A preconditioned fast finite difference scheme for space-fractional diffusion equations in convex domains

A fast finite difference method is developed for solving space-fractional diffusion equations with variable coefficient in convex domains using a volume penalization approach. The resulting coefficient matrix can be written as the discretized matrix from the extended rectangular domain plus a diagonal matrix with jumping entries due to the penalization parameter. An efficient preconditioner is constructed based on the combination of two approximate inverse circulant matrices. The preconditioned BiCGSTAB method, with the proposed preconditioner, is implemented for solving the resulting linear system. Numerical results are carried out to demonstrate the utility of the proposed algorithm.