Solving elliptic PDEs in unbounded domains

TL;DR

This paper extends a method for accurately solving elliptic PDEs in unbounded domains, removing the need for periodic coefficients and revealing conditions for fast convergence based on Fourier domain properties.

Contribution

It generalizes previous approaches to non-periodic elliptic problems and analyzes the Fourier domain properties influencing convergence speed.

Findings

Method achieves exponential accuracy decay with domain size.

Convergence depends on Fourier domain properties of the right-hand side.

Extension to non-periodic coefficients broadens applicability.

Abstract

An accurate approximation of solutions to elliptic problems in infinite domains is challenging from a computational point of view. This is due to the need to replace the infinite domain with a sufficiently large and bounded computational domain, and posing artificial boundary conditions on the boundary of the truncated computational geometry, which will then pollute the solution in an interior region of interest. For elliptic problems with periodically varying coefficients (with a possibly unknown period), a modelling strategy based on exponentially regularized elliptic problem was previously developed and analysed. The main idea was to replace the infinite domain periodic problem with a regularized elliptic problem posed over a finite domain, while retaining an accuracy decaying exponentially with respect to the size of the truncated domain. In this article, we extend the analysis to problems, where no structural assumptions on the coefficient are assumed. Moreover, the analysis here uncovers an interesting property of the right hand side in the Fourier domain for the method to converge fast for problems beyond periodicity.