Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T14:17:16.641Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum-norm resolvent estimatesfor ellipticfinite element operators on nonquasiuniform triangulations

Published online by Cambridge University Press:  16 January 2007

Nikolai Yu. Bakaev ,
Michel Crouzeix  and
Vidar Thomée
Nikolai Yu. Bakaev
Affiliation:
Department of Economic Dynamics, EAI, Moscow Engineering Physics Institute (State University), Kashirskoe Shosse 31, Moscow 115409, Russia. bakaev@postman.ru
Michel Crouzeix
Affiliation:
IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France. michel.crouzeix@univ-rennes1.fr
Vidar Thomée
Affiliation:
Department of Mathematics, Chalmers University of Technology, 41296 Göteborg, Sweden. thomee@math.chalmers.se
Get access

Abstract


In recent years several papers have been devoted to stabilityand smoothing properties in maximum-norm offinite element discretizations of parabolic problems.Using the theory of analytic semigroups it has been possibleto rephrase such properties as bounds for the resolventof the associated discrete elliptic operator. In all thesecases the triangulations of the spatial domain has beenassumed to be quasiuniform. In the present paper weshow 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 containsa logarithmic factor.


Keywords

Resolvent estimatesstabilitysmoothingmaximum-normellipticparabolicfiniteelementsnonquasiuniform triangulations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 5 , September 2006 , pp. 923 - 937
Copyright
© EDP Sciences, SMAI, 2007

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

Bakaev, N.Yu., Maximum norm resolvent estimates for elliptic finite element operators. BIT 41 (2001) 215–239. CrossRef
Bakaev, N.Yu., Larsson, S. and Thomée, V., 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.
N.Yu. Bakaev, V. Thomée and L.B. Wahlbin, Maximum-norm estimates for resolvents of elliptic finite element operators. Math. Comp. 72 (2002) 1597–1610.
P. Chatzipantelidis, R.D. Lazarov, V. Thomée and L.B. Wahlbin, Parabolic finite element equations in nonconvex polygonal domains. BIT (to appear).
Crouzeix, M. and Thomée, V., The stability in L p and $W_p^1$ of the L 2-projection onto finite element function spaces. Math. Comp. 48 (1987) 521–532.
Crouzeix, M. and Thomée, V., Resolvent estimates in l p for discrete Laplacians on irregular meshes and maximum-norm stability of parabolic finite difference schemes. Comput. Meth. Appl. Math. 1 (2001) 3–17. CrossRef
Crouzeix, M., Larsson, S. and Thomée, V., Resolvent estimates for elliptic finite element operators in one dimension. Math. Comp. 63 (1994) 121–140. CrossRef
Ouhabaz, E.L., Gaussian estimates and holomorphy of semigroups. Proc. Amer. Math. Soc. 123 (1995) 1465–1474. CrossRef
Schatz, A.H., Thomée, V. and Wahlbin, L.B., Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980) 265–304. CrossRef
Schatz, A.H., Thomée, V. and Wahlbin, L.B., Stability, analyticity, and almost best approximation in maximum-norm for parabolic finite element equations. Comm. Pure Appl. Math. 51 (1998) 1349–1385. 3.0.CO;2-1>CrossRef
Stewart, H.B., Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974) 141–161. CrossRef
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, New York (1997).
Thomée, V. and Wahlbin, L.B., Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable. Numer. Math. 41 (1983) 345–371. CrossRef