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 x^{α - 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), H^α/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 x^{α - 2}. Extensive numerical results confirm the theoretical study and efficiency of the proposed approach.