Error Estimates with Spherical Basis Functions

We employ spherical basis functions for the discretization method we develop to approximate the solution to the problem Lu = f . Consider a spherical basis function φ on S2 that is either positive definite or conditionally positive definite. Let X be a collection of quasi-uniformly scattered centers on S2 _{and let}

VX := Vφ,X =  X ξ∈X aξφ(·, ξ) : X ξ aξp(ξ) = 0 for all p ∈ Π  + Π

where Π is the space of continuous functions corresponding to the conditionally positive definite kernel φ. For this section, we assume Π is the span of spherical harmonics up to a fixed degree L (not to be confused with the differential operator L). We place some restrictions on the behavior of the kernel to guarantee that we may employ known interpolation error estimates. Let τ > 1 and assume that there exists positive constants c, C so that the Fourier coefficients of φ satisfy

c(1 + λ`)−τ ≤ ˆφ` ≤ C(1 + λ`)−τ for all ` ≥ L + 1 (5.5)

where L is the degree of the highest order spherical harmonic in Π.

Let Hk := W₂k_(S2) for shorthand. In the weak formulation, the Hilbert space we search in is H1. For sufficient τ > 1, the space VX ⊂ H1, and consequently we may

considered the discretized following discretized problem: Find uX ∈ VX such that

for all vX ∈ VX

a(uX, vX) = `(vX).

Since VX ⊂ H1, the discretization is conforming, and consequently an application

of Lax-Milgram once again guarantees such a solution exists. We refer to the so- lution uX as the discrete solution in contrast to the solution (or exact solution) u.

Furthermore, we know by Cea’s Lemma that the discrete solution is a near-optimal approximation to u in the discretization space VX. That is,

ku − uXkH1 ≤ C inf

vX∈VX

ku − vXkH1.

Consequently, by choosing the interpolant IXu ∈ VX, it follows that

By identical reasoning, for the problem Lw = u − uX, we know the Galerkin solution

wX ∈ VX satisfies

kw − wXkH1 ≤ Ckw − I_Xwk_H1.

Regarding the regularity of each solution, if the data f ∈ Hs_{, then by the regularity}

result Proposition 15, it follows that u ∈ Hs+2_{. For τ > 1, the interpolant I} Xu is

in the space Hτ +α _{for α < τ − 1. As a result, the difference u − I}

Xu ∈ Hσ where

σ = min(s + 2, τ + α). We may once again apply elliptic regularity Proposition 15 to the problem Lw = u − IXu to find that since u − IXu ∈ Hσ, w ∈ Hσ+2.

We consider the kernel φ = κm, the restricted surface spline that is in the Sobolev

space Wm

2 (S2). Let χξ denote the Lagrange function centered at ξ constructed by

linear combinations of κm(·, η) plus an appropriate polynomial from Πm, the space of

degree m spherical polynomials. The space VX is the space we employ to discretize

the problem (5.1). With these regularity results combined with the approximation powers of the discretization spaces, we may derive error estimates for the problem. Lemma 16. Let τ > n_{2 (for S}n _{or τ > 1 for S}2). Let X be a collection of scattered centers with mesh norm h. Let w be the solution to the problem Lw = u − IXu.

Then,

kw − wXkH1 ≤ Ch2ku − I_Xuk_H3.

Proof. We know w ∈ Hσ+2 _{where σ = min(s + 2, τ + α) for τ >} n

2 and α < τ − n 2.

Then, as a result, σ > 1, so σ + 2 > 3. Therefore, w ∈ H3_{. As a result, we may}

apply the interpolation error estimates for the kernel φ, which allows us to estimate

Furthermore, by the regularity result Proposition 15, it follows

kwkH3 ≤ CkLwk_H1 = Cku − I_Xuk_H1.

We focus on the specific case of n = 2 since the sphere S2 _{is our primary manifold}

of interest. For general Sn_{, the key replacements are that τ >} n

2 and α < τ − n 2.

Theorem 5. Let n = 2 and let Lu = f for f ∈ Hs _{and s ≥ 0. Let u}

X denote the

Galerkin solution to the problem Lu = f in the space VX constructed from scattered

centers with mesh norm h and a kernel φ that satisfies (5.5). Then,

ku − uXkL2 ≤        Chs+2_kuk Hs+2 for s ≤ 2τ − 2 Ch2τ_kuk H2τ for 2τ − 2 < s (5.6)

Proof. The interpolation error estimates derived in [25] (Proposition A.3) show that on Sn, if τ > n₂ and n₂ < µ ≤ 2τ , and β ≤ min(µ, τ ), then

ku − IXukHβ ≤ Chµ−βkuk_Hµ. (5.7)

In the context of our problem, u ∈ Hs+2 _{where s ≥ 0. The highest order approxi-}

mation power possible is 2τ , which occurs in the event 2τ < s + 2, and consequently, the two cases to consider are s ≤ 2τ − 2 and s > 2τ − 2.

By Cea’s Lemma, it follows that

ku − uXkH1 ≤ Cku − I_Xuk_H1.

u − IXu in the H1 norm. In the language of the interpolation error estimate (5.7), β = 1, and we see ku − IXukH1 ≤        Chs+1_kuk Hs+1 for s ≤ 2τ − 2, Ch2τ −1_kuk H2τ for s > 2τ − 2. (5.8)

To recover L2 _{error estimates, we need to apply the Nitsche trick to recover an}

additional order of h. By Corollary 3,

ku − uXk2L2 ≤ kw − I_Xwk_H1ku − I_Xuk_H1.

By applying Lemma 16, we may replace the w − IXw norm by

ku − uXk2L2 ≤ Ch2ku − I_Xuk2_H1

which implies that

ku − uXkL2 ≤ Chku − I_Xuk_H1.

Finally, we may apply (5.8) to acquire the approximation error rates

ku − IXukL2 ≤        Chs+2_kuk Hs+1 for s ≤ 2τ − 2 Ch2τ_kuk H2τ for s > 2τ − 2. (5.9)

The error estimate we derived is under the assumption that the stiffness matrix is assembled precisely with no quadrature error. The issue of quadrature is technical and difficult and we study this problem in a later section. The assumptions on the

kernel are given that the kernel is (conditionally) positive definite and the conditions in (5.5). We remark that our particular interest is in the surface spline φm(x, y) :=

(−1)m_{(1 − x · y)}m−1_{log(1 − x · y).}