H(div)-postprocessing and hybridization of the continuous Galerkin finite element method for second-order elliptic and linear elasticity problems

The standard continuous Galerkin (CG) finite element method for second order elliptic problems suffers from its inability to provide locally conservative flux approximations, a much needed quantity in many applications. In the first part of this thesis, we show how to overcome this shortcoming by using a two step postprocessing. The first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous Galerkin methods. This computation is non-local in that it requires the solution of a symmetric positive definite system, but the system is well conditioned independently of mesh size, so it can be solved at asymptotically optimal cost. The second step is a local element by element postprocessing of the CG solution incorporating the result of the first step. This leads to a locally conservative flux approximation with continuous normal components. This postprocessing applies for the CG method in its standard form or for a hybridized version of it. We present the hybridized version since it allows easy handling of variable-degree polynomials and hanging nodes. Furthermore, we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the Raviart-Thomas mixed method of corresponding degree. In the second part of the thesis, we extend the postprocessing procedure to linear elasticity. Although the new stress is not symmetric, its asymmetry can be controlled by a small term which is of the same order as that of the error of the approximate stress itself.