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IMA Journal of Numerical Analysis Advance Access published online on March 4, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn049
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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.

An h-narrow band finite-element method for elliptic equations on implicit surfaces

Klaus Deckelnick

Institut für Analysis und Numerik, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany

Gerhard Dziuk

Abteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder Strasse 10, 79104 Freiburg, Germany

Charles M. Elliott{dagger}

Mathematics Institute and Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, UK

Claus-Justus Heine

Abteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder Strasse 10, 79104 Freiburg, Germany

{dagger} Corresponding author. Email: c.m.elliott{at}warwick.ac.uk

Received on 14 December 2007. Revised on 24 June 2008.


  Abstract

In this article we define a finite-element method for elliptic partial differential equations (PDEs) on curves or surfaces, which are given implicitly by some level set function. The method is specially designed for complicated surfaces. The key idea is to solve the PDE on a narrow band around the surface. The width of the band is proportional to the grid size. We use finite-element spaces that are unfitted to the narrow band, so that elements are cut off. The implementation nevertheless is easy. We prove error estimates of optimal order for a Poisson equation on a surface and provide numerical tests and examples.

Key Words: elliptic equations; implicit surfaces; level sets; unfitted mesh finite-element method


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