Multigrid preconditioners for symmetric Sinc systems

The symmetric Sinc-Galerkin method applied to a separable secondorder self-adjoint elliptic boundary value problem gives rise to a system of linear equations (Ψx ⊗ Dy + Dx ⊗ Ψy) u = g where ⊗ is the Kronecker product symbol, Ψx and Ψyare Toeplitz-plus-diagonal matrices, and Dx and Dy 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.