An Eulerian level set method for partial differential equations on evolving surfaces

In this article we define an Eulerian level set method for a partial differential equation on an evolving hypersurface Γ (t) contained in a domain Ω ⊂ R. The key idea is based on formulating the partial differential equations on all level set surfaces of a prescribed time dependent function Φ whose zero level set is Γ (t). The resulting equation is then solved in one dimension higher but can be solved on a fixed grid. In particular we formulate an Eulerian version of a scalar conservation law on Γ (t) and, in the case of a diffusive flux, derive an Eulerian transport and diffusion equation. Using Eulerian surface gradients to define weak forms of elliptic operators naturally generates weak formulations of elliptic and parabolic equations. The finite element method is applied to the weak form of the conservation equation yielding an Eulerian Evolving Surface Finite Element Method (EESFEM). The computation of the mass and element stiffness matrices are simple and straightforward. Numerical experiments are described which indicate the power of the method. We describe how this framework may be employed in applications. Gerhard Dziuk Abteilung für Angewandte Mathematik University of Freiburg Hermann-Herder-Straße 10 D–79104 Freiburg i. Br. Germany Tel.: +49 761-2035628 Fax: +49 761-2035632 E-mail: [email protected] Charles M. Elliott Department of Mathematics University of Sussex Falmer, Brighton BN1 9RF United Kingdom Tel.: +4