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.

我们考虑了一种具有奇异性重建的有限元方法，用于分数阶对流扩散方程，其中涉及\(\alpha \in (1,2)\)阶的黎曼-刘维尔导数。金等人。(ESAIM Math. Model. Numer. Anal. 49(5):1261–1283, 2015) 开发了分数反应扩散方程的奇异点重建策略，其中将解分成奇异部分\(x^{\alpha - 1}\)和常规部分\(u^r\)与\(u^r(0)=u^r(1)=0\)。在本文中，我们将原问题转化为单点边值问题，其解u满足条件\(u(0)=({}_0D_x^{\alpha -1}u)(0)=0\). 在域\({\tilde{H}}_L^{\alpha /2}(\Omega ) \times {\tilde{H}}_R^{\alpha /2} (\Omega )\)，在此基础上建立有限元近似方案。分析了连续情况和离散情况的inf-sup条件，从而验证了相应的适定性。的\（L ^ 2 \） -error估计是通过考虑衍生的伴变制剂，其是从原始Petrov的辽金弱形式不同。给出了分段线性和二次有限元的一些数值结果以验证理论发现。