A parametric finite element method for fourth order geometric evolution equations

Authors:
John W. Barrett;Harald Garcke;Robert Nürnberg
Affiliations:
Department of Mathematics, Imperial College, London SW7 2AZ, UK;NWF I - Mathematik, Universität Regensburg, 93040 Regensburg, Germany;Department of Mathematics, Imperial College, London SW7 2AZ, UK
Venue:
Journal of Computational Physics
Year:
2007

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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.