Amixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations

摘要

We consider the variable-coefficient fractional diffusion equations with two-sided fractional derivative. By introducing an intermediate variable, we propose amixed-type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over H-0(1) (Omega) X H1-beta/2 (Omega). On the basis of the formulation, we develop a mixed-type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz-like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from O(N-2)/ to O(N-2) and the computing cost from O(N-3) to O(N log N). Numerical experiments are included to verify our theoretical findings. Copyright (C) 2017 John Wiley & Sons, Ltd.