On Continuous, Discontinuous, Mixed, And Primal Hybrid Finite Element Methods For Second-Order Elliptic Problems

Finite element formulations for second-order elliptic problems, including the classic H-1-conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three-dimensional test problem with smooth solution, the simulations are performed with h-refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two-dimensional problem, the results are for approximation spaces based on given sets of hp-refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L-2-convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.