Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T19:22:16.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptivity and variational stabilization for convection-diffusion equations∗

Published online by Cambridge University Press:  27 March 2012

Albert Cohen ,
Wolfgang Dahmen  and
Gerrit Welper
Albert Cohen
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France. [email protected]
Wolfgang Dahmen
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 52056 Aachen, Germany; [email protected]; [email protected]
Gerrit Welper
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 52056 Aachen, Germany; [email protected]; [email protected]
Get access

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.

Keywords

Variational problemsadaptivitya-posteriori error estimatorsstabilization
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1247 - 1273
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Bramble, J.H. and Pasciak, J.E., A new approximation technique for div-curl systems. Math. Comp. 73 (2004) 1739–1762. Google Scholar
Bramble, J.H., Lazarov, R.D. and Pasciak, J.E., Least-squares methods for linear elasticity based on a discrete minus one inner product. Comput. Methods Appl. Mech. Eng. 191 (2001) 727–744. Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Comput. Math. 15 (1991).
Brezzi, F., Hughes, T.J.R., Marini, L.D., Russo, A. and Süli, E., A priori analysis of residual-free bubbles for advection-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 1933–1948. Google Scholar
Cohen, A., Dahmen, W. and DeVore, R., Adaptive wavelet methods II – beyond the elliptic case. Found. Comput. Math. 2 (2002) 203–245. Google Scholar
Cohen, A., Dahmen, W. and DeVore, R., Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal. 41 (2003) 1785–1823. Google Scholar
Dahlke, S., Dahmen, W. and Urban, K., Adaptive wavelet methods for saddle point problems – convergence rates. SIAM J. Numer. Anal. 40 (2002) 1230–1262. Google Scholar
Dahmen, W., Müller, S. and Schlinkmann, T., On an adaptive multigrid solver for convection-dominated problems, SIAM J. Sci. Comput. 23 (2001) 781–804. Google Scholar
W. Dahmen, C. Huang, C. Schwab and G. Welper, Adaptive Petrov-Galerkin methods for first order transport equations. IGPM Report 321, RWTH Aachen (2011).
Demkowicz, L. and Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. Part II : Optimal test functions. Numer. Methods Partial Differ. Equ. 27 (2011) 70–105. Google Scholar
L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part III : Adaptivity. To appear in Appl. Numer. Math. (2012).
Guermond, J.L., Oden, J.T. and Prudhomme, S., An interpretation of the Navier-Stokes-alpha model as a frame-indifferent Leray regularization. Physica D 177 (2003) 23–30. Google Scholar
Guermond, J.-L., Oden, J.T. and Prudhomme, S., Mathematical perspectives on large eddy simulation models for turbulent flows. J. Math. Fluid Mech. 6 (2004) 194–248. Google Scholar
Hughes, T. and Sangalli, G., Variational multiscale analysis : the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45 (2007) 539–557. Google Scholar
John, V., Kaya, S. and Layton, W., A two-level variational multiscale method for convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 195 (2006) 4594–4603. Google Scholar
Lee, E. and Manteuffel, T.A., FOSLL* method for the eddy current problem with three-dimensional edge singularities. SIAM J. Numer. Anal. 45 (2007) 787–809. Google Scholar
Manteuffel, T., McCormick, S., Ruge, J. and Schmidt, J.G., First-order system ℒℒ∗ (FOSLL)∗ for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43 (2005) 2098-2120. Google Scholar
Sangalli, G., A uniform analysis of non-symmetric and coercive linear operators. SIAM J. Math. Anal. 36 (2005) 2033–2048. Google Scholar
Sangalli, G., Robust a-posteriori estimators for advection-diffusion-reaction problems. Math. Comput. 77 (2008) 41–70. Google Scholar
Verfürth, R., Robust a-posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math. 78 (1998) 479–493. Google Scholar
Verfürth, R., Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766–1782. Google Scholar