Adaptivity and variational stabilization for convection-diffusion equations∗

Albert Cohen; Wolfgang Dahmen; Gerrit Welper

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 5, page 1247-1273
  • ISSN: 0764-583X

Abstract

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In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on other specifications, are explored and illustrated by numerical experiments.

How to cite

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Cohen, Albert, Dahmen, Wolfgang, and Welper, Gerrit. "Adaptivity and variational stabilization for convection-diffusion equations∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1247-1273. <http://eudml.org/doc/222112>.

@article{Cohen2012,
abstract = {In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on other specifications, are explored and illustrated by numerical experiments.},
author = {Cohen, Albert, Dahmen, Wolfgang, Welper, Gerrit},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Variational problems; adaptivity; a-posteriori error estimators; stabilization; variational problems; a posteriori error estimators; convection-diffusion problems; numerical experiments},
language = {eng},
month = {3},
number = {5},
pages = {1247-1273},
publisher = {EDP Sciences},
title = {Adaptivity and variational stabilization for convection-diffusion equations∗},
url = {http://eudml.org/doc/222112},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Cohen, Albert
AU - Dahmen, Wolfgang
AU - Welper, Gerrit
TI - Adaptivity and variational stabilization for convection-diffusion equations∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/3//
PB - EDP Sciences
VL - 46
IS - 5
SP - 1247
EP - 1273
AB - In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on other specifications, are explored and illustrated by numerical experiments.
LA - eng
KW - Variational problems; adaptivity; a-posteriori error estimators; stabilization; variational problems; a posteriori error estimators; convection-diffusion problems; numerical experiments
UR - http://eudml.org/doc/222112
ER -

References

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  1. J.H. Bramble and J.E. Pasciak, A new approximation technique for div-curl systems. Math. Comp.73 (2004) 1739–1762.  
  2. J.H. Bramble, R.D. Lazarov and J.E. Pasciak, Least-squares methods for linear elasticity based on a discrete minus one inner product. Comput. Methods Appl. Mech. Eng.191 (2001) 727–744.  
  3. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Comput. Math.15 (1991).  
  4. F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Russo and E. Süli, A priori analysis of residual-free bubbles for advection-diffusion problems. SIAM J. Numer. Anal.36 (1999) 1933–1948.  
  5. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods II – beyond the elliptic case. Found. Comput. Math.2 (2002) 203–245.  
  6. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal.41 (2003) 1785–1823.  
  7. S. Dahlke, W. Dahmen and K. Urban, Adaptive wavelet methods for saddle point problems – convergence rates. SIAM J. Numer. Anal.40 (2002) 1230–1262.  
  8. W. Dahmen, S. Müller and T. Schlinkmann, On an adaptive multigrid solver for convection-dominated problems, SIAM J. Sci. Comput.23 (2001) 781–804.  
  9. W. Dahmen, C. Huang, C. Schwab and G. Welper, Adaptive Petrov-Galerkin methods for first order transport equations. IGPM Report 321, RWTH Aachen (2011).  
  10. L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part II : Optimal test functions. Numer. Methods Partial Differ. Equ.27 (2011) 70–105.  
  11. L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part III : Adaptivity. To appear in Appl. Numer. Math. (2012).  
  12. J.L. Guermond, J.T. Oden and S. Prudhomme, An interpretation of the Navier-Stokes-alpha model as a frame-indifferent Leray regularization. Physica D177 (2003) 23–30.  
  13. J.-L. Guermond, J.T. Oden and S. Prudhomme, Mathematical perspectives on large eddy simulation models for turbulent flows. J. Math. Fluid Mech.6 (2004) 194–248.  
  14. T. Hughes and G. Sangalli, Variational multiscale analysis : the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal.45 (2007) 539–557.  
  15. V. John, S. Kaya and W. Layton, A two-level variational multiscale method for convection-diffusion equations. Comput. Methods Appl. Mech. Eng.195 (2006) 4594–4603.  
  16. E. Lee and T.A. Manteuffel, FOSLL* method for the eddy current problem with three-dimensional edge singularities. SIAM J. Numer. Anal.45 (2007) 787–809.  
  17. T. Manteuffel, S. McCormick, J. Ruge and J.G. Schmidt, First-order system ℒℒ∗ (FOSLL)∗ for general scalar elliptic problems in the plane. SIAM J. Numer. Anal.43 (2005) 2098-2120.  
  18. G. Sangalli, A uniform analysis of non-symmetric and coercive linear operators. SIAM J. Math. Anal.36 (2005) 2033–2048.  
  19. G. Sangalli, Robust a-posteriori estimators for advection-diffusion-reaction problems. Math. Comput.77 (2008) 41–70.  
  20. R. Verfürth, Robust a-posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math.78 (1998) 479–493.  
  21. R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal.43 (2005) 1766–1782.  

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