belong. By a standard colouring argument wehave

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a(u, u) � ‖u‖2H1/2(S)

≤ ‖u0‖2H1/2(S)

+∥∥∥∥∥

J∑i=1

ui

∥∥∥∥∥2

H1/2(S)

� ‖u0‖2H1/2(S)

+ MJ∑

i=1

‖ui‖2H1/2(S)

�J∑

i=0

a(ui , ui ).

The lemma is proved. ��In the following subsection, we prove (3.7) for both odd and even polynomial

degrees d. In Sect. 5.2, a better estimate of κ(P) is established for even degrees dwhen the quasi-interpolation operators I h and I H reproduce constant functions.

5.1 A general result for both odd and even degrees

To prove (3.7), we need to introduce an operator PH from H1/2(S) into Srd(�H )

defined by

a(PH u, v) = a(u, v) ∀v ∈ Srd(�H )

for any u ∈ H1/2(S). Standard finite element arguments yield

‖PH u − u‖H1/2(S) � ‖u − v‖H1/2(S) ∀v ∈ Srd(�H )

‖PH u − u‖H1/2(S) � ‖u‖H1/2(S)

‖PH u‖H1/2(S) � ‖u‖H1/2(S)

‖PH u − u‖L2(S) � H1/2‖u‖H1/2(S).

(5.1)

Lemma 5.2 There exists a positive constant C depending on the smallest angle of �h

and the polynomial degree d such that for any u ∈ V there exist ui ∈ Vi , i = 0, . . . , J ,satisfying u = ∑J

i=0 ui and

J∑i=0

a(ui , ui ) � H

ha(u, u).

Proof Let u′0 := PH u and u0 = I hu′

0. Let {θi }Ji=1 be a partition of unity defined

on S satisfying supp(θi ) = �i , for i = 1, . . . , J . We define w := u − u0, andui := I h(θiw) for i = 1, . . . , J . Then we can split u ∈ V by

u = u0 + u − u0 = u0 + I h(u − u0) = u0 + I h

(J∑

i=1

θiw

)= u0 + u1 + · · · u J .

It is clear that ui ∈ Vi for all i = 0, . . . , J . By Lemma 2.2, we have

‖ I h(PH u − u)‖H1/2(S) � h−1/2‖PH u − u‖L2(S). (5.2)

123



By writing u0 = I h(PH u −u)+u and using the triangular inequality, (5.2) and (5.1),we have

a(u0, u0) ‖u0‖2H1/2(S)

≤(‖ I h(PH u − u)‖H1/2(S) + ‖u‖H1/2(S)

)2

�(

h−1/2‖PH u − u‖L2(S) + ‖u‖H1/2(S)

)2

� H

h‖u‖2

H1/2(S) H

ha(u, u). (5.3)

Applying Lemma 2.2 and noting that θi vanishes outside �i , we obtain

a(ui , ui ) ‖ I h(θiw)‖2H1/2(S)

� h−1‖θiw‖2L2(S) h−1‖θiw‖2

L2(�i ). (5.4)

This together with the fact that ‖θiw‖L2(�i ) ≤ ‖w‖L2(�i ) implies

a(ui , ui ) � h−1‖w‖2L2(�i )

.

Summing up the above inequality over all subdomains, we obtain

J∑i=1

a(ui , ui ) � h−1‖w‖2L2(S). (5.5)

Noting that w ∈ Srd(�h) then I hw = w and by applying Lemma 2.2 and the last

inequality in (5.1), we infer

‖w‖L2(S) = ‖ I h(u − I h PH u)‖L2(S) = ‖ I h(u − PH u)‖L2(S)

� ‖u − PH u‖L2(S) � H1/2‖u‖H1/2(S).

This together with (5.5) gives

J∑i=1

a(ui , ui ) � H

h‖u‖2

H1/2(S) H

ha(u, u).

From this and (5.3) we infer

J∑i=0

a(ui , ui ) � H

ha(u, u),

completing the proof. ��Combining the results in Lemmas 5.1, 5.2 and 3.1 we obtain a bound for the con-

dition number κ(P) of the additive Schwarz operator.

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Fig. 1 Extended sphericaltriangle τ i

H,h

Theorem 5.3 The condition number of the additive Schwarz operator P is boundedby

κ(P) � H

h,

where the constant depends on the smallest angle in �h and the polynomial degree d.

Remark 5.4 Recall that we have chosen the overlap δ to be proportional to h.

