Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations
Nikolai Yu. Bakaev; Michel Crouzeix; Vidar Thomée
ESAIM: Mathematical Modelling and Numerical Analysis (2007)
- 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 40.5 (2007): 923-937. <http://eudml.org/doc/194341>.
@article{Bakaev2007,
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.
},
author = {Bakaev, Nikolai Yu., Crouzeix, Michel, Thomée, Vidar},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
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},
month = {1},
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/194341},
volume = {40},
year = {2007},
}
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
DA - 2007/1//
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/194341
ER -
References
top- N.Yu. Bakaev, Maximum norm resolvent estimates for elliptic finite element operators. BIT41 (2001) 215–239.
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- 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.
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- M. Crouzeix, S. Larsson and V. Thomée, Resolvent estimates for elliptic finite element operators in one dimension. Math. Comp.63 (1994) 121–140.
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- 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.
- 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.
- H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc.199 (1974) 141–161.
- V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, New York (1997).
- 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.
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