Numerical Analysis of an H1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

Abstract EN

We discuss and analyze an H1-Galerkin mixed finite element (H1-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 H1-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 H1-GMFE method.

Based on the discussion on the theoretical error analysis in L2-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 H1-norm.

Moreover, we derive and analyze the stability of H1-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.