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.