Finite Volume Method on Moving Surfaces

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 with a sketch of the proof. Finally, first numerical results are discussed.