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IMA Journal of Numerical Analysis Advance Access published online on July 16, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp005
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Numerical approximation of gradient flows for closed curves in Rd

John W. Barrett{dagger}

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

Harald Garcke

NWF I—Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Robert Nürnberg

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

{dagger} Corresponding author. Email: j.barrett{at}imperial.ac.uk

Received on 18 April 2008. Revised on 9 October 2008.


  Abstract

We present parametric finite-element approximations of curvature flows for curves in Rd, where d ≥ 2, as well as for curves on two-dimensional manifolds in R3. Here we consider the curve shortening flow, the curve diffusion and the elastic flow. It is demonstrated that the curve shortening and the elastic flows on manifolds can be used to compute nontrivial geodesics and that the corresponding geodesic curve diffusion flow leads to solutions of partitioning problems on two-dimensional manifolds in R3. In addition, we extend these schemes to anisotropic surface energy densities. The presented schemes have very good properties with respect to stability and the distribution of mesh points, and hence no remeshing is needed in practice.

Key Words: curve shortening flow; geodesic curvature flows; curve diffusion; surface diffusion; elastic flow; Willmore flow; geodesics; parametric finite elements; anisotropy; tangential movement

Dedicated to the memory of A. R. Mitchell, 1921–2007.


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