NDLI: An Eulerian approach to transport and diffusion on evolving implicit surfaces

Content Provider	Springer Nature Link
Author	Dziuk, G. Elliott, C. M.
Copyright Year	2008
Abstract	In this article we define a level set method for a scalar conservation law with a diffusive flux on an evolving hypersurface Γ(t) contained in a domain $${\Omega \subset \mathbb R^{n+1}}$$ . The partial differential equation is solved on all level set surfaces of a prescribed time dependent function Φ whose zero level set is Γ(t). The key idea lies in formulating an appropriate weak form of the conservation law with respect to time and space. A major advantage of this approach is that it avoids the numerical evaluation of curvature. The resulting equation is then solved in one dimension higher but can be solved on a fixed grid. In particular we formulate an Eulerian transport and diffusion equation on evolving implicit surfaces. 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. The computation of the mass and element stiffness matrices, depending only on the gradient of the level set function, 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.
Starting Page	17
Ending Page	28
Page Count	12
File Format	PDF
ISSN	14329360
Journal	Computing and Visualization in Science
Volume Number	13
Issue Number	1
e-ISSN	14330369
Language	English
Publisher	Springer-Verlag
Publisher Date	2008-07-24
Publisher Place	Berlin, Heidelberg
Access Restriction	Subscribed
Subject Keyword	Calculus of Variations and Optimal Control; Optimization Numerical Analysis Visualization Algorithms Computer Applications in Chemistry Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Engineering Computational Theory and Mathematics Computer Vision and Pattern Recognition Modeling and Simulation Software

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