author = {Lenz, Martin and Nemadjieu, Simplice Firmin and Rumpf, Martin},
  title = {A Convergent Finite Volume Scheme for Diffusion on Evolving Surfaces},
  journal = {SIAM Journal on Numerical Analysis},
  year = {2011},
  volume = {49},
  pages = {15--37},
  number = {1},
  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 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.},
  doi = {10.1137/090776767},
  pdf = { 1}