| The standard continuous Galerkin finite element method (CGFEM) is a widely-utilized and well-understood numerical method for solving a large class of partial differential equations (PDEs). Their numerical solutions, however, are not locally conservative. In this dissertation, we develop post-processing techniques for CGFEM to obtain locally conservative fluxes.;The post-processing technique proposed in this dissertation consists of two steps. The first step is to construct a dual mesh consisting of control volumes from the original mesh such that local conservation holds on these control volumes. For linear CGFEM, a linear dual mesh is created, while for k-th order method where k>1, either linear dual mesh (a coarser mesh) or k-th order dual mesh (a finer mesh) depending on the demand of the application is created. This is an easy but fundamental step. The second step is to set and solve, on each element independently, a local/elemental problem. It is a pure Neumann boundary value type problem which yields a very low dimensional linear algebra system. Existence and uniqueness are established by verifying compatibility condition as well as the invertibility of the derived linear algebra system. Also, the technique has a flexibility to construct different types of dual meshes and a flexibility to choose different approximation spaces to search for the post-processed locally conservative numerical solutions.;The post-processing technique is validated both theoretically and numerically. Analysis on the technique is given. On one hand, the post-processed solutions are proved to satisfy the local conservation. On the other hand, optimal convergence rates of errors of the post-processed solutions in both L 2 norm and in H1 semi-norm are achieved. Numerically, various examples and experiments, including simulations of multiphase flow in porous media, elasticity problems, and electron motion in semiconductor material (governed by drift-diffusion equations), are presented to demonstrate the performance of the post-processing techniques. |
