Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory

Akira Mizutani; Norikazu Saito; Takashi Suzuki

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 4, page 755-780
  • ISSN: 0764-583X

Abstract

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Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L 1 contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L 1 and L ∞ , respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L 1 convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.

How to cite

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Mizutani, Akira, Saito, Norikazu, and Suzuki, Takashi. "Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.4 (2005): 755-780. <http://eudml.org/doc/244767>.

@article{Mizutani2005,
abstract = {Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and $L^1$ contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in $L^1$ and $L^\infty $, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive $L^1$ convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.},
author = {Mizutani, Akira, Saito, Norikazu, Suzuki, Takashi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite element method; degenerate parabolic equation; nonlinear semigroup; order-preserving; contraction properties; rate of convergence},
language = {eng},
number = {4},
pages = {755-780},
publisher = {EDP-Sciences},
title = {Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory},
url = {http://eudml.org/doc/244767},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Mizutani, Akira
AU - Saito, Norikazu
AU - Suzuki, Takashi
TI - Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 4
SP - 755
EP - 780
AB - Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and $L^1$ contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in $L^1$ and $L^\infty $, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive $L^1$ convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.
LA - eng
KW - finite element method; degenerate parabolic equation; nonlinear semigroup; order-preserving; contraction properties; rate of convergence
UR - http://eudml.org/doc/244767
ER -

References

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