A spectral method for elliptic equations: the Dirichlet problem

TL;DR

This paper introduces a spectral Galerkin method for solving elliptic PDEs with Dirichlet boundary conditions by transforming the problem to the unit ball, demonstrating rapid convergence and analyzing numerical stability.

Contribution

It presents a novel spectral approach for elliptic equations on the unit ball, with proven super-polynomial convergence and empirical analysis of the linear system condition number.

Findings

Method converges faster than any polynomial rate for smooth problems.

Condition number of the linear system grows linearly with system size.

Numerical examples in 2D and 3D validate the approach.

Abstract

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.