Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations

Nikolai Yu. Bakaev; Michel Crouzeix[1]; Vidar Thomée

  • [1] Université de Rennes 1 IRMAR, UMR 6625 Campus de Beaulieu 35042 Rennes Cedex FRANCE

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2006)

  • Volume: 40, Issue: 5, page 923-937
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Bakaev, Nikolai Yu., Crouzeix, Michel, and Thomée, Vidar. "Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 40.5 (2006): 923-937. <http://eudml.org/doc/245316>.

@article{Bakaev2006,
abstract = {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.},
affiliation = {Université de Rennes 1 IRMAR, UMR 6625 Campus de Beaulieu 35042 Rennes Cedex FRANCE},
author = {Bakaev, Nikolai Yu., Crouzeix, Michel, Thomée, Vidar},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {resolvent estimates; stability; smoothing; maximum-norm; elliptic; parabolic; finite elements; nonquasiuniform triangulations; parabolic equation; elliptic operator; resolvent estimate; maximum-norm estimate; finite element method; smoothing property; non-uniform mesh; semidiscretisation},
language = {eng},
number = {5},
pages = {923-937},
publisher = {EDP-Sciences},
title = {Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations},
url = {http://eudml.org/doc/245316},
volume = {40},
year = {2006},
}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2006
PB - EDP-Sciences
VL - 40
IS - 5
SP - 923
EP - 937
AB - 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.
LA - eng
KW - resolvent estimates; stability; smoothing; maximum-norm; elliptic; parabolic; finite elements; nonquasiuniform triangulations; parabolic equation; elliptic operator; resolvent estimate; maximum-norm estimate; finite element method; smoothing property; non-uniform mesh; semidiscretisation
UR - http://eudml.org/doc/245316
ER -

References

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  1. [1] N.Yu. Bakaev, Maximum norm resolvent estimates for elliptic finite element operators. BIT 41 (2001) 215–239. Zbl0979.65097
  2. [2] N.Yu. Bakaev, S. Larsson and V. Thomée, 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. Zbl0913.65139
  3. [3] 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. Zbl1028.65113
  4. [4] P. Chatzipantelidis, R.D. Lazarov, V. Thomée and L.B. Wahlbin, Parabolic finite element equations in nonconvex polygonal domains. BIT (to appear). Zbl1108.65097MR2283311
  5. [5] M. Crouzeix and V. Thomée, 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. Zbl0637.41034
  6. [6] M. Crouzeix and V. Thomée, 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. Zbl0987.65093
  7. [7] M. Crouzeix, S. Larsson and V. Thomée, Resolvent estimates for elliptic finite element operators in one dimension. Math. Comp. 63 (1994) 121–140. Zbl0806.65096
  8. [8] E.L. Ouhabaz, Gaussian estimates and holomorphy of semigroups. Proc. Amer. Math. Soc. 123 (1995) 1465–1474. Zbl0829.47032
  9. [9] A.H. Schatz, V. Thomée and L.B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980) 265–304. Zbl0414.65066
  10. [10] A.H. Schatz, V. Thomée and L.B. Wahlbin, Stability, analyticity, and almost best approximation in maximum-norm for parabolic finite element equations. Comm. Pure Appl. Math. 51 (1998) 1349–1385. Zbl0932.65103
  11. [11] H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974) 141–161. Zbl0264.35043
  12. [12] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, New York (1997). Zbl0884.65097MR1479170
  13. [13] V. Thomée and L.B. Wahlbin, Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable. Numer. Math. 41 (1983) 345–371. Zbl0515.65082

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