A comparative numerical study of different finite element formulations for 2D model elliptic problems: Continuous and discontinuous Galerkin, mixed and hybrid methods

摘要

In this work, different finite element formulations for elliptic problems are implemented and compared, in terms of accuracy versus number of required degrees of freedom. The implemented formulations are: (a) the classical H-1 weak formulation (continuous); (b) the non-symmetric discontinuous Galerkin formulation by Baumann, Oden and Babuska; (c) a mixed discontinuous Galerkin formulation, known as Local Discontinuous Galerkin (LDG); (d) a mixed H(div)-conforming formulation; (e) a primal hybrid formulation. In order to compare the methods, two 2-dimensional test problems are approximated, one having a smooth solution and the second one presenting a square root singularity in a boundary node. The different formulations are compared in terms of the L-2 norm of the approximation errors in the solution and in its gradient (the flux). The tests are performed with h refinement with constant order of approximation p, as well as for a given hp refinement procedure. For the problem with a smooth solution, the results confirm convergence orders predicted by theoretical a priori error estimates. As expected, the application of hp refinement to the singular problem improves considerably the performance of all methods. Furthermore, due to the type of the singularity (square root), the efficiency of continuous and discontinuous Galerkin formulations is further improved by using enriched spaces with quarter-point elements. Regarding continuous, hybrid and mixed formulations, the effect of using static condensation of element equations is also analysed, in order to illustrate the reduction in the global system of equations in each case. A third comparison is given in terms of the conservation of the flux over a curve around a singularity. (C) 2016 Elsevier B.V. All rights reserved.