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

Abstract

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 L 2 (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 L 2 (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.