# 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

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

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topBakaev, 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

top- [1] N.Yu. Bakaev, Maximum norm resolvent estimates for elliptic finite element operators. BIT 41 (2001) 215–239. Zbl0979.65097
- [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] 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] 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] 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] 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] 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] E.L. Ouhabaz, Gaussian estimates and holomorphy of semigroups. Proc. Amer. Math. Soc. 123 (1995) 1465–1474. Zbl0829.47032
- [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] 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] H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974) 141–161. Zbl0264.35043
- [12] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, New York (1997). Zbl0884.65097MR1479170
- [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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