Conservative P1 conforming and nonconforming Galerkin FEMS: Effective fluxevaluation via a nonmixed method approach

Given a P1 conforming or nonconforming Galerkin finite element method (GFEM ) solution p(h), which approximates the exact solution p of the diffusion-r eaction equation -del.K del p + alpha p = f with full tensor variable coeff icient K, we evaluate the approximate flux u(h) to the exact flux u = -K de l p by a simple but physically intuitive formula over each finite element. The flux is sought in the continuous ( in normal component) or the disconti nuous Raviart-Thomas space. A systematic way of deriving such a formula is introduced. This direct method retains local conservation property at the e lement level, typical of mixed methods (finite element or finite volume typ e), but avoids solving an indefinite linear system. In short, the present m ethod retains the best of the GFEM and the mixed method but without their s hortcomings. Thus we view our method as a conservative GFEM and demonstrate its equivalence to a certain mixed finite volume box method. The equivalen ce 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; Omega) norm. Numerical results are provided to support the theory.