Adaptive finite element methods for elliptic problems : abstract framework and applications
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, p. 485-508

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.

DOI : https://doi.org/10.1051/m2an/2010010
Classification:  65N30,  65N15,  65N50
Keywords: a posteriori estimator, adaptive FEM, discontinuous Galerkin FEM
@article{M2AN_2010__44_3_485_0,
author = {Nicaise, Serge and Cochez-Dhondt, Sarah},
title = {Adaptive finite element methods for elliptic problems : abstract framework and applications},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {3},
year = {2010},
pages = {485-508},
doi = {10.1051/m2an/2010010},
zbl = {1191.65158},
mrnumber = {2666652},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_3_485_0}
}

Nicaise, Serge; Cochez-Dhondt, Sarah. Adaptive finite element methods for elliptic problems : abstract framework and applications. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, pp. 485-508. doi : 10.1051/m2an/2010010. http://www.numdam.org/item/M2AN_2010__44_3_485_0/

[1] M. Ainsworth, A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 45 (2007) 1777-1798 (electronic). | Zbl 1151.65083

[2] M. Ainsworth, A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements. SIAM J. Sci. Comput. 30 (2009) 189-204. | Zbl 1159.65353

[3] M. Ainsworth and J.T. Oden, A Posterior Error Estimation in Finite Element Analysis. Wiley, New York, USA (2000). | Zbl 1008.65076

[4] D.G. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749-1779. | Zbl 1008.65080

[5] I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984) 75-102. | Zbl 0574.65098

[6] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283-301. | Zbl 0569.65079

[7] R. Becker, P. Hansbo and M.G. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Meth. Appl. Mech. Engrg. 192 (2003) 723-733. | Zbl 1042.65083

[8] P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219-268. | Zbl 1063.65120

[9] S. Cochez and S. Nicaise, A posteriori error estimators based on equilibrated fluxes. CMAM (to appear). | Zbl 1283.65107

[10] S. Cochez-Dhondt and S. Nicaise, Equilibrated error estimators for discontinuous Galerkin methods. Numer. Meth. PDE 24 (2008) 1236-1252. | Zbl 1160.65056

[11] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems. ESAIM: M2AN 33 (1999) 627-649. | Numdam | Zbl 0937.78003

[12] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | Zbl 0854.65090

[13] A. Ern and A.F. Stephansen, A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods. J. Comput. Math. 26 (2008) 488-510. | Zbl 1174.65034

[14] A. Ern, S. Nicaise and M. Vohralík, An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. Math. Acad. Sci. Paris 345 (2007) 709-712. | Zbl 1129.65085

[15] A. Ern, A.F. Stephansen and P. Zunino, 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. | Zbl 1165.65074

[16] 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). | Zbl 1190.65165

[17] P. Houston, I. Perugia and D. Schötzau, Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator. Comput. Meth. Appl. Mech. Engrg. 194 (2005) 499-510. | Zbl 1063.78021

[18] O.A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order problems. SIAM J. Numer. Anal. 41 (2003) 2374-2399. | Zbl 1058.65120

[19] O.A. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 45 (2007) 641-665 (electronic). | Zbl 1140.65083

[20] K.Y. Kim, A posteriori error analysis for locally conservative mixed methods. Math. Comp. 76 (2007) 43-66 (electronic). | Zbl 1121.65112

[21] K.Y. Kim, A posteriori error estimators for locally conservative methods of nonlinear elliptic problems. Appl. Numer. Math. 57 (2007) 1065-1080. | Zbl 1125.65098

[22] P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20 (1983) 485-509. | Zbl 0582.65078

[23] K. Mekchay and R.H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43 (2005) 1803-1827 (electronic). | Zbl 1104.65103

[24] P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466-488 (electronic). | Zbl 0970.65113

[25] P. Morin, R.H. Nochetto and K.G. Siebert, 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).] | Zbl 1016.65074

[26] B. Rivière and M. Wheeler, A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46 (2003) 141-163. | Zbl 1059.65098

[27] D. Schötzau and L. Zhu, A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math. 59 (2009) 2236-2255. | Zbl 1169.65108

[28] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Chichester-Stuttgart (1996). | Zbl 0853.65108