Optimal order error estimates of a mixed discontinuous Galerkin method with BDF-2 scheme for transient Darcy flow

In this paper, we study the numerical solution of Darcy flow via a mixed discontinuous Galerkin method. The two-step Euler backward formula is used for the time variable discretization and a mixed discontinuous Galerkin method is applied to spatial variable discretization. The stability results are established for our mixed discontinuous Galerkin numerical scheme in a fully discrete framework. Optimal error estimates for our unconditional scheme are derived for both the velocity and pressure variables. Finally, a numerical experiment is provided to approve the accuracy and efficiency of the proposed numerical method.