On the Application of the Continuous Galerkin Finite Element Method for Conservation Problems

One major drawback that prevents the use of the standard continuous Galerkin finite element method in solving conservation problems is its lack of a locally conservative flux. Our present work has developed a simple postprocessing for the continuous Galerkin finite element method resulting in a locally conservative flux 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. These local problems are independent of each other and in two dimensions involve solving only a 3-by-3 system in the case of triangular elements and a 4-by-4 system for quadrilateral elements. A convergence analysis for the method is provided and its performance is demonstrated through numerical examples of multiphase flow with triangular and quadrilateral elements along with a description of its parallel implementation.