Numerical behaviour of multigrid methods for symmetric Sinc–Galerkin systems

The symmetric Sinc–Galerkin method developed by Lund (Math. Comput. 1986; 47 :571–588), when applied to second‐order self‐adjoint boundary value problems on d dimensional rectangular domains, gives rise to an N × N positive definite coefficient matrix which can be viewed as the sum of d Kronecker products among d − 1 real diagonal matrices and one symmetric Toeplitz‐plus‐diagonal matrix. Thus, the resulting coefficient matrix has a strong structure so that it can be advantageously used in solving the discrete system. The main contribution of this paper is to present and analyse a multigrid method for these Sinc–Galerkin systems. In particular, we show by numerical examples that the solution of a discrete symmetric Sinc–Galerkin system can be obtained in an optimal way only using O ( N log N ) arithmetic operations. Numerical examples concerning one‐ and two‐dimensional problems show that the multigrid method is practical and efficient for solving the above symmetric Sinc–Galerkin linear system. Copyright © 2004 John Wiley & Sons, Ltd.