A new generalization of the P1 non-conforming FEM to higher polynomial degrees
A new generalization of the P1 non-conforming FEM to higher polynomial degrees
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2015Publisher
Institut für Numerische SimulationPublisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11870
Abstract
This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.
Subjects
non-conforming FEM, Helmholtz decomposition, mixed FEM, adaptive FEM, optimality
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.1515/cmam-2016-0031Part of Series
INS Preprints ; 1507Schedensack, Mira: A new generalization of the P1 non-conforming FEM to higher polynomial degrees. In: INS Preprints, 1507.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11870
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11870
@unpublished{handle:20.500.11811/11870,
author = {{Mira Schedensack}},
title = {A new generalization of the P1 non-conforming FEM to higher polynomial degrees},
publisher = {Institut für Numerische Simulation},
year = 2015,
INS Preprints},
volume = 1507,
note = {This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.},
url = {https://hdl.handle.net/20.500.11811/11870}
}
author = {{Mira Schedensack}},
title = {A new generalization of the P1 non-conforming FEM to higher polynomial degrees},
publisher = {Institut für Numerische Simulation},
year = 2015,
INS Preprints},
volume = 1507,
note = {This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.},
url = {https://hdl.handle.net/20.500.11811/11870}
}
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