Error and stability estimates for surface-divergence free RBF interpolants on the sphere

Abstract

Abstract. Recently, a new class of surface-divergence free radial basis function interpolants has been developed for surfaces in R 3 . In this paper, several approximation results for this class of interpolants will be derived in the case of the sphere, S 2 . In particular, Sobolev-type error estimates are obtained, as well as optimal stability estimates for the associated interpolation matrices. In addition, a Bernstein estimate and an inverse theorem are also derived. Numerical validation of the theoretical results is also given.

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

“…It can be shown by a similar argument as in [12] that, in terms of the power of the separation radius, these estimates are the best possible. Also note that Theorem 1, when combined with the analogous divergence-free result in [12], automatically gives us stability estimates for A X,Ψ , which we state below.…”

“…Also note that Theorem 1, when combined with the analogous divergence-free result in [12], automatically gives us stability estimates for A X,Ψ , which we state below. (10), then the smallest eigenvalue of the interpolation matrix A X,Ψ can be bounded by…”

“…Following [12], we will show that every eigenvalue of A X,Ψ curl is also bounded from below by λ min (A X,Φ curl ). We will transition from intrinsic to extrinsic coordinates on the sphere using the notation in Section 3.…”

“…[31] in which a new class of positive definite kernels based on RBFs were introduced that yield divergence-free approximants to tangent vector fields on S 2 (and general manifolds). For the case that these divergence-free approximants interpolate the data, stability and Sobolev error estimates have since been given [12]. We use a similar methodology and introduce a kernel from which one can build curl-free approximants and a "full" kernel that can be used for fitting and decomposing a vector field into its divergence-free and curl-free components.…”

“…We use a similar methodology and introduce a kernel from which one can build curl-free approximants and a "full" kernel that can be used for fitting and decomposing a vector field into its divergence-free and curl-free components. Following [12], we will present stability estimates for the interpolation matrices associated with these new kernels. Further, for the full kernel we will present error estimates for both the interpolant and its vector decomposition.…”