Inhomogeneous Dirichlet Boundary-Value Problems of Space-Fractional Diffusion Equations and their Finite Element Approximations

We prove the wellposedness of the Galerkin weak formulation and Petrov--Galerkin weak formulation for inhomogeneous Dirichlet boundary-value problems of constant- or variable-coefficient conservative Caputo space-fractional diffusion equations. We also show that the weak solutions to their Riemann--Liouville analogues do not exist, in general. In addition, we develop an indirect finite element method for the Dirichlet boundary-value problems of Caputo fractional differential equations, which reduces the computational work for the numerical solution of variable-coefficient fractional diffusion equations from $O(N^3)$ to $O(N)$ and the memory requirement from $O(N^2)$ to $O(N)$ on any quasiuniform space partition. We further prove a nearly sharp error estimate for the method, which is expressed in terms of the smoothness of the prescribed data of the problem only. We carry out numerical experiments to investigate the performance of the method in comparison with the Galerkin finite element method.