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

**Abstract.**
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 post-processing 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 post-processing 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 multi-phase flow with triangular and quadrilateral elements along
with a description of its parallel implementation.