Abstract. We identify discontinuous Galerkin methods for second-order elliptic problems in several space dimensions having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potential as well as the flux. We show that the approximate flux converges in L 2 with the optimal order of k + 1, and that the approximate potential and its numerical trace superconverge, in L 2 -like norms, to suitably chosen projections of the potential, with order k + 2. We also apply element-by-element postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have normal components continuous across inter-element boundaries, to converge in L 2 with order k + 1, and to have a divergence converging in L 2 also with order k + 1. The new approximate potential is proven to converge with order k + 2 in L 2 . Numerical experiments validating these theoretical results are presented.
The standard continuous Galerkin (CG) finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations, a much needed quantity in many applications. 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 nonlocal 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 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.
Introduction.In this paper, we revisit the classical finite element method [13,20], otherwise known as the continuous Galerkin (CG) method, for second order elliptic problems, with the intention of showing how to overcome what is perhaps its main disadvantage, namely, the discontinuity of the normal component of the approximate flux across element interfaces. We show how to achieve this by means of an efficient postprocessing of the approximate solution provided by the CG method. We also show that the postprocessed flux is competitive with the flux provided by the Raviart-Thomas mixed method of corresponding degree.We illustrate our technique in the framework of the model second order elliptic boundary value problem
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