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.
@article{M2AN_2012__46_5_1247_0,
author = {Cohen, Albert and Dahmen, Wolfgang and Welper, Gerrit},
title = {Adaptivity and variational stabilization for convection-diffusion equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {1247-1273},
doi = {10.1051/m2an/2012003},
mrnumber = {2916380},
zbl = {1270.65065},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2012__46_5_1247_0}
}
Cohen, Albert; Dahmen, Wolfgang; Welper, Gerrit. Adaptivity and variational stabilization for convection-diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 1247-1273. doi : 10.1051/m2an/2012003. http://gdmltest.u-ga.fr/item/M2AN_2012__46_5_1247_0/
[1] and , A new approximation technique for div-curl systems. Math. Comp. 73 (2004) 1739-1762. | MR 2059734 | Zbl 1049.78026
[2] , and , Least-squares methods for linear elasticity based on a discrete minus one inner product. Comput. Methods Appl. Mech. Eng. 191 (2001) 727-744. | MR 1870517 | Zbl 0999.74107
[3] and , Mixed and Hybrid Finite Element Methods. Springer Series in Comput. Math. 15 (1991). | MR 1115205 | Zbl 0788.73002
[4] , , , and , A priori analysis of residual-free bubbles for advection-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 1933-1948. | MR 1712145 | Zbl 0947.65115
[5] , and , Adaptive wavelet methods II - beyond the elliptic case. Found. Comput. Math. 2 (2002) 203-245. | MR 1907380 | Zbl 1025.65056
[6] , and , Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal. 41 (2003) 1785-1823. | MR 2035007 | Zbl 1057.65031
[7] , and , Adaptive wavelet methods for saddle point problems - convergence rates. SIAM J. Numer. Anal. 40 (2002) 1230-1262. | MR 1951893 | Zbl 1024.65101
[8] , and , On an adaptive multigrid solver for convection-dominated problems, SIAM J. Sci. Comput. 23 (2001) 781-804. | MR 1860964 | Zbl 1004.65114
[9] , , and , Adaptive Petrov-Galerkin methods for first order transport equations. IGPM Report 321, RWTH Aachen (2011). | MR 3022225 | Zbl 1260.65091
[10] and , A class of discontinuous Petrov-Galerkin methods. Part II : Optimal test functions. Numer. Methods Partial Differ. Equ. 27 (2011) 70-105. | MR 2743600 | Zbl 1208.65164
[11] and , A class of discontinuous Petrov-Galerkin methods. Part III : Adaptivity. To appear in Appl. Numer. Math. (2012). | MR 2899253 | Zbl pre06030368
[12] , and , An interpretation of the Navier-Stokes-alpha model as a frame-indifferent Leray regularization. Physica D 177 (2003) 23-30. | MR 1965324 | Zbl 1082.35120
[13] , and , Mathematical perspectives on large eddy simulation models for turbulent flows. J. Math. Fluid Mech. 6 (2004) 194-248. | MR 2053583 | Zbl 1094.76030
[14] and , Variational multiscale analysis : the fine-scale Green's function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45 (2007) 539-557. | MR 2300286 | Zbl 1152.65111
[15] , and , A two-level variational multiscale method for convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 195 (2006) 4594-4603. | MR 2229851 | Zbl 1124.76028
[16] and , FOSLL* method for the eddy current problem with three-dimensional edge singularities. SIAM J. Numer. Anal. 45 (2007) 787-809. | MR 2300297 | Zbl 1137.78342
[17] , , and , First-order system ℒℒ∗ (FOSLL)∗ for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43 (2005) 2098-2120. | MR 2192333 | Zbl 1103.65117
[18] , A uniform analysis of non-symmetric and coercive linear operators. SIAM J. Math. Anal. 36 (2005) 2033-2048. | MR 2178232 | Zbl 1114.35060
[19] , Robust a-posteriori estimators for advection-diffusion-reaction problems. Math. Comput. 77 (2008) 41-70. | MR 2353943 | Zbl 1130.65083
[20] , Robust a-posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math. 78 (1998) 479-493. | MR 1603287 | Zbl 0887.65108
[21] , Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766-1782. | MR 2182149 | Zbl 1099.65100