11 research outputs found
A spectral method for elliptic equations: the Dirichlet problem
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.Comment: This is latex with the standard article style, produced using
Scientific Workplace in a portable format. The paper is 22 pages in length
with 8 figure