An H¹-Galerkin mixed finite element method for parabolic partial differential equations

Abstract

In this paper, an H¹-Galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts. Compared to the standard H¹-Galerkin procedure, C¹-continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method. Moreover, it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method, but without LBB consistency condition and quasiuniformity requirement on the finite element mesh. Finally, a better rate of convergence for the flux in L²-norm is derived using a modified H¹-Galerkin mixed method in two and three space dimensions, which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution.