On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

Summary Finite element formulations for second‐order elliptic problems, including the classic H1‐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. Excepti...

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Bibliographic Details
Published in:International journal for numerical methods in engineering 2018-08, Vol.115 (9), p.1083-1107
Main Authors: Devloo, P. R. B., Faria, C. O., Farias, A. M., Gomes, S. M., Loula, A. F. D., Malta, S. M. C.
Format: Article
Language:English
Subjects:
adaptivity
Approximation
Computation
Computer simulation
Condensation
continuous Galerkin
discontinuous Galerkin
Finite element method
finite element methods
Formulations
Galerkin method
hybrid formulations
Mathematical analysis
mixed formulation
Numerical methods
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Summary:Summary Finite element formulations for second‐order elliptic problems, including the classic H1‐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 L2‐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.
ISSN:0029-5981
1097-0207
DOI:10.1002/nme.5836