5.2 A better estimate for even degrees

In this section we assume that the degree d of the spherical splines is even. Recallingthe construction of the subdomains in Sect. 4, for each triangle τ i

H in the coarse mesh�H , we defined a corresponding subdomain �i . We now define the triangle τ i

H,hsatisfying

τ iH ⊂ τ i

H,h ⊂ �i , i = 1, . . . , J. (5.6)

Without loss of generality we can assume that the edges of τ iH,h are parallel and of

distance h from those of τ iH (see Fig. 1). (We can always choose �i to be a bigger

domain so that the assumption holds.)For simplicity of presentation, we assume that the spherical triangles τ i

H , i =1, . . . , J , are equilateral triangles. Then so are the spherical triangles τ i

H,h . The set

{τ iH,h : i = 1, . . . , J } is a set of overlapping spherical triangles which covers the

sphere S (Fig. 2). We will define a partition of unity {θi }Ji=1, whose definition is given

123



Fig. 2 Six overlapping extended spherical triangles (triangles with dotted edges τ iH )

in Appendix B, satisfying

supp(θi ) ⊂ τ iH,h, i = 1, . . . , J. (5.7)

In the sequel, we denote

Ti := supp(θi ), i = 1, . . . , J, (5.8)

and Wi the smallest rectangle which contains Ti and shares a common edge with Ti

(see Fig. 4).

Lemma 5.5 For any v ∈ H1/2(S) there holds

J∑i=1

�θiv�2H1/2(Ti )

�J∑

i=1

(1 + log

Hi

δ

)2

�v�2H1/2(Wi )

. (5.9)

Proof See Appendix B. ��Lemma 5.6 Assume that the polynomial degree d is even. For any u ∈ V , there existui ∈ Vi for i = 0, . . . , J satisfying u = ∑J

i=0 ui and

J∑i=0

a(ui , ui ) �(

1 + log2 H

δ

)a(u, u), (5.10)

where the constant depends only on the smallest angle of the triangulations.

123



Proof Recall that the support Ti of the partition of unity function θi satisfies Ti ⊂ �i

(see (5.6)). We now define a decomposition of u ∈ V such that (5.10) holds. Forany u ∈ V let u0 := I h PH u ∈ V0 and ui = I h(θiw) ∈ Vi , i = 1, . . . , J , wherew = u − u0. It is clear that u = ∑J

i=1 ui . By using (2.17) and (5.1), we obtain

a(u0, u0) ‖u0‖2H1/2(S)

= ‖ I h PH u‖2H1/2(S)

� ‖u‖2H1/2(S)

a(u, u) (5.11)

‖w‖H1/2(S) = ‖ I h(u − PH u)‖H1/2(S) � ‖u − PH u‖H1/2(S) � ‖u‖H1/2(S) (5.12)

and

‖w‖L2(S) = ‖ I h(u − PH u)‖L2(S) � ‖u − PH u‖L2(S) � H1/2‖u‖H1/2(S). (5.13)

By Lemma 6 in [1] there holds

‖θiw‖H1/2(S) �θiw�H1/2(Ti ). (5.14)

By using successively (2.17), (5.14), (5.9) and (2.7), we obtain

J∑i=1

a(ui , ui ) J∑

i=1

‖ui‖2H1/2(S)

�J∑

i=1

‖θiw‖2H1/2(S)

J∑

i=1

�θiw�2H1/2(Ti )

�J∑

i=1

(1 + log

Hi

δ

)2

�w�2H1/2(Wi )

(

1 + logH

δ

)2 J∑i=1

(1

H‖w‖2

L2(Wi )+ |w|2H1/2(Wi )

). (5.15)

It is obvious that∑J

i=1 ‖w‖2L2(Wi )

‖w‖L2(S) and by the definition of the seminorm|·|H1/2(Wi )

, it is clear that

J∑i=1

|w|2H1/2(Wi )� |w|2H1/2(S)

. (5.16)

This together with (5.15), (5.13) and (5.12) implies

J∑i=1

a(ui , ui ) �(

1 + logH

δ

)2 (1

H‖w‖2

L2(S) + |w|2H1/2(S)

)

�(

1 + logH

δ

)2

‖u‖2H1/2(S)

(

1 + logH

δ

)2

a(u, u),

completing the proof of the lemma. ��

123



Combining the results in Lemmas 5.1, 5.6 and 3.1 we obtain a bound for the con-dition number κ(P) of the additive Schwarz operator.

Theorem 5.7 Assume that the polynomial degree d is even. The condition number ofthe additive Schwarz operator P is bounded by

κ(P) �(

1 + log2 H

δ

)

where the constant depends on the smallest angle in �h and the polynomial degree d.

