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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
Published electronically: January 8, 2003
MathSciNet review: 1972735
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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 $M_h(U_h)\nabla P_h$ will be identified with the flux $\mathcal M(u)\nabla(W'(u)-\Delta u)$ 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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Günther Grün
Affiliation: Universität Bonn, Institut für Angewandte Mathematik, Beringstr. 6, 53115 Bonn, Germany

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