An Effective Finite Element Method with Singularity Reconstruction for Fractional Convection-diffusion Equation

Abstract

We consider a finite element method with singularity reconstruction for fractional convection-diffusion equation involving Riemann-Liouville derivative of order $\alpha \in (1,2)$. Jin et al. (ESAIM Math. Model. Numer. Anal. 49(5):1261–1283, 2015) developed a singularity reconstruction strategy for fractional reaction-diffusion equation in which the solution was split into a singular part $x^{\alpha -1}$ and a regular part $u^r$ with $u^r(0)=u^r(1)=0$. In this paper we transform the original problem into a one-point boundary-value problem whose solution u satisfies the condition $u(0)=({}_0D_x^{\alpha -1}u)(0)=0$. A novel Petrov–Galerkin variational formulation is developed on the domain ${\tilde{H}}_L^{\alpha /2}(\Omega ) \times {\tilde{H}}_R^{\alpha /2}(\Omega )$, based on which the finite element approximation scheme is established. The inf-sup conditions for both continuous case and discrete case are analyzed thus the corresponding well-posedness is verified. The $L^2$-error estimate is derived by considering an adjoint variational formulation which is different from the original Petrov–Galerkin weak form. Some numerical results for piecewise linear and quadratic finite elements are presented to verify the theoretical findings.