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September 2006 PDE's on surfaces---a diffuse interface approach
Andreas Rätz, Axel Voigt
Commun. Math. Sci. 4(3): 575-590 (September 2006).


We introduce a new approach to deal with the numerical solution of partial differential equations on surfaces. Thereby we reformulate the problem on a larger domain in one higher dimension and introduce a diffuse interface region of a phase-field variable, which is defined in the whole domain. The surface of interest is now only implicitly given by the $1=2$-level set of this phase-field variable. Formal matched asymptotics show the convergence of the reformulated problem to the original PDE on the surface, as the diffuse interface width shrinks to zero. The main advantage of the approach is the possibility to formulate the problem on a Cartesian grid. With adaptive grid refinement the additional computational cost resulting from the higher dimension can be significantly reduced. Examples on linear diffusion and nonlinear phase separation demonstrate the wide applicability of the method.


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Andreas Rätz. Axel Voigt. "PDE's on surfaces---a diffuse interface approach." Commun. Math. Sci. 4 (3) 575 - 590, September 2006.


Published: September 2006
First available in Project Euclid: 5 April 2007

zbMATH: 1113.35092
MathSciNet: MR2247931

Primary: 35K57
Secondary: 35B40 , 65Mxx , 82C24

Keywords: adaptive finite elements , asymptotic analysis , Cahn-Hilliard equation , implicit surfaces , PDEs on surfaces , phase-field approximation

Rights: Copyright © 2006 International Press of Boston

Vol.4 • No. 3 • September 2006
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