A hybrid preconditioner of banded matrix approximation and alternating direction implicit iteration for symmetric Sinc–Galerkin linear systems

The symmetric Sinc-Galerkin method applied to a sparable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations (Ψ_x⊗D_y+D_x⊗Ψ_y)u=g, where⊗ is the Kronecker product symbol, Ψ_x and Ψ_y are Toeplitz-plus-diagonal matrices, and D_x and D_y are diagonal matrices. The main contribution of this paper is to present and analyze a two-step preconditioning strategy based on the banded matrix approximation (BMA) and the alternating direction implicit (ADI) iteration for these Sinc-Galerkin systems. In particular, we show that the two-step preconditioner is symmetric positive definite, and the condition number of the preconditioned matrix is bounded by the convergence factor of the involved ADI iteration. Numerical examples show that the new preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc-Galerkin linear system.