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

Mizutani, Akira; Saito, Norikazu; Suzuki, Takashi

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 755-780.

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.

Classification : 35K65, 47H20, 65M60
Mots clés : finite element method, degenerate parabolic equation, nonlinear semigroup

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