Remark 5.8 It can be seen that the boundedness in the H1/2(S)-norm of the quasi-interpolation operators I h and I H is crucial for the proof of Lemma 5.6. The proofof this boundedness (Lemma 2.3) requires the property that I h and I H reproduceconstant functions, which does not hold in the case of odd degree splines.

6 Numerical results

We solved (1.1),

− N u + ω2∫

S

u dσ = f on S, (6.1)

with

f (x) = f (x, y, z) = ex (2x − xz).

by using spherical spline spaces S0d (�h), in which �h are spherical triangulations of

the following types:

• Type 1: Uniform triangulations which are generated as follows. First, we start with aspherical triangulation whose vertices are (1, 0, 0), (0, 1, 0), (0, 0, 1), (−1, 0, 0),

(0,−1, 0), and (0, 0,−1). The following finer meshes are obtained by dividingeach triangle in the previous triangulation into four equal equilateral triangles (byconnecting the midpoints of the three edges), resulting in triangulations with thenumber of vertices being 18, 66, 258, 1,026, and 4,098.

• Type 2: Triangulations with vertices obtained from magsat satellite data. The freestripack package is used to generate the triangulations from these vertices. Thenumber of vertices to be used are 204, 414, 836, and 1,635.

Assume that B1, B2, . . . , BM are the basis functions for the approximation spaceS0

d (�). The entry Ai j , for i, j = 1, . . . , M , of the stiffness matrix A in (2.21) iscomputed by

Ai j = −∫

S

(N Bi )(x)B j (x) dσx + ω2∫

S

Bi (x) dσx

∫

S

B j (x) dσx . (6.2)

123



The first integral in (6.2) is computed by using the following formula

−∫

S

(N u)v dσ = 1

4π

∫

S

∫

S

−→curlSu(x) · −→

curlSv( y)|x − y| dσx dσ y

for any smooth functions u and v; see [13, Theorem 3.3.2]. Here,−→curlSv is the vectorial

surface rotation defined by

−→curlSv = −∂v

∂θ

−→eϕ + 1

sin θ

∂v

∂ϕ

−→eθ ,

where −→eϕ ,−→eθ are the two unit vectors corresponding to the Euler angles. Therefore

−∫

S

(N Bi )B j dσ = 1

4π

∫

S

∫

S

−→curlS Bi (x) · −→

curlS B j ( y)

|x − y| dσx dσ y

= 1

4π

∑τ∈�

∑τ ′∈�

∫

τ

∫

τ ′

−→curlS Bi (x) · −→

curlS B j ( y)

|x − y| dσx dσ y. (6.3)

Computation of the double integrals in (6.3) requires evaluation of integrals of thetype

∫

τ (1)

∫

τ (2)

f1(x) f2( y)|x − y| dσx dσ y,

where τ (1) and τ (2) are spherical triangles in �h and the functions f1 and f2 are ana-lytic for all x ∈ τ (1) and y ∈ τ (2). For more details about the above evaluation, pleaserefer to [15].

The right hand side of the linear system (2.21) has entries given by

bi =∫

S

Bi (x) f (x) dσx =∑τ∈�

∫

τ

Bi (x) f (x) dσx, i = 1, . . . , M. (6.4)

The computation of the right hand side as seen in (6.4) includes the evaluation ofintegrals of a smooth function g over a spherical triangle τ . This computation wasdiscussed in [15].

We now discuss the overlapping additive Schwarz algorithm. The algorithm con-sists of constructing the subdomains, the subproblem for each subdomain, the stiffnessmatrix for the coarse mesh, the transformation matrices between the coarse and finemeshes, and finally solving the problem with the preconditioned conjugate gradientmethod. Recall that the subdomains are constructed by

�i = ∪{τ ∈ �h : τ ∩ τ iH �= ∅}, i = 1, . . . , J,

123



Table 2 Condition numberswhen d = 1, ω2 = 0.01 withuniform triangulations

DoF h κ(A) H κ(P)

66 0.3536 20.90 0.7071 5.37

258 0.1768 39.22 0.3536 6.05

0.7071 6.45

1,026 0.0883 75.93 0.1768 6.46

0.3536 6.68

0.7071 7.45

4,098 0.0442 149.40 0.0883 6.71

0.1768 6.80

0.3536 7.90

0.7071 9.05


DoF h κ(A) H κ(P)

