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

Jin, Bangti; Zhou, Zhi

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1261-1283.

We consider a two-point boundary value problem involving a Riemann−Liouville fractional derivative of order $\alpha \in (1,2)$ 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 $x^{\alpha -1}$ 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 $L^2(0,1)$, $H^{\alpha /2}(0,1)$ 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 $x^{\alpha -2}$. Extensive numerical results confirm the theoretical study and efficiency of the proposed approach.

Classification : 65M60, 65N30, 45J05
Mots clés : Finite element method, Riemann−Liouville derivative, fractional boundary value problem, error estimate, singularity reconstruction

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