Kronecker product based preconditioners for boundary value method discretizations of space fractional diffusion equations

This paper is concerned with the construction of efficient preconditioners for systems arising from boundary value methods time discretization of space fractional diffusion equations. The boundary value methods lead to a coupled block system which is in the form of the sum of two Kronecker products. Our approach is based on an alternating Kronecker product splitting technique which leads to a splitting iteration method. We show that the splitting iteration converges to the unique solution of the linear system and derive the optimal values of the involved iteration parameters. The splitting iteration is then accelerated by a Krylov subspace method like GMRES. One component of the Kronecker product preconditioners has the same structure as the matrix derived from implicit Euler discretization of the problem. Therefore, we can reuse the available high performance of implicit Euler discretization preconditioners as the building block for our preconditioners. Several numerical experiments are presented to show the effectiveness of our approaches.