Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error
Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error
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2016Publisher
Institut für Numerische Simulation (INS)Publisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11847
Abstract
Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H1 norm best approximation error estimates for H2 functions hold for arbitrary triangulations. However, similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on the example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.
Subjects
nonconforming P1 elements, lowest order Raviart-Thomas elements, discrete energy norm estimates, divergence of finite element methods, maximum angle condition, distorted triangulations
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.21136/AM.2017.0150-17Part of Series
INS Preprints ; 1618Oswald, Peter: Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error. In: INS Preprints, 1618.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11847
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11847
@unpublished{handle:20.500.11811/11847,
author = {{Peter Oswald}},
title = {Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
INS Preprints},
volume = 1618,
note = {Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H1 norm best approximation error estimates for H2 functions hold for arbitrary triangulations. However, similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on the example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.},
url = {https://hdl.handle.net/20.500.11811/11847}
}
author = {{Peter Oswald}},
title = {Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
INS Preprints},
volume = 1618,
note = {Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H1 norm best approximation error estimates for H2 functions hold for arbitrary triangulations. However, similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on the example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.},
url = {https://hdl.handle.net/20.500.11811/11847}
}
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