258 0.3536 736.53 0.7071 7.24

1,026 0.1768 1,422.77 0.3536 6.92

0.7071 7.06

4,098 0.0883 2,798.35 0.1768 7.22

0.3536 6.90

0.7071 6.59

where τ iH ∈ �H . A pseudo–code to construct the overlapping subdomains is as

follows.Input: Sets of triangles of the coarse mesh �H and fine mesh �h .Output: A set of subdomains {�i : i = 1, . . . , J } where each subdomain

consists of triangles of the fine mesh.foreach τ i

H ∈ �H do� j = τ i

Hforeach τ := 〈x1, x2, x3〉 ∈ �h do

if at least one of x1, x2, x3 belongs to τ iH then

�i = �i ∪ τ ;end

endend

We note that this construction yields overlapping subdomains with overlap size δ

proportional to h.The results for uniform triangulations are presented in Tables 1, 2, 3 and 4. In

Table 1 we computed the experimented rate of increasing κ(A) = O(hα), and thenumbers show that α ≈ −1 as predicted by Proposition 2.6. Tables 2, 3 and 4 showthe significant improvement of our preconditioner.

The results for triangulations generated from satellite data are presented in Tables 5,6 and 7. Again the advantage of the preconditioner can be observed.

123




DoF h κ(A) H κ(P)

578 0.3536 25,735.72 0.7071 412.96

2,306 0.1768 50,877.82 0.3536 13.49

0.7071 24.62

9,218 0.0883 100,876.63 0.1768 7.94

0.3536 8.62

0.7071 8.13

Table 5 Condition numberswhen d = 1, ω2 = 0.001 withmagsat satellite data

DoF h κ(A) H κ(P)

204 0.511 390.4 1.307 19.5

1.670 19.2

2.292 26.7

414 0.370 509.9 1.307 17.2

1.670 17.0

2.292 27.1

836 0.280 785.2 1.307 13.8

1.670 14.9

2.292 28.3

1,635 0.184 1,048.5 1.307 11.1

1.670 16.3

2.292 31.5


DoF h κ(A) H κ(P)

810 0.511 1,170.9 1.307 23.2

1.670 11.7

2.292 5.7

1,650 0.370 1,616.9 1.307 17.7

1.670 9.0

2.292 10.0

3,338 0.280 2,230.5 1.307 13.2

1.670 11.1

2.292 8.1

6,534 0.184 3,046.7 1.307 13.1

1.670 9.0

2.292 6.3

7 Appendix A

The proof for the boundedness of the quasi-interpolation operators in H1/2(S)-norm(Lemma 2.3) requires spherical versions of Poincaré and Friedrichs type inequalities,which have been proved for open subsets of R

n ; see e.g. [17].

123




DoF h κ(A) H κ(P)

1,820 0.511 3,322.0 1.307 23.9

1.670 10.9

2.292 116.6

3,710 0.370 4,696.2 1.307 16.5

1.670 7.7

2.292 186.9

7,508 0.280 6,211.0 1.307 12.1

1.670 12.0

2.292 167.6

Proposition 7.1 Let � ⊆ S be a Lipschitz domain and let fi , i = 1, . . . , L , L ≥ 1,be functionals (not necessarily linear) in H1(�), such that, if v is constant in �,

L∑i=1

| fi (v)|2 = 0 ⇔ v = 0.

Then, there exist constants depending only on � and the functionals fi , such that, foru ∈ H1(�),

‖u‖2L2(�) ≤ C1|u|2H1(�)

+ C2

L∑i=1

| fi (u)|2.

Proof For any u ∈ H1(�), we denote by u ∈ H1(R(�)) the restriction on R(�) (see(2.2)) of the homogeneous extension of degree 0 of u. It is obvious that u = u ◦ R�

where R� is the radial projection defined in (2.3). For any i = 1, . . . , L , we definethe functional fi : H1(R(�)) → R by

fi (u) = fi (u) ∀u ∈ H1(R(�)),

where u = u ◦ R−1� . Let v be a constant function in T�. Then v = v ◦ R−1

� is a constantfunction in �. We have

L∑i=1

| fi (v)|2 = 0 ⇔L∑

i=1

| fi (v)|2 = 0

⇔ v = 0

⇔ v = 0.

By Theorem A.12 in [17], there exist constants C ′1 and C ′

2 depending only on R(�)

and functional fi , such that, for u ∈ H1(R(�)),

‖u‖2L2(R(�)) ≤ C ′

1|u|2H1(R(�))+ C ′

2

L∑i=1

| fi (u)|2.

123



By the definition of fi and noting that ‖u‖L2(�) ‖u‖L2(R(�)) and |u|H1(�) |u|H1(R(�)), see [12], we have

‖u‖2L2(�) ≤ C1|u|2H1(�)

+ C2

L∑i=1

| fi (u)|2.

The lemma is proved. ��

The above lemma, together with a scaling argument as in the case of open sets inR

n (see e.g. [17]), yields the following spherical versions of Poincaré and Friedrichstype inequalities.

