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An Eulerian approach to transport and diffusion on evolving implicit surfaces. (English) Zbl 1220.65132

We define a level set method for a scalar conservation law with a diffusive flux on an evolving hypersurface \(\Gamma (t)\) 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 \(\Phi \) whose zero level set is \(\Gamma (t)\). 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. The finite element method is applied to the weak form of the conservation equation yielding an Eulerian evolving surface finite element method. Numerical experiments are described which indicate the power of the method.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics

Software:

ALBERTA
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References:

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