Optimal Preconditioning Techniques for Finite Volume Approximation of Three-Dimensional Conservative Space-Fractional Diffusion Equations

A Crank–Nicolson finite volume approximation for three-dimensional conservative space-fractional diffusion equation results in large and dense three-level Toeplitz discrete linear systems. Preconditioned Krylov subspace methods with sine transform–based preconditioners are developed to solve these systems, including the preconditioned conjugate gradient (PCG) method for the symmetric case and the preconditioned generalized minimal residual (PGMRES) method for the nonsymmetric case. Moreover, we provide detailed analysis of the convergence of these Krylov subspace methods. Specifically, for the symmetric case, we prove the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2), which results in a linear convergence rate of the PCG method. For the nonsymmetric case, we demonstrate that the PGMRES method also achieves a linear convergence rate independent of discretization stepsizes from the residual point of view. These results imply that the iteration counts of the PCG and PGMRES methods are uniformly bounded and independent of the matrix sizes. Numerical experiments in both symmetric and nonsymmetric cases in two- and three-dimensions are conducted to confirm the optimal performance of the proposed preconditioned Krylov subspace methods.