M. Lenz, S. F. Nemadjieu, and M. Rumpf.
A convergent finite volume scheme for diffusion on evolving surfaces.
SIAM Journal on Numerical Analysis, 49(1):15-37, 2011.
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A finite volume scheme for transport and diffusion problems on evolving hypersurfaces is discussed. The underlying motion is assumed to be described by a fixed, not necessarily normal, velocity field. The ingredients of the numerical 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. The semi-implicit scheme is derived via a discretization of the underlying conservation law, and discrete counterparts of continuous a priori estimates in this geometric setup are proved. The continuous solution on the continuous family of evolving surfaces is compared to the finite volume solution on the discrete sequence of simplicial surfaces and convergence of the family of discrete solutions on successively refined meshes is proved under suitable assumptions on the geometry and the discrete meshes. Furthermore, numerical results show remarkable aspects of transport and diffusion phenomena on evolving surfaces and experimentally reflect the established convergence results. Finally, we discuss how to combine the presented scheme with a corresponding finite volume scheme for advective transport on the surface via an operator splitting and present some applications.