A Convergent Finite Volume Scheme for Diffusion on Evolving Surfaces

Abstract. 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 p...

• Type: Journal
• Year: 2011
• Where: SIAMNUM
• Authors: Martin Lenz, Simplice Firmin Nemadjieu, Martin Rumpf
• Keywords: SIAMNUM 2011, Software Engineering, Surfaces, finite Volume, finite Volume Scheme