A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems

TL;DR

This paper revisits a Steklov-spectral method for solving Laplace- Robin boundary value problems, demonstrating exponential convergence for smooth cases and polynomial convergence for non-smooth cases, with new regularity results.

Contribution

It extends Steklov spectral methods to non-tensorial domains, proves a new regularity theorem for eigenfunctions, and compares three numerical approaches for eigenfunction computation.

Findings

Exponential convergence for smooth domains and boundary data.

Polynomial convergence for non-smooth cases.

Three numerical methods for computing Steklov eigenfunctions.

Abstract

In this paper we revisit an approach pioneered by Auchmuty to approximate solutions of the Laplace- Robin boundary value problem. We demonstrate the efficacy of this approach on a large class of non-tensorial domains, in contrast with other spectral approaches for such problems. We establish a spectral approximation theorem showing an exponential fast numerical evaluation with regards to the number of Steklov eigenfunctions used, for smooth domains and smooth boundary data. A polynomial fast numerical evaluation is observed for either non-smooth domains or non-smooth boundary data. We additionally prove a new result on the regularity of the Steklov eigenfunctions, depending on the regularity of the domain boundary. We describe three numerical methods to compute Steklov eigenfunctions.