Lemma 7.2 Let � ⊆ S be a Lipschitz domain with diameter H. Then, there existconstants C, C1 and C2, depending only on the shape of � but not on H, such that

‖u‖L2(�) ≤ C H |u|H1(�)

for u ∈ H1(�) with vanishing mean value on �(i.e.∫S

u dσ = 0). Moreover, if� ⊆ ∂� has a length of order H, then

‖u‖2L2(�) ≤ C1 H2|u|2H1(�)

+ C2 H‖u‖2L2(�)

for u ∈ H1(�).

8 Appendix B

We first define a partition of unity {θi }Ji=1. Note here that the partition of unity will

be defined for the extended spherical triangles {τ iH,h : i = 1, . . . , J } and it is also a

partition of unity for the overlapping subdomains {�i : i = 1, . . . , J }.Let x be a point on the sphere S. Then θi (x), for i = 1, . . . , J , are defined as

follows:

• Case 1: x belongs to only one triangle τi0H,h for some i0 ∈ {1, . . . , J } (e.g. x ∈ A

in Fig. 3). Then

θi0(x) := 1 and θi (x) := 0, for all i �= i0.

• Case 2: x belongs to exactly two extended triangles τi0H,h and τ

i1H,h (e.g. x ∈ B in

Fig. 3). Then

123



Fig. 3 τi : equilateral triangles, i = 1, . . . , 6

Fig. 4 Supports Ti and rectangles Wi

θi0(x) := δ−1 dist(x, ∂τi0H,h)

θi1(x) := δ−1 dist(x, ∂τi1H,h)

θi (x) := 0, i /∈ {i0, i1},

where ∂τ iH,h is the boundary of τ i

H,h .

123



• Case 3: x belongs to the intersection of more than two extended triangles (i.e.x ∈ U � in Fig. 3). For simplicity of notation we denote the six triangles in Fig. 3by τ1, . . . , τ6. Then

θi (x) := 0, i /∈ {1, 2, 3, 4, 5, 6}

θ1(x) =⎧⎨⎩

δ−1 dist(x, ∂τ1), if x ∈ U3 ∪ U7δ−2 dist(x, ∂τ1 ∩ int(τ5)) dist(x, ∂τ1 ∩ int(τ3)), if x ∈ U1 ∪ U2 ∪ U50, if x ∈ U4 ∪ U6,

θ2(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

δ−2 dist(x, ∂τ2 ∩ int(τ6)) dist(x, ∂τ2 ∩ int(τ4)), if x ∈ U3δ−2 dist(x, ∂τ2) dist(x, ∂τ1 ∩ int(τ5)), if x ∈ U2 ∪ U4δ−3 dist(x, ∂τ2 ∩ int(τ6))

× dist(x, ∂τ2 ∩ int(τ4)) dist(x, ∂τ1 ∩ int(τ5)), if x ∈ U10, if x ∈ U5 ∪ U6 ∪ U7,

Here, int(τi ) denotes the interior of τi . The function θ4(x) is defined similarly toθ1(x), and θ3(x), θ5(x), θ6(x) similarly to θ2(x).

The supports Ti of these functions θi can be one of the four shapes in Fig. 4. Therectangles Wi mentioned in Lemma 5.5 are also depicted in Fig. 4.

We will frequently use the following results; see [9].

Lemma 8.1 Let 0 < α < min{β, β ′}. If v ∈ H1/2([0, β ′] × [0, β]) then

β∫

0

⎛⎜⎝

β ′∫

α

|v(x, y)|2x

dx

⎞⎟⎠ dy �

(1 + log2 β ′

α

)�v�H1/2([0,β ′]×[0,β]), (8.1)

and

1

α‖v‖2

L2([0,α]×[0,β]) �(

1 + logβ ′

α

)2

�v�2H1/2([0,β ′]×[0,β]). (8.2)

We note that even though (8.1) and (8.2) are proved in [9] for β = β ′, the results stillhold for β �= β ′.

Recall that for each subset � ⊂ S, R(�) is the image of � under the inverse of theradial projection R�; see (2.2) and (2.3). It was shown in [12, Lemma 3.1] that

‖v‖L2(�) ‖v0‖L2(R(�)) and ‖v‖H1(�) ‖v0‖H1(R(�))

for any v belongs to L2(�) and H1(�), respectively. Using interpolation (see [11,Theorem B2]), we obtain

‖v‖H1/2(�) ‖v0‖H1/2(R(�)), v ∈ H1/2(�). (8.3)

123



From (2.7), (2.8) and (2.9) and repeating the argument used in the proof of [12, Lemma3.1], we obtain that for any v ∈ H1/2(�), there hold

�v�H1/2(�) �v0�H1/2(R(�))

�v�H1/2(�) �v0�H1/2(R(�))

|v|H1/2(�) |v0|H1/2(R(�)),

(8.4)

where the constants are independent of the size of �.

