Development of a New Spectral Collocation Method Using Laplacian Eigenbasis for Elliptic Partial Differential Equations in an Extended Domain

TL;DR

This paper introduces a novel spectral collocation method using Laplacian eigenbasis in extended domains, improving convergence and regularity for solving elliptic PDEs in complex geometries.

Contribution

The paper proposes a new spectral collocation approach based on basis functions in extended domains, enhancing convergence and regularity for complex geometrical problems.

Findings

Retains exponential decay convergence for analytical solutions.

Demonstrates effectiveness on 2D Poisson and convection-diffusion equations.

Shows improved handling of complex geometries compared to traditional methods.

Abstract

The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle geometrically complicated problems. However, the convergence is deteriorated when embedded boundary strategies are employed. Owing to the loss of regularity, in this paper we propose a new spectral collocation method which retains the regularity of solutions to solve differential equations in the case of complex geometries. The idea is rooted in the basis functions defined in an extended domain, which leads to a useful upper bound of the Lebesgue constant with respect to the Fourier best approximation. In particular, how the stretching of the domain defining basis functions affects the convergence rate directly is detailed. Error estimates chosen in our proposed method show that the exponential decay convergence for problems with analytical solutions can be retained. Moreover, two-dimensional Poisson equations and convection-diffusion equations with simple and complex geometrical domains will be simulated. The predicted results justify the advantages of applying our method to tackle geometrically complicated problems.