A spectral method for elliptic equations: the Dirichlet problem
For researchers solving elliptic PDEs on smooth domains, this provides a spectral method with proven high-order convergence and well-conditioned linear systems.
The paper converts an elliptic PDE with Dirichlet boundary conditions to an equivalent equation on the unit ball and applies a spectral Galerkin method using multivariate polynomials. The method converges faster than any power of 1/n, and the condition number grows linearly with system size.
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.