Efficient implementation of mixed finite element methods for parabolic problems

Abstract

It is well-known that conventional mixed finite element method (FEM) results in a large saddle point system or non-symmetric system. A splitting mixed FEM is introduced by Yang (Numer. Methods Partial Differential Equations 17: 229–249, 2001), which computes the flux variable by solving a symmetric positive definite system. Then, the original unknown is solved based on the computed flux. In this paper, we prove that the splitting mixed FEM proposed by Yang is equivalent to the conventional mixed FEM provided that the initial data is carefully taken. Moreover, we show that the equivalence also holds for more complicated models, such as splitting mixed FEM for the compressible flow in porous media. Therefore, all available theoretical results of conventional mixed FEMs hold for the splitting ones. Numerical comparisons are provided for the conventional mixed FEM and the splitting mixed FEM. Efficiency of the splitting mixed method is clearly demonstrated. Corresponding results can be extended to many other mixed FEMs, such as Brezzi–Douglas–Marini mixed FEM.