Superconvergence of FEM for Distributed Order Time Fractional Variable Coefficient Diffusion Equations

In this paper, a numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional variable coefficient diffusion equations is developed and a complete error analysis is provided. The weighted and shifted Grunwald formula is applied for the time-fractional derivative and finite element approach for the spatial discretization. The unconditional stability and the global superconvergence estimate of the fully discrete scheme are proved rigorously. Extensive numerical experiments are presented to illustrate the accuracy and efficiency of the scheme, and to verify the convergence theory.