As modern variant of nonconforming schemes, discontinuous Galerkin finite element methods appear to be highly attractive for fourth-order elliptic PDEs. There exist various modifications and the most prominent versions with first-order convergence properties are the symmetric interior penalty DG method and the interior penalty method which may compete with the classical Morley nonconforming FEM on triangles. Those schemes differ in their various jump and penalisation terms and also in the norms. This paper proves that the best-approximation errors of all the three schemes are equivalent in the sense that their minimal error in the respective norm and the optimal choice of a discrete approximation can be bounded from below and above by each other. The equivalence constants do only depend on the minimal angle of the triangulation and the penalisation parameter of the schemes; they are independent of any regularity requirement and hold for an arbitrarily coarse mesh.
Keywords: Medius error analysis, Morley element, interior penalty, discontinuous Galerkin method, biharmonic, comparison
Carstensen, Carsten 1 ; Gallistl, Dietmar 2 ; Nataraj, Neela 3
@article{M2AN_2015__49_4_977_0,
author = {Carstensen, Carsten and Gallistl, Dietmar and Nataraj, Neela},
title = {Comparison {Results} of {Nonstandard} $P_{2}$ {Finite} {Element} {Methods} for the {Biharmonic} {Problem}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {977--990},
year = {2015},
publisher = {EDP Sciences},
volume = {49},
number = {4},
doi = {10.1051/m2an/2014062},
mrnumber = {3371900},
zbl = {1327.65211},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2014062/}
}
TY - JOUR
AU - Carstensen, Carsten
AU - Gallistl, Dietmar
AU - Nataraj, Neela
TI - Comparison Results of Nonstandard $P_{2}$ Finite Element Methods for the Biharmonic Problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
PY - 2015
SP - 977
EP - 990
VL - 49
IS - 4
PB - EDP Sciences
UR - https://www.numdam.org/articles/10.1051/m2an/2014062/
DO - 10.1051/m2an/2014062
LA - en
ID - M2AN_2015__49_4_977_0
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%A Nataraj, Neela
%T Comparison Results of Nonstandard $P_{2}$ Finite Element Methods for the Biharmonic Problem
%J ESAIM: Mathematical Modelling and Numerical Analysis
%D 2015
%P 977-990
%V 49
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%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2014062/
%R 10.1051/m2an/2014062
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Carstensen, Carsten; Gallistl, Dietmar; Nataraj, Neela. Comparison Results of Nonstandard $P_{2}$ Finite Element Methods for the Biharmonic Problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 977-990. doi: 10.1051/m2an/2014062
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