Numerical methods for fourth order nonlinear degenerate diffusion problems

Jürgen Becker; Günther Grün; Martin Lenz; Martin Rumpf

Applications of Mathematics (2002)

  • Volume: 47, Issue: 6, page 517-543
  • ISSN: 0862-7940

Abstract

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Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach.

How to cite

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Becker, Jürgen, et al. "Numerical methods for fourth order nonlinear degenerate diffusion problems." Applications of Mathematics 47.6 (2002): 517-543. <http://eudml.org/doc/33129>.

@article{Becker2002,
abstract = {Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach.},
author = {Becker, Jürgen, Grün, Günther, Lenz, Martin, Rumpf, Martin},
journal = {Applications of Mathematics},
keywords = {thin film; fourth order degenerate parabolic equation; nonnegativity preserving scheme; surfactant driven flow; finite element method; finite volume method; thin film; fourth order degenerate parabolic equation; surfactant driven flow; finite element method; finite volume method},
language = {eng},
number = {6},
pages = {517-543},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Numerical methods for fourth order nonlinear degenerate diffusion problems},
url = {http://eudml.org/doc/33129},
volume = {47},
year = {2002},
}

TY - JOUR
AU - Becker, Jürgen
AU - Grün, Günther
AU - Lenz, Martin
AU - Rumpf, Martin
TI - Numerical methods for fourth order nonlinear degenerate diffusion problems
JO - Applications of Mathematics
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 6
SP - 517
EP - 543
AB - Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach.
LA - eng
KW - thin film; fourth order degenerate parabolic equation; nonnegativity preserving scheme; surfactant driven flow; finite element method; finite volume method; thin film; fourth order degenerate parabolic equation; surfactant driven flow; finite element method; finite volume method
UR - http://eudml.org/doc/33129
ER -

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