A spectral method for elliptic equations: the Dirichlet problem

Details A spectral method for elliptic equations: the Dirichlet problem A spectral method for elliptic equations: the Dirichlet problem

2008-08-29

arxiv.org/abs/0809.0021v1

Advances in Computational Mathematics, 33 (2010), pp. 169-189

Mathematics

Numerical Analysis

This is latex with the standard article style, produced using Scientific Workplace in a portable format. The paper is 22 pages

Scientific paper

10.1007/s10444-009-9125-8

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.

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