The Time Discontinuous H1-Galerkin Mixed Finite Element Method for Linear Sobolev Equations

We combine the H-1-Galerkin mixed finite element method with the time discontinuous Galerkin method to approximate linear Sobolev equations. The advantages of these two methods are fully utilized. The approximate schemes are established to get the approximate solutions by a piecewise polynomial of degree at most q - 1 with the time variable. The existence and uniqueness of the solutions are proved, and the optimal H-1-norm error estimates are derived. We get high accuracy for both the space and time variables.