In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions, under weaker conditions on the triangulations than quasiuniformity. In the two-dimensional case, the bound for the resolvent contains a logarithmic factor.
Mots clés : resolvent estimates, stability, smoothing, maximum-norm, elliptic, parabolic, finite elements, nonquasiuniform triangulations
@article{M2AN_2006__40_5_923_0,
author = {Bakaev, Nikolai Yu. and Crouzeix, Michel and Thom\'ee, Vidar},
title = {Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {923--937},
publisher = {EDP-Sciences},
volume = {40},
number = {5},
year = {2006},
doi = {10.1051/m2an:2006040},
mrnumber = {2293252},
zbl = {1116.65108},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006040/}
}
TY - JOUR AU - Bakaev, Nikolai Yu. AU - Crouzeix, Michel AU - Thomée, Vidar TI - Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 923 EP - 937 VL - 40 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006040/ DO - 10.1051/m2an:2006040 LA - en ID - M2AN_2006__40_5_923_0 ER -
%0 Journal Article %A Bakaev, Nikolai Yu. %A Crouzeix, Michel %A Thomée, Vidar %T Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 923-937 %V 40 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006040/ %R 10.1051/m2an:2006040 %G en %F M2AN_2006__40_5_923_0
Bakaev, Nikolai Yu.; Crouzeix, Michel; Thomée, Vidar. Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 5, pp. 923-937. doi : 10.1051/m2an:2006040. http://www.numdam.org/articles/10.1051/m2an:2006040/
[1] , Maximum norm resolvent estimates for elliptic finite element operators. BIT 41 (2001) 215-239. | Zbl
[2] , and, Long-time behavior of backward difference type methods for parabolic equations with memory in Banach space. East-West J. Numer. Math. 6 (1998) 185-206. | Zbl
[3] , and, Maximum-norm estimates for resolvents of elliptic finite element operators. Math. Comp. 72 (2002) 1597-1610. | Zbl
[4] ,, and, Parabolic finite element equations in nonconvex polygonal domains. BIT (to appear). | MR | Zbl
[5] and, The stability in and of the -projection onto finite element function spaces. Math. Comp. 48 (1987) 521-532. | Zbl
[6] and, Resolvent estimates in for discrete Laplacians on irregular meshes and maximum-norm stability of parabolic finite difference schemes. Comput. Meth. Appl. Math. 1 (2001) 3-17. | Zbl
[7] , and, Resolvent estimates for elliptic finite element operators in one dimension. Math. Comp. 63 (1994) 121-140. | Zbl
[8] , Gaussian estimates and holomorphy of semigroups. Proc. Amer. Math. Soc. 123 (1995) 1465-1474. | Zbl
[9] , and, Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980) 265-304. | Zbl
[10] , and, Stability, analyticity, and almost best approximation in maximum-norm for parabolic finite element equations. Comm. Pure Appl. Math. 51 (1998) 1349-1385. | Zbl
[11] , Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974) 141-161. | Zbl
[12] , Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, New York (1997). | MR | Zbl
[13] and, Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable. Numer. Math. 41 (1983) 345-371. | Zbl
Cité par Sources :
