A spectral collocation method for elliptic PDEs in irregular domains with Fourier extension

TL;DR

This paper introduces a Fourier extension-based spectral collocation method for solving elliptic PDEs in irregular domains, emphasizing low computational cost and effectiveness despite some ill-conditioning issues.

Contribution

It presents a novel oversampling collocation approach leveraging Fourier extension for elliptic PDEs in irregular domains, with analysis of its accuracy and computational advantages.

Findings

Fast approximation error plateau due to frame ill-conditioning

Method effective on irregular domains with low computational cost

Numerical experiments confirm accuracy and efficiency

Abstract

Based on the Fourier extension, we propose an oversampling collocation method for solving the elliptic partial differential equations with variable coefficients over arbitrary irregular domains. This method only uses the function values on the equispaced nodes, which has low computational cost and versatility. While a variety of numerical experiments are presented to demonstrate the effectiveness of this method, it shows that the approximation error fast reaches a plateau with increasing the degrees of freedom, due to the inherent ill-conditioned of frames.