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

TL;DR

This paper introduces a finite element method with singularity reconstruction for fractional boundary value problems, significantly improving convergence rates by explicitly handling solution singularities.

Contribution

A novel singularity reconstruction strategy is developed for fractional boundary value problems, enhancing finite element method accuracy by explicitly capturing singularities.

Findings

Improved convergence rates in multiple norms compared to standard methods.

The approach effectively handles explicit singularities and extends to Neumann boundary conditions.

Numerical results confirm theoretical convergence and efficiency.

Abstract

We consider a two-point boundary value problem involving a Riemann-Liouville fractional derivative of order $\al\in (1,2)$ in the leading term on the unit interval $(0,1)$. Generally 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 $x^{\al-1}$ 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 establish that the Galerkin approximation of the regular part can achieve a better convergence order in the $L^2(0,1)$, $H^{\al/2}(0,1)$ and $L^\infty(0,1)$-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 $x^{\al-2}$. Extensive numerical results confirm the theoretical study and efficiency of the proposed approach.