Superconvergence of Finite Element Approximations for the Fractional Diffusion-Wave Equation

In this paper, the error estimates of fully discrete finite element approximation for the time fractional diffusion-wave equation are discussed. Based on the standard Galerkin finite element method approach for the spatial discretization and the L1 formula for the approximation of the time fractional derivative, the fully discrete scheme for solving the constant coefficient fractional diffusion-wave equation is obtained and the superconvergence estimate is proposed and analyzed. Further, a fully discrete finite element scheme is presented for solving the variable coefficient fractional diffusion-wave equation and the corresponding error estimates are also established. Finally, numerical experiments are included to support the theoretical results.