Analysis On H~1-Galerkin Mixed Finite Element Method For Second Order Parabolic Problems

In this paper, we consider H~1-Galerkin mixed finite element and H~1-Galerkin expanded mixed finite element simulations for second order linear parabolic problems. Optimal error estimates for the unknown function and the adjoint vector function are proved without introducing the curl operator.The followling second order parabolic problemsare numerically simulated by an H~1-Galerkin mixed finite element procedure. The equivalence between the initial-boundary value problems and its mixed formulation are presented, also the stability, existence and the uniqueness of the solutions to the mixed form are discussed. Further, the solvability and optimal error estimates both for the semi-discrete scheme and the fully-discrete scheme are proved without introducing the curl operator.Then the followling second order parabolic problems are also numerically simulated by an H~1-Galerkin expanded mixed finite element procedure which inheits the advantages of H~1-Galerkin mixed finite element, such as approximating well the unknown function, its gradient and the adjoint vector-function, and without satisfying the LBB-Constraint. In addition, the new mixed formulation can approximate the solutions well when the coefficient a(x,(?)) becomes very small. The equivalence, stability, existence and uniqueness both for mixed formulation and for discrete procedure are discussed. The optimal error estimates for the unknown function, its gradient and the adjoint vector function are proved without introducing the curl operator. A numerical example confirms the efficiency of our method.