Multigrid preconditioners for symmetric Sinc systems

Abstract

The symmetric Sinc-Galerkin method applied to a separable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations ( ? x ?D y + D x ?? y ) u = g where ? is the Kronecker product symbol, ? x and ? y are Toeplitz-plus-diagonal matrices, and D x and D y are diagonal matrices. The main contribution of this paper is to present a two-step preconditioning strategy based on the banded matrix approximation and the multigrid iteration for these Sinc-Galerkin systems. Numerical examples show that the multigrid preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc-Galerkin linear system.