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 (2010)

  • 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 L1 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 L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 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 39.4 (2010): 755-780. <http://eudml.org/doc/194285>.

@article{Mizutani2010,
abstract = { Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L1 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 L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 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},
keywords = {Finite element method; degenerate parabolic equation; nonlinear semigroup.; order-preserving; contraction properties; rate of convergence},
language = {eng},
month = {3},
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/194285},
volume = {39},
year = {2010},
}

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
DA - 2010/3//
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 L1 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 L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 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/194285
ER -

References

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