Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

Abstract

The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation.

Solutions to Pseudodifferential Equations Using Spherical Radial Basis Functions

“…These equations arise, for example, when one solves, 474 T. D. Pham and T. Tran [2] in the exterior of the sphere, the Dirichlet problem with the Laplacian; see [3,10]. The novelty of our work compared to [4,5] lies not only in the negativity of the orders of the operators, but also in the analysis of the error of the approximation. It is well known that in general the collocation method is easier to implement but elicits a more complicated analysis than the Galerkin method.…”

“…Collocation solutions to pseudodifferential equations on the sphere by using spherical radial basis functions have been studied by Morton and Neamtu [5]. Error bounds were later improved by Morton [4].…”

“…Error bounds were later improved by Morton [4]. These papers emphasize operators of positive orders, and the crux of the analysis therein is the transformation of the collocation problem to a Lagrange or Hermite interpolation problem.…”

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

“…(12) Update the pseudoresidual vector p = p + E 0 (y g ). (13) If iter > 0, then set ζ 0 = ζ 1 . (14) Set ζ 1 = p · r. (15) Update the counter, iter = iter +1.…”

“…This elliptic equation arises, for example, when one discretizes in time the diffusion equation on the sphere. When solving elliptic PDEs on the unit sphere based on scattered measured data, with the approximate solution constructed using shifts of a strictly positive definite kernel on the sphere, a very ill-conditioned linear system results, whether a Galerkin method [8] or a collocation method [13] is used. This is due to the separation radius of the scattered data [11], which can be very small for a large set of scattered data.…”