Construction of Locally Conservative Fluxes for High Order Continuous Galerkin Finite Element Methods

We propose a simple post-processing technique for linear and high order continuous Galerkin Finite Element Methods (CGFEMs) to obtain locally conservative flux field. The post-processing technique requires solving an auxiliary problem on each element independently which results in solving a linear algebra system whose size is low for any order CGFEM. The post-processing could have been done directly from the finite element solution that results in locally conservative flux on the element. However, the normal flux is not continuous at the elemental boundary. To construct locally conservative flux field whose normal component is also continuous, we propose to do the post-processing on the nodal-centered control volumes which are constructed from the original finite element mesh. We show that the post-processed solution converges in an optimal fashion to the true solution in an H1 semi-norm. We present various numerical examples to demonstrate the performance of the post-processing technique.