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Krylov subspace spectral (KSS) methods have been demonstrated to be effective tools for solving time-dependent variable-coefficient PDE. They employ techniques developed by Golub and Meurant for computing elements of functions of matrices to approximate each Fourier coef- ficient of the solution using a Gaussian quadrature rule that is tailored to that coefficient. In this paper, we apply this same approach to time-independent PDE of the form Lu = g, where L is an elliptic differential operator. Numerical results demonstrate the effectiveness of this approach, in conjunction with residual correction applied on progressively finer grids, for Poisson's equation and the Helmholtz equation.

A Multigrid Block Krylov Subspace Spectral Method for Variable-Coefficient Elliptic PDE