Spectral methods with sparse matrices

Abstract

Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N 4) for polynomials of degree ?N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N 2). Certain "algebraic spectral multigrid methods" can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.