Conservative P1 Conforming and Nonconforming Galerkin FEMs: Effective Flux Evaluation via a Nonmixed Method Approach

Given a P1 conforming or nonconforming Galerkin finite element method (GFEM) solution ph, which approximates the exact solution p of the diffusion-reaction equation −∇ · K∇p+ αp = f with full tensor variable coefficient K, we evaluate the approximate flux uh to the exact flux u = −K∇p by a simple but physically intuitive formula over each finite element. The flux is sought in the continuous (in normal component) or the discontinuous Raviart–Thomas space. A systematic way of deriving such a formula is introduced. This direct method retains local conservation property at the element level, typical of mixed methods (finite element or finite volume type), but avoids solving an indefinite linear system. In short, the present method retains the best of the GFEM and the mixed method but without their shortcomings. Thus we view our method as a conservative GFEM and demonstrate its equivalence to a certain mixed finite volume box method. The equivalence theorems explain how the pressure can decouple basically cost free from the mixed formulation. The accuracy in the flux is of first order in the H(div; Ω) norm. Numerical results are provided to support the theory.