Superconvergence Error Estimate of a Finite Element Method on Nonuniform Time Meshes for Reaction–Subdiffusion Equations

Abstract

In this paper, we consider superconvergence error estimates of finite element method approximation of Caputo’s time fractional reaction–subdiffusion equations under nonuniform time meshes. For the standard Galerkin method we see that the optimal order error estimate of temporal direction cannot be derived from the weak formulation of the problem. We establish a time-space error splitting argument, which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully based on an improved discrete Gronwall inequality. We obtain the sharp temporal $$H^1$$ -norm error estimates with respect to the convergence order of the approximate solution and $$H^1$$ -norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global $$H^1$$ -norm superconvergence results are presented. Finally, we present some numerical results that give insight into the reliability of the theoretical analysis.