Stability and preconditioning for a hybrid approximation on the sphere

TL;DR

This paper introduces a new preconditioning method for saddle-point linear systems from a hybrid spherical approximation scheme, improving computational stability and efficiency.

Contribution

It develops an optimal block-diagonal preconditioner for the hybrid spherical approximation scheme, leveraging recent inf-sup stability results.

Findings

Numerical experiments confirm the effectiveness of the preconditioner.

The scheme ensures stability and convergence of the linear system.

Preconditioning improves computational performance for non-uniform data distributions.

Abstract

This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. Making use of a recently derived inf-sup condition [13] and the Brezzi stability and convergence theorem for this approximation scheme, we show that the linear system can be optimally preconditioned with a suitable block-diagonal preconditioner. Numerical experiments with a non-uniform distribution of data points support the theoretical conclusions.