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.
Mots-clés : Medius error analysis, Morley element, interior penalty, discontinuous Galerkin method, biharmonic, comparison
@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},
publisher = {EDP-Sciences},
volume = {49},
number = {4},
year = {2015},
doi = {10.1051/m2an/2014062},
mrnumber = {3371900},
zbl = {1327.65211},
language = {en},
url = {http://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
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PB - EDP-Sciences
UR - http://www.numdam.org/articles/10.1051/m2an/2014062/
DO - 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. http://www.numdam.org/articles/10.1051/m2an/2014062/
, and , Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117–137. | DOI | MR | Zbl
, Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31 (1977) 45–59. | DOI | MR | Zbl
, and , Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. | DOI | MR | Zbl
, An a posteriori error estimate and a comparison theorem for the nonconforming element. Calcolo 46 (2009) 149–155. | DOI | MR | Zbl
, and , interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23 (2005) 83–118. | MR | Zbl
, and , An a posteriori error estimator for a quadratic -interior penalty method for the biharmonic problem. IMA J. Numer. Anal. 30 (2010) 777–798. | DOI | MR | Zbl
, and , A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles. Numer. Math. 124 (2013) 309–335. | DOI | MR | Zbl
, and , A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes. Comput. Math. Appl. 68 (2014) 2167–2181. | DOI | MR | Zbl
, and , Comparison results of finite element methods for the Poisson model problem. SIAM J. Numer. Anal. 50 (2012) 2803–2823. | DOI | MR | Zbl
P.G. Ciarlet, The finite element method for elliptic problems. Vol. 4. of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl
, , , , and , Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg. 191 (2002) 3669–3750. | DOI | MR | Zbl
and , Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn–Hilliard equation of phase transition. Math. Comp. 76 (2007) 1093–1117. | DOI | MR | Zbl
, and , An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems. IMA J. Numer. Anal. 31 (2011) 281–298. | DOI | MR | Zbl
P. Grisvard, Singularities in boundary value problems. Vol. 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris (1992). | MR | Zbl
, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comp. 79 (2010) 2169–2189. | DOI | MR | Zbl
, and , Convergence and optimality of the adaptive Morley element method. Numer. Math. 121 (2012) 731–752. | DOI | MR | Zbl
and , A priori error analysis for the -version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math. 3 (2003) 596–607. | DOI | MR | Zbl
, The completion of locally refined simplicial partitions created by bisection. Math. Comp. 77 (2008) 227–241. | DOI | MR | Zbl
, Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. (2015). | DOI | MR
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