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Superconvergence error estimate of a finite element method on nonuniform time meshes for reaction-subdiffusion equations. (English) Zbl 07259465
Summary: 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 Grönwall 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.
##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals
FODE
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