A Finite Element Method with Singularity Reconstruction for Fractional Boundary Value Problems

We consider a two-point boundary value problem involving a Riemann−Liouville fractional derivative of order α ∈ (1,2) in the leading term on the unit interval (0,1). The standard Galerkin finite element method can only give a low-order convergence even if the source term is very smooth due to the presence of the singularity term xα − 1 in the solution representation. In order to enhance the convergence, we develop a simple singularity reconstruction strategy by splitting the solution into a singular part and a regular part, where the former captures explicitly the singularity. We derive a new variational formulation for the regular part, and show that the Galerkin approximation of the regular part can achieve a better convergence order in the L2(0,1), Hα/ 2(0,1) and L∞(0,1)-norms than the standard Galerkin approach, with a convergence rate for the recovered singularity strength identical with the L2(0,1) error estimate. The reconstruction approach is very flexible in handling explicit singularity, and it is further extended to the case of a Neumann type boundary condition on the left end point, which involves a strong singularity xα − 2. Extensive numerical results confirm the theoretical study and efficiency of the proposed approach.