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A Review of Unified A Posteriori Finite Element Error Control

Published online by Cambridge University Press:  28 May 2015

C. Carstensen ,
M. Eigel ,
R. H. W. Hoppe  and
C. Löbhard
C. Carstensen*
Affiliation:
Department of Mathematics, Humboldt Universität zu Berlin, D-10099 Berlin, Germany Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, Korea
M. Eigel*
Affiliation:
Department of Mathematics, Humboldt Universität zu Berlin, D-10099 Berlin, Germany
R. H. W. Hoppe*
Affiliation:
Department of Mathematics, University of Houston, Houston TX 77204-3008, USA Institute of Mathematics, University of Augsburg, D-86159 Augsburg, Germany
C. Löbhard*
Affiliation:
Department of Mathematics, Humboldt Universität zu Berlin, D-10099 Berlin, Germany
*
Corresponding author.Email address:cc@math.hu-berlin.de
Corresponding author.Email address:eigel@math.hu-berlin.de
Corresponding author.Email address:hoppe@math.uni-augsburg.de
Corresponding author.Email address:loebhard@math.hu-berlin.de

Abstract

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This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations. In the abstract setting of mixed formulations, a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms. Notably, this can be done with an approach which is applicable in the same way to conforming, nonconforming and mixed discretisations. Subsequently, the unified approach is applied to various model problems. In particular, we consider the Laplace, Stokes, Navier-Lamé, and the semi-discrete eddy current equations.

Keywords

65N3065N1535J25A posteriorierror analysisfinite element methodnonconforming finite element methodmixed finite element methodadaptive algorithmPoisson equationLamé equationsStokes equationsMaxwell equationsunified a posteriori error analysisdiscontinuous Galerkinresidual estimator
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 4 , November 2012 , pp. 509 - 558
Copyright
Copyright © Global Science Press Limited 2012

