Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutions

Abstract

In this work, anisotropic bilinear finite element and second‐order temporal approximation are adopted to establish a fully discrete scheme for the fractional substantial diffusion equation with variable coefficient. We prove the proposed discrete scheme is unconditionally stable in the senses of L2$$ {L}^2 $$‐norm and H1$$ {H}^1 $$‐norm. By introducing a new projection, the optimal convergence error in L2$$ {L}^2 $$‐norm and the superclose property in H1$$ {H}^1 $$‐norm are obtained under the condition of u∈H3false(normalΩfalse)$$ u\in {H}^3\left(\Omega \right) $$, where u$$ u $$ is the exact solution of the problem. Through interpolation post processing technique, we arrive at the global superconvergence of the interpolation. Furthermore, an improved algorithm is proposed to solve the problem with nonsmooth solution. Finally, numerical tests are given to illustrate the validity and efficiency of our theoretical analysis.