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

Serge Nicaise; Sarah Cochez-Dhondt

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 3, page 485-508
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Nicaise, Serge, and Cochez-Dhondt, Sarah. "Adaptive finite element methods for elliptic problems: Abstract framework and applications." ESAIM: Mathematical Modelling and Numerical Analysis 44.3 (2010): 485-508. <http://eudml.org/doc/250819>.

@article{Nicaise2010,
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. },
author = {Nicaise, Serge, Cochez-Dhondt, Sarah},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori estimator; adaptive FEM; discontinuous Galerkin FEM; elliptic boundary value problems; a posteriori estimator; finite element method; convergence; continuous elliptic problems; Hilbert spaces; numerical examples; convergence; adaptive algorithm; Dirichlet boundary value problem; linear convection-diffusion-reaction problems; discontinuous Galerkin method},
language = {eng},
month = {4},
number = {3},
pages = {485-508},
publisher = {EDP Sciences},
title = {Adaptive finite element methods for elliptic problems: Abstract framework and applications},
url = {http://eudml.org/doc/250819},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Nicaise, Serge
AU - Cochez-Dhondt, Sarah
TI - Adaptive finite element methods for elliptic problems: Abstract framework and applications
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/4//
PB - EDP Sciences
VL - 44
IS - 3
SP - 485
EP - 508
AB - 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.
LA - eng
KW - A posteriori estimator; adaptive FEM; discontinuous Galerkin FEM; elliptic boundary value problems; a posteriori estimator; finite element method; convergence; continuous elliptic problems; Hilbert spaces; numerical examples; convergence; adaptive algorithm; Dirichlet boundary value problem; linear convection-diffusion-reaction problems; discontinuous Galerkin method
UR - http://eudml.org/doc/250819
ER -

References

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  1. M. Ainsworth, A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal.45 (2007) 1777–1798 (electronic).  Zbl1151.65083
  2. M. Ainsworth, A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements. SIAM J. Sci. Comput.30 (2009) 189–204.  Zbl1159.65353
  3. M. Ainsworth and J.T. Oden, A Posterior Error Estimation in Finite Element Analysis. Wiley, New York, USA (2000).  Zbl1008.65076
  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.  Zbl1008.65080
  5. I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math.44 (1984) 75–102.  Zbl0574.65098
  6. R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput.44 (1985) 283–301.  Zbl0569.65079
  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.  Zbl1042.65083
  8. P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math.97 (2004) 219–268.  Zbl1063.65120
  9. S. Cochez and S. Nicaise, A posteriori error estimators based on equilibrated fluxes. CMAM (to appear).  Zbl1283.65107
  10. S. Cochez-Dhondt and S. Nicaise, Equilibrated error estimators for discontinuous Galerkin methods. Numer. Meth. PDE24 (2008) 1236–1252.  Zbl1160.65056
  11. M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems. ESAIM: M2AN33 (1999) 627–649.  Zbl0937.78003
  12. W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal.33 (1996) 1106–1124.  Zbl0854.65090
  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.  Zbl1174.65034
  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. Paris345 (2007) 709–712.  Zbl1129.65085
  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.  Zbl1165.65074
  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).  Zbl1190.65165
  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.  Zbl1063.78021
  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.  Zbl1058.65120
  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).  Zbl1140.65083
  20. K.Y. Kim, A posteriori error analysis for locally conservative mixed methods. Math. Comp.76 (2007) 43–66 (electronic).  Zbl1121.65112
  21. K.Y. Kim, A posteriori error estimators for locally conservative methods of nonlinear elliptic problems. Appl. Numer. Math.57 (2007) 1065–1080.  Zbl1125.65098
  22. P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal.20 (1983) 485–509.  Zbl0582.65078
  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).  Zbl1104.65103
  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).  Zbl0970.65113
  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).]  Zbl1016.65074
  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.  Zbl1059.65098
  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.  Zbl1169.65108
  28. R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley-Teubner, Chichester-Stuttgart (1996).  Zbl0853.65108

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