M. Lenz, S. F. Nemadjieu, and M. Rumpf.
Finite volume method on moving surfaces.
In R. Eymard and J.-M. Hérald, editors, Finite Volumes for
Complex Applications V, pages 561-576. Wiley, 2008.
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In this paper an evolving surface finite volume method is introduced for the numerical resolution of a transport diffusion problem on a family of moving hypersurfaces. These surfaces are assumed to evolve according to a given motion field. The ingredients of the method are an approximation of the family of surfaces by a family of interpolating simplicial meshes, where grid vertices move on motion trajectories, a consistent finite volume discretization of the induced transport on the simplices, and a proper incorporation of a diffusive flux balance at simplicial faces. Existence, uniqueness and a priori estimates are proved for the discrete solution. Furthermore, a convergence result is formulated together a sketch of the proof. Finally, first numerical results are discussed.