On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions

Author:
Günther Grün

Journal:
Math. Comp. **72** (2003), 1251-1279

MSC (2000):
Primary 35K35, 35K55, 35K65, 35R35, 65M12, 65M60, 76D08

DOI:
https://doi.org/10.1090/S0025-5718-03-01492-3

Published electronically:
January 8, 2003

MathSciNet review:
1972735

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Abstract | References | Similar Articles | Additional Information

Abstract: We present nonnegativity-preserving finite element schemes for a general class of thin film equations in multiple space dimensions. The equations are fourth order degenerate parabolic, and may contain singular terms of second order which are to model van der Waals interactions. A subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estimates found in the continuous setting. By use of the entropy estimate, strong convergence results for discrete solutions are obtained. In particular, the limit of discrete fluxes will be identified with the flux in the continuous setting. As a by-product, first results on existence and positivity almost everywhere of solutions to equations with singular lower order terms can be established in the continuous setting.

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Additional Information

**Günther Grün**

Affiliation:
Universität Bonn, Institut für Angewandte Mathematik, Beringstr. 6, 53115 Bonn, Germany

Email:
gg@iam.uni-bonn.de

DOI:
https://doi.org/10.1090/S0025-5718-03-01492-3

Keywords:
{Lubrication approximation,
fourth order degenerate parabolic equations,
nonnegativity preserving,
finite elements}

Received by editor(s):
August 14, 2000

Received by editor(s) in revised form:
September 21, 2001

Published electronically:
January 8, 2003

Article copyright:
© Copyright 2003
American Mathematical Society