Superconvergence analysis of an H1-Galerkin mixed finite element method for Sobolev equations

An H1-Galerkin mixed finite element method (MFEM) is discussed for the Sobolev equations with the bilinear element and zero order Raviart-Thomas element (Q11+Q10×Q01). The existence and uniqueness of the solutions about the approximation scheme are proved. Two new important lemmas are given by using the properties of the integral identity and the Bramble-Hilbert lemma, which lead to the superclose results of order O(h2) for original variable u in H1 norm and flux q→ in H(div;Ω) norm under semi-discrete scheme. Furthermore, two new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained. On the other hand, a second order fully-discrete scheme with superclose property O(h2+τ2) is also proposed. At last, numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.