References

[1]Ainsworth, M. and J.Oden, T., A Posteriori Error Estimation in Finite Element Analysis, Wiley Interscience, John Wiley & Sons, New York, 2000.CrossRefGoogle Scholar
[2]Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), pp. 823–864.3.0.CO;2-B>CrossRefGoogle Scholar
[3]Arnold, D.N., Brezzi, F. and Douglas, J., PEERS: A new finite element for plane elasticity, Japan J. Appl. Math, 1(2) (1984), pp. 347–367.CrossRefGoogle Scholar
[4]Arnold, D.N., Brezzi, F., Cockburn, B. and Donatella Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39(5) (2002), pp. 1749–1779.CrossRefGoogle Scholar
[5]Arnold, D.N., Falk, R.S. and Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), pp. 1–155.CrossRefGoogle Scholar
[6]Arnold, D.N., Falk, R.S. and Winther, R., Finite element exterior calculus: From hodge theory to numerical stability, Bull. Amer. Math. Soc., 47 (2010), pp. 281–354.CrossRefGoogle Scholar
[7]Arnold, D.N. and Winther, R., Mixed finite elements for elasticity, Numer. Math., 92 (2002), pp. 401–419.CrossRefGoogle Scholar
[8]Babuška, I. and Strouboulis, T., The Finite Element Method and its Reliability, The Clarendon Press, Oxford University Press, 2001.CrossRefGoogle Scholar
[9]Beck, R., Hiptmair, R., R.H.Hoppe, W. and Wohlmuth, B., Residual based a posteriori error estimators for eddy current computation, ESAIM: Modeling and Numer. Anal., 34 (2000), pp. 159–182.CrossRefGoogle Scholar
[10]Binev, P., Dahmen, W. and Devore, R., Quasi-optimal convergence rate of an adaptive discontinuous galerkin method, Numer. Math., 97 (2004), pp. 219–268.Google Scholar
[11]Bonito, A. and Nochetto, R., Quasi-optimal convergence rate of an adaptive discontinuous galerkin method, SIAM J. Numer. Anal., 48 (2010), pp. 734–771.CrossRefGoogle Scholar
[12]Braess, D., Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 3rd edition, 2007.CrossRefGoogle Scholar
[13]Braess, D., Carstensen, C. and B.Reddy, D., Uniform convergence and a posteriori error estimators for the enhanced strain finite element method, Numer. Math., 96 (2004), pp. 461–479.Google Scholar
[14]Brenner, S.C. and Scott, L.R., The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics, Springer Verlag, New York, 1994.Google Scholar
[15]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
[16]Carstensen, C., A posteriori error estimate for the mixed finite element method, Math. Comp., 66 (1997), pp. 465–476.CrossRefGoogle Scholar
[17]Carstensen, C., Quasi-interpolation and a posteriori error analysis in finite element method, Math. Model. Numer. Anal., 33(6) (1999), pp. 1187–1202.CrossRefGoogle Scholar
[18]Carstensen, C., A unifying theory of a posteriori finite element error control, Numer. Math., 100(4) (2005), pp. 617–637.CrossRefGoogle Scholar
[19]Carstensen, C. and Dolzmann, G., A posteriori error estimates for mixed FEM in elasticity, Numer. Math., 81(2) (1998), pp. 187–209.CrossRefGoogle Scholar
[20]Carstensen, C., Dolzmann, G., Funken, S.A. and D.Helm, S., Locking-free adaptive mixed finite element methods in linear elasticity, Comput. Methods Appl. Mech. Engrg., 190(13–14) (2000), pp. 1701–1718.CrossRefGoogle Scholar
[21]Carstensen, C., Günther, D., Reininghaus, J. and Thiele, J., The Arnold-Winther mixed FEM in linear elasticity. Part I: Implementation and numerical verification, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 3014–3023.CrossRefGoogle Scholar
[22]Carstensen, C. and R.H.Hoppe, W., Unified framework for an a posteriori error analysis of non-standard finite element approximations of H(curl)-elliptic problems, J. Numer. Math., 17 (2009), pp. 27–44.CrossRefGoogle Scholar
[23]Carstensen, C. and Hu, J., A unifying theory of a posteriori error control for nonconforming finite element methods, Numer. Math., 107(3) (2007), pp. 473–502.CrossRefGoogle Scholar
[24]Carstensen, C., Hu, J. and Orlando, A., Framework for the a posteriori error analysis of non-conforming finite element, SIAM J. Numer. Anal., 45(1) (2007), pp. 68–82.CrossRefGoogle Scholar
[25]Carstensen, C., Gudi, T. and Jensen, M., A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Numer. Math., 112(3) (2009), pp. 363–379.CrossRefGoogle Scholar
[26]Cascon, J.M., Kreuzer, C., Nochetto, R.H. and K.Siebert, G., Quasi-optimal rate of convergence of adaptive finite element methods, SIAM J. Numer. Anal., 46 (2008), pp. 2524–2550.CrossRefGoogle Scholar
[27]Ciarlet, P.G., Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, Studies in Mathematics and its Applications. SIAM, 1988.Google Scholar
[28]Ciarlet, P.G., The Finite Element Method for Elliptic Problems, SIAM, 2002.CrossRefGoogle Scholar
[29]Clement, C., Approximation by finite element functions using local regularization, RAIRO Analyse Numérique, 9 (1975), pp. 77–84.Google Scholar
[30]Cockburn, B., Schötzau, D. and Wang, J., Discontinuous Galerkin methods for incompressible elastic materials, Comput. Methods Appl. Mech. Engrg., 195(25-28) (2006), pp. 3184–3204.CrossRefGoogle Scholar
[31]Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124.CrossRefGoogle Scholar
[32]Hansbo, P. and Larson, M.G., Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comput. Methods Appl. Mech. Engrg., 191(17-18) (2002), pp. 1895–1908.CrossRefGoogle Scholar
[33]Hoppe, R.H.W. and Schöberl, J., Convergence of adaptive edge element methods for the 3D eddy currents equations, J. Comp. Math., 27 (2009), pp. 657–676.Google Scholar
[34]Houston, P., Perugia, I. and Schötzau, D., A posteriori error estimation for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations, IMA Journal of Numerical Analysis, 27 (2007), pp. 122–150.CrossRefGoogle Scholar
[35]Kouhia, R. and Stenberg, R., A linear nonconforming finite element method for nearly incompressible elasticity and stokes flow, Comput. Methods Appl. Mech. Engrg., 124(3) (1995), pp. 195–212.CrossRefGoogle Scholar
[36]Monk, P., Finite Element Methods for Maxwell’s equations, Clarendon Press, Oxford, 2003.CrossRefGoogle Scholar
[37]Nédélec, J.C., Mixed finite elements in M3, Numer. Math., 35 (1980), pp. 315–341.CrossRefGoogle Scholar
[38]Nédélec, J.C., A new family of mixed finite elements in M3, Numer. Math., 50 (1986), pp. 57–81.CrossRefGoogle Scholar
[39]Payne, L.E. and Weinberger, H.F., An optimal poincaré inequality for convex domains, Numer. Math., 5 (1960), pp. 286–292.Google Scholar
[40]Schöberl, J., A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), pp. 633–649.CrossRefGoogle Scholar
[41]Schxötzau, D., Schwab, C. and Toselli, A., Mixed hp-DGFEMfor incompressible flows, SIAM J. Numer. Anal., 40(6) (2003), pp. 2171–2194.CrossRefGoogle Scholar
[42]Stevenson, R., Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), pp. 245–269.CrossRefGoogle Scholar
[43]Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996.Google Scholar
[44]Wohlmuth, B. and Hoppe, R.H.W., A comparison of a posteriori error estimators for mixed finite element discretizations, Math. Comp., 82 (1999), pp. 253–279.Google Scholar
[45]Zhong, L., Chen, L., Shu, S., Wittum, G. and Xu, J., Quasi-Optimal Convergence of Adaptive Edge Finite Element Methods for Three Dimensional Indefinite Time-Harmonic Maxwell’s Equations, Tech. Rep., Department of Mathematics, University of California at Irvine, 2010.Google Scholar