Proof for Lemma 5.5 Recall that we need to prove

J∑i=1


�J∑

i=1

(1 + log

Hi

δ

)2

�v�2H1/2(Wi )

,

where for any i = 1, . . . , J, Ti is the support of the partition function θi and Wi is therectangle which contains Ti and shares a common edge with Ti (see Fig. 4). We willprove


�(

1 + logHi

δ

)2

�v�2H1/2(Wi )

(8.5)

for Ti being of the first shape in Fig. 4. The cases when Ti is of other shapes can beproved in the same manner.

Equivalences (8.4) and (8.3) allow us to prove instead of (8.5) the followinginequality

�θiv0�2H1/2(R(Ti ))

�(

1 + logHi

δ

)2

�v0�2H1/2(R(Wi ))

. (8.6)

For notational convenience, in this proof we write Ti , Wi , θi , and v instead ofR(Ti ), R(Wi ), ¯(θi )0, and v0, namely we think of Ti and Wi as planar regions, and θi

and v as two variables functions. Here Hi and δ are the size of Ti and the size of theoverlap.

It is noted that Wi = [0, Hi ] × [0, H ′i ] where H ′

i = √3/2Hi . Recall that


= |θiv|2H1/2(Ti )+

∫

Ti

[θiv(x)]2

dist(x, ∂Ti )dx, (8.7)

where

|θiv|2H1/2(Ti )=

∫

Ti

∫

Ti

|θiv(x) − θiv(x′)|2|x − x′|3 dx dx′. (8.8)

123



Fig. 5 Ti and Wi

We first estimate the second term in the right hand side of (8.7), which is split into asum of integrals over the triangles T �

i := A�∪B�∪C�∪D� (see Fig. 5), for � = 1, 2, 3,as follows:

∫

Ti

[θiv(x)]2

dist(x, ∂Ti )dx =

3∑�=1

∫

T �i

[θiv(x)]2

dist(x, ∂Ti )dx.

We only need to estimate the integral over T 1i , the other two can be bounded similarly.

Recall that

θi (x) =

⎧⎪⎪⎨⎪⎪⎩

1, if x ∈ A1,

δ−1 dist(x, ∂Ti ), if x ∈ B1,

δ−2 dist(x, ∂Ti ) dist(x, �1), if x ∈ C1,

δ−2 dist(x, ∂Ti ) dist(x, �2), if x ∈ D1.

Since δ−1 dist(x, �k) ≤ 1 for k = 1, 2 and x, respectively belongs to C1 and D1, thereholds

θi (x) ≤ δ−1 dist(x, ∂Ti ) ∀x ∈ B1 ∪ C1 ∪ D1.

123



Hence

∫

T 1i

[θiv(x)]2

dist(x, ∂Ti )dx =

∫

A1

[θiv(x)]2

dist(x, ∂Ti )dx +

∫

B1∪C1∪D1

[θiv(x)]2

dist(x, ∂Ti )dx

�∫

A1

|v(x)|2y

dx + 1

δ

∫

B1∪C1∪D1

|v(x)|2 dx

≤Hi∫

0

⎛⎜⎝

H ′i∫

δ

|v(x, y)|2y

dy

⎞⎟⎠ dx + 1

δ

Hi∫

0

δ∫

0

|v(x, y)|2 dy dx .

It follows from Lemma 8.1 and H ′i Hi that

∫

T 1i

[θiv(x)]2

dist(x, ∂Ti )dx �

(1 + log

Hi

δ

)2

�v�2H1/2(Wi )

. (8.9)

We now need to use (2.14) to estimate the double integral in (8.8). This motivates usto transform the integral over the triangle Ti into the integral over the rectangle Wi .To do so, we first introduce an extension θi of θi over the rectangle Wi as follows (seeFig. 5):

θi (x) :=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

θi (x), if x ∈ Ti ,

δ−1 dist(x, �k), if x ∈ Fk, k = 1, 2,

δ−2 dist(x, �k) dist(x, �3), if x ∈ Ek, k = 1, 2,

δ−2 dist(x, �1) dist(x, �2), if x ∈ Hk, k = 1, 2,

1, if x ∈ Gk, k = 1, 2.

