A parametric finite element method for fourth order geometric evolution equations

descriptionPublicationkeyboard_double_arrow_right Article 01 Mar 2007 Italy, United Kingdom English Publisher:Elsevier BVJournal:Journal of Computational Physics, volume 222, pages 441-467 (issn: 0021-9991,

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Authors: John W. Barrett 0001; Harald Garcke; Robert Nürnberg;

doi: 10.1016/j.jcp.2006.07.026 , 10.5283/epub.13831

handle: 11572/283501 , 10044/1/641

A parametric finite element method for fourth order geometric evolution equations

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Abstract

We present a finite element approximation of motion by minus the Laplacian of curvature and related flows. The proposed scheme covers both the closed curve case, and the case of curves that are connected via triple junctions. On introducing a parametric finite element approximation, we prove stability bounds and compare our scheme with existing approaches. It turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points. We state also an extension of our scheme to Willmore flow of curves and discuss possible further generalizations.

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Italy, United Kingdom

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Keywords

fourth order parabolic problem, convergence, 65M12, 510 Mathematik, stability, surface diffusion, Fourth order parabolic problem; Parametric finite elements; Schur complement; Surface diffusion; Tangential movement; Triple junctions; Willmore flow, evolution equation, surface diffusion, Willmore flow, triple junctions, fourth order parabolic problem, parametric finite elements, Schur complement, tangential movement., 35K55, Nonlinear parabolic equations, parametric finite elements, Schur complement, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, numerical experiments, triple junctions, 65M60

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