Mixed Finite Element Methods with Piece-Wise Constant Fluxes

In this dissertation, we consider a new mixed finite element discretization, its error estimation, monotonicity and the approaches to implement local refinement. We also do some numerical experiments to verify error estimates and to see the effect of distorted faces.

In the first part, we introduce a discontinuous Galerkin method based on piece-wise constant fluxes, we elaborate its construction and discretization on triangular meshes. We then consider the monotonicity of this method, compare it with classical RT0 method and extend to KR methods. Finally, error estimation is investigated.

In the second part, we start from reviewing traditional approaches to implement local refinement for the new mixed finite element method. Because of disappearance of monotonicity, we then try to find an alternative way to do local refinement to keep monotonicity.

In the third part, we implement this method on a specially constructed prismatic grid. Numerical results are provided. We then verify the error estimates from the numerical results.