On the postprocessing techniques of the continuous Galerkin finite element

The standard continuous Galerkin finite element method (FEM) is a versatile and well understood method of solving partial differential equations. However, one shortcoming of the method is lack of continuity of derivatives of the approximate solution at element boundaries. This leads to the undesirable consequences for a variety of problems such as a lack of local conservation, which is needed in many problems. In order to find a solution to this shortcoming, a postprocessing has been developed in order to obtain a local conservation with the stand continuous Galerkin finite element method on a vertex centered dual mesh relative to the finite element mesh. The postprocessing requires an auxiliary fully Neumann problem to be solved on each finite element where local problems are independent of each other and involve solving a small system whose size ranges from 3-by-3 to 8-by-8 for the problems presented here depending on the discretization used and the particular partial differential equation being solved. The method is presented for multiphase flow problems using FEM as well as the Generalized Multiscale finite element method (GMsFEM) in the context of multi-phase flow problems. The method is also applied to parabolic problems coupled to multiphase flow using FEM. Finally the postprocessing is used with FEM to solve displacement based linear elasticity problems in order to recover the stress.