  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.