(8.10)

The extension θi is defined in order to preserve the continuity and symmetry of θi

across the edges �1 and �2 of Ti . Since Ti ⊂ Wi and θi = θi on Ti , there holds

∫

Ti

∫

Ti

|θiv(x) − θiv(x′)|2|x − x′|3 dx dx′ ≤

∫

Wi

∫

Wi

|θiv(x) − θiv(x′)|2|x − x′|3 dx dx′.

Noting (2.14), we have

∫

Wi

∫

Wi

|θiv(x) − θiv(x′)|2|x − x′|3 dx dx′

∫

I

∫

I

‖θiv(x, ·) − θiv(x ′, ·)‖2L2(I ′)

|x − x ′|2 dx dx ′

+∫

I ′

∫

I ′

‖θiv(·, y) − θiv(·, y′)‖2L2(I )

|y − y′|2 dy dy′

=: I1 + I2, (8.11)

123



where I = [0, Hi ], I ′ = [0, H ′i ], x = (x, y) and x′ = (x ′, y′). We will show that I1

is bounded by

(1 + log

Hi

δ

)2

�v�2H1/2(Wi )

.

The term I2 can be estimated in the same manner. By using the triangular inequality,and noting that θi ≤ 1, we have

I1 ≤∫

I

∫

I

‖[θi (x, ·) − θi (x ′, ·)]v(x, ·)‖2L2(I ′)

|x − x ′|2 dx dx ′

+∫

I

∫

I

‖v(x, ·) − v(x ′, ·)‖2L2(I ′)

|x − x ′|2 dx dx ′. (8.12)

It follows from (2.14) that the last integral in (8.12) is bounded by �v�2H1/2(Wi )

. Westill need to estimate

AI,I (θiv) :=∫

I

∫

I

‖[θi (x, ·) − θi (x ′, ·)]v(x, ·)‖2L2(I ′)

|x − x ′|2 dx dx ′.

We denote I1 := [0, δ√

3], I2 := [δ√3, Hi − δ√

3], and I3 := [Hi − δ√

3, Hi ].In order to show that AI,I (θiv) is bounded by |v|2

H1/2(Wi ), we will prove that AIk ,I� (θiv)

are bounded by �v�2H1/2(Wi )

, for k, � = 1, 2, 3. By the symmetry of θi , it is sufficient

to prove the estimation for AIk ,I� (θiv) for (k, �) ∈ {(1, 1), (1, 2), (1, 3), (2, 2)}.Let x, x ′ ∈ I1. Elementary calculation (though tedious) reveals

maxy∈I ′ |θi (x, y) − θi (x ′, y)| � δ−1|x − x ′|.

Thus

AI1,I1(θiv) � 1

δ

∫

I1

‖v(x, ·)‖2L2(I ′) dx = 1

δ‖v(·, ·)‖2

L2(I1×I ′).

Noting that the size of I1 is defined to be proportional to δ, and by using (8.2), wededuce

AI1,I1(θiv) �(

1 + logHi

δ

)2

�v�2H1/2(Wi )

. (8.13)

123



We next estimate AI1,I2∪I3(θiv) by estimating the two integral AI1,[δ√

3, 2δ√

3](θiv) and

AI1,[2δ√

3, Hi ](θiv). Repeating the argument used in estimating AI1,I1(θiv), we obtain

AI1,[δ√

3, 2δ√

3](θiv) �(

1 + logHi

δ

)2

�v�2H1/2(Wi )

. (8.14)

We note here that

maxy∈I ′ [θi (x, y) − θi (x ′, y)]2 ≤ 4, ∀x, x ′ ∈ I, (8.15)

and

|x − 2δ√

3| ≥ δ√

3 ∀x ∈ I1. (8.16)

We then deduce from (8.15), (8.16) and (8.2) that

AI1,[2δ√

3, Hi ](θiv) �∫

I1

‖v(x, ·)‖2L2(I ′)

∫

[2δ√

3, Hi ]

1

(x ′ − x)2 dx ′ dx

�∫

I1

Hi − 2δ√

3

(2δ√

3 − x)(Hi − x)‖v(x, ·)‖2

L2(I ′) dx

� 1

δ‖v(·, ·)‖2

L2(I1×I ′)

�(

1 + logHi

δ

)2

�v�2H1/2(Wi )

. (8.17)

We note here that the estimation for AI1,I1(θiv) is based on the fact that the sizeof I1 is proportional to δ and

|θi (x, y) − θi (x ′, y)| � δ−1|x − x ′| ∀x, x ′ ∈ I1, ∀y ∈ I ′. (8.18)

