Received: 18 July 2016 Revised: 1 August 2017 Accepted: 30 April 2018 DOI: 10.1002/nme.5836 RESEARCH ARTICLE On continuous, discontinuous, mixed, and primal hybrid finite element methods for second-order elliptic problems P. R. B. Devloo 1 C. O. Faria 2 A. M. Farias 3 S. M. Gomes 4 A. F. D. Loula 5 S. M. C. Malta 6 1 Faculdade de Engenharia Civil, Universidade Estadual de Campinas, Campinas, Brazil 2 Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil 3 Departamento de Matemática, Instituto Federal do Norte de Minas Gerais, Salinas, Brazil 4 Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, Brazil 5 Coordenação de Modelagem Computacional, Laboratório Nacional de Computação Científica, Petrópolis, Brazil 6 Coordenação de Métodos Matemáticos e Computacionais, Laboratório Nacional de Computação Científica, Petrópolis, Brazil Correspondence Cristiane O. Faria, ANMAT, Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, Brazil. Email: cofaria@ime.uerj.br Funding information CNPq, the Brazilian Research Council, Grant/Award Number: 310369/2006-1, 308632/2006-0, 352991/92-5, and 475259/2013-0 Summary 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 numer- ically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static con- densation 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 singu- lar two-dimensional problem, the results are for approximation spaces based on given sets of hp-refined quadrilateral and triangular meshes adapted to an inter- nal 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. KEYWORDS adaptivity, continuous Galerkin, discontinuous Galerkin, finite element methods, hybrid formulations, mixed formulation 1 INTRODUCTION This work intends to be a survey on six finite element methods for second-order elliptic problems. A comparison study is presented in terms of stability, accuracy, flexibility in the choice of approximation spaces, solution strategy, and also some aspects of computational efficiency. The considered formulations use approximation spaces according to their original weak forms as follows: 1. single field formulations (discontinuous Galerkin (DG) and continuous Galerkin methods), Int J Numer Methods Eng. 2018;1–25. wileyonlinelibrary.com/journal/nme Copyright © 2018 John Wiley & Sons, Ltd. 1