Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients

Efficient spectral-Galerkin algorithms are developed to solve multi-dimensional fractional elliptic equations with variable coefficients in conserved form as well as non-conserved form. These algorithms are extensions of the spectral-Galerkin algorithms for usual elliptic PDEs developed in [24]. More precisely, for separable FPDEs, we construct a direct method by using a matrix diagonalization approach, while for non-separable FPDEs, we employ a preconditioned BICGSTAB method with a suitable separable FPDE with constant-coefficients as preconditioner. The cost of these algorithms is of O(Nd+1) flops where d is the space dimension. We derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries. We also present ample numerical results to validate the algorithms and error estimates.