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Adaptive finite element methods for elliptic problems: Abstract framework and applications

Published online by Cambridge University Press:  04 February 2010

Serge Nicaise  and
Sarah Cochez-Dhondt
Serge Nicaise
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France. Serge.Nicaise@univ-valenciennes.fr; Sarah.Cochez@univ-valenciennes.fr
Sarah Cochez-Dhondt
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France. Serge.Nicaise@univ-valenciennes.fr; Sarah.Cochez@univ-valenciennes.fr
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Abstract

We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convection-reaction-diffusion problems approximated by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.

Keywords

A posteriori estimatoradaptive FEMdiscontinuous Galerkin FEM
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 3 , May 2010 , pp. 485 - 508
Copyright
© EDP Sciences, SMAI, 2010

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References

Ainsworth, M., A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 45 (2007) 1777–1798 (electronic). CrossRef
Ainsworth, M., A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements. SIAM J. Sci. Comput. 30 (2009) 189–204. CrossRef
M. Ainsworth and J.T. Oden, A Posterior Error Estimation in Finite Element Analysis. Wiley, New York, USA (2000).
Arnold, D.G., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749–1779. CrossRef
Babuška, I. and Vogelius, M., Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984) 75–102. CrossRef
Bank, R.E. and Weiser, A., Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283–301. CrossRef
Becker, R., Hansbo, P. and Larson, M.G., Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Meth. Appl. Mech. Engrg. 192 (2003) 723–733. CrossRef
Binev, P., Dahmen, W. and DeVore, R., Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. CrossRef
S. Cochez and S. Nicaise, A posteriori error estimators based on equilibrated fluxes. CMAM (to appear).
Cochez-Dhondt, S. and Nicaise, S., Equilibrated error estimators for discontinuous Galerkin methods. Numer. Meth. PDE 24 (2008) 1236–1252. CrossRef
Costabel, M., Dauge, M. and Nicaise, S., Singularities of Maxwell interface problems. ESAIM: M2AN 33 (1999) 627–649. CrossRef
Dörfler, W., A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. CrossRef
Ern, A. and Stephansen, A.F., A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods. J. Comput. Math. 26 (2008) 488–510.
Ern, A., Nicaise, S. and Vohralík, M., An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. Math. Acad. Sci. Paris 345 (2007) 709–712. CrossRef
Ern, A., Stephansen, A.F. and Zunino, P., A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Anal. 29 (2009) 235–256. CrossRef
A. Ern, A.F. Stephansen and M. Vohralík, Guaranteed and robust discontinuous galerkin a posteriori error estimates for convection-diffusion-reaction problems. JCAM (to appear).
Houston, P., Perugia, I. and Schötzau, D., Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator. Comput. Meth. Appl. Mech. Engrg. 194 (2005) 499–510. CrossRef
Karakashian, O.A. and Pascal, F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. CrossRef
Karakashian, O.A. and Pascal, F., Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 45 (2007) 641–665 (electronic). CrossRef
Kim, K.Y., A posteriori error analysis for locally conservative mixed methods. Math. Comp. 76 (2007) 43–66 (electronic). CrossRef
Kim, K.Y., A posteriori error estimators for locally conservative methods of nonlinear elliptic problems. Appl. Numer. Math. 57 (2007) 1065–1080. CrossRef
Ladevèze, P. and Leguillon, D., Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20 (1983) 485–509. CrossRef
Mekchay, K. and Nochetto, R.H., Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43 (2005) 1803–1827 (electronic). CrossRef
Morin, P., Nochetto, R.H. and Siebert, K.G., Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466–488 (electronic). CrossRef
Morin, P., Nochetto, R.H. and Siebert, K.G., Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631–658 (electronic). [Revised reprint of “Data oscillation and convergence of adaptive FEM”. SIAM J. Numer. Anal. 38 (2001) 466–488 (electronic).] CrossRef
Rivière, B. and Wheeler, M., A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46 (2003) 141–163. CrossRef
Schötzau, D. and Zhu, L., A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math. 59 (2009) 2236–2255. CrossRef
R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley-Teubner, Chichester-Stuttgart (1996).