Numerical Analysis of an H (1)-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

We discuss and analyze an H (1)-Galerkin mixed finite element (H (1)-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H (1)-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H (1)-GMFE method. Based on the discussion on the theoretical error analysis in L (2)-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H (1)-norm. Moreover, we derive and analyze the stability of H (1)-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.