The proof AI1,[δ√

3, Hi ](θiv) is then obtained by first splitting it into AI1,[δ√

3, 2δ√

3](θiv)

and AI1,[2δ√

3, Hi ](θiv) in which the former is bounded by using similar argument as in

the proof for AI1,I1(θiv), requiring the sizes of I1 and [δ√3, 2δ√

3] to be proportionalto δ and (8.18) to hold for x ∈ I1, x ′ ∈ [δ√3, 2δ

√3]. The latter is estimated by first

using (8.15) to write it in the form containing the integral

∫

[2δ√

3,Hi ]

1

|x − x ′|2 dx ′,

123



which is then proved to be bounded by cδ−1 for some constant c > 0. This procedurecan be used in estimating AI2,I2(θiv) by first writing

AI2,I2(θiv) =∫

I2

∫

I2

∫

[0,H ′i ]

[θi (x, y) − θi (x ′, y)]2v2(x, y)

|x − x ′|2 dy dx dx ′.

The integral is then split into sum of integrals over subregions in which (x, y) and(x ′, y) can belong to one of the following sets G1, F1 ∪ H1, B3 ∪C3 ∪ D2 ∪ H2, H1 ∪C3 ∪ D2 ∪ B2, H2 ∪ F2, G2, and A1 ∪ A2 ∪ A3 in Fig. 5. The integral when (x, y)

and (x ′, y) belong to A1 ∪ A2 ∪ A3 is easily bounded by �v�H1/2(Wi )by using (2.14),

noting that θi (x, y) = θi (x ′, y) = 1. The other integrals can be estimated by usingsimilar argument as used in the proof for AI1,I (θiv).

Combining all these we obtain (8.5). ��Acknowledgments We thank the anonymous referees whose constructive remarks improve the presen-tation of this paper.

References

1. Ainsworth, M., Guo, B.: Analysis of iterative sub-structuring techniques for boundary element approx-imation of the hypersingular operator in three dimensions. Appl. Anal. 81, 241–280 (2002)

2. Alfeld, P., Neamtu, M., Schumaker, L.L.: Bernstein–Bézier polynomials on spheres and sphere-likesurfaces. Comput. Aided Geom. Des. 13, 333–349 (1996)

3. Alfeld, P., Neamtu, M., Schumaker, L.L.: Dimension and local bases of homogeneous splinespaces. SIAM J. Math. Anal. 27, 1482–1501 (1996)

4. Alfeld, P., Neamtu, M., Schumaker, L.L.: Fitting scattered data on sphere-like surfaces using sphericalsplines. J. Comput. Appl. Math. 73, 5–43 (1996)

5. Baramidze, V., Lai, M.-J.: Error bounds for minimal energy interpolatory spherical splines. In: Approx-imation Theory XI: Gatlinburg 2004, Mod. Methods Math., pp 25–50. Nashboro Press, Brentwood(2005)

6. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin (1976)7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer,

Berlin (2002)8. de Klerk, J.: Hypersingular integral equations—past, present, future. Nonlinear Anal. 63, 533–540

(2005)9. Dryja, M., Widlund, O.B.: Domain decomposition algorithms with small overlap. SIAM J. Sci. Com-

put. 15, 604–620 (1994)10. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer,

New York (1972)11. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. CUP, Cambridge (2000)12. Neamtu, M., Schumaker, L.L.: On the approximation order of splines on spherical triangulations. Adv.

Comput. Math. 21, 3–20 (2004)13. Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Springer, New York (2000)14. Necas, J.: Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris (1967)15. Pham, T.D., Tran, T., Chernov, A.: Pseudodifferential equations on the sphere with spherical splines.

M3AS (in press). http://www.maths.unsw.edu.au/applied/pubs/apppreprints2010.html16. Stephan, E.P.: Boundary integral equations for screen problems in R

3. Integral Equ. Operator Theory10, 236–257 (1987)

17. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. In: SpringerSeries in Computational Mathematics, vol. 34. Springer, Berlin (2005)

123


http://www.maths.unsw.edu.au/applied/pubs/apppreprints2010.html


18. Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Preconditioners for pseudodifferential equa-tions on the sphere with radial basis functions. Numer. Math. 115, 141–163 (2010). doi:10.1007/s00211-009-0269-8

19. Tran, T., Stephan, E.P.: Additive Schwarz methods for the h version boundary element method. Appl.Anal. 60, 63–84 (1996)

20. Tran, T., Stephan, E.P.: An overlapping additive Schwarz preconditioner for boundary element approx-imations to the Laplace screen and Lamé crack problems. J. Numer. Math. 12, 311–330 (2004)

21. Wendland, W.L., Stephan, E.P.: A hypersingular boundary integral method for two-dimensional screenand crack problems. Arch. Ration. Mech. Anal. 112, 363–390 (1990)

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