A Convergent Finite Volume Scheme for Diffusion on Evolving Surfaces

Abstract

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.

A Fully Discrete Evolving Surface Finite Element Method

“…In particular, stability and optimal order error bounds for the ordinary differential equation systems arising from ESFEM approximation of advection-diffusion equations on moving surfaces are obtained. In [13] Lenz, Nemadjieu, and Rumpf proved L 2 and H 1 error bounds for their proposed fully discrete time implicit finite volume scheme. The purpose of this paper is to extend the results of [5] to the case when the time discretization is based on a backward Euler scheme.…”

“…Numerical approaches include level set methods, surface finite elements, finite volume methods, and diffuse interface methods; see [1,4,5,9,13,15,17] and the references cited therein.…”

A 3D front-tracking approach for simulation of a two-phase fluid with insoluble surfactant

“…By assuming that adjacent triangles S k and S k i share the same such edge points, as displayed in Fig. 2(a), Lenz et al [29] introduce a finite volume scheme for (6) for which the interface itself (and not the polyhedron given by the triangularization) is split into finite volume cells. Defining the lifting operator P(·), described in Section 3.4, as the orthogonal projection in the direction of the normal to the surface, Fig.…”

“…Lenz et al [29] extended to finite volumes ideas of Dziuk and Elliott [13]. The key idea is to use Leibniz (Transport) Formula for the time derivative of integrals over moving surfaces…”

“…The evolving separation interface is represented by an unstructured, curved triangular mesh where a finite volume scheme is employed to solve the equation for the surfactant concentration. The main numerical ingredients forming the methodology presented were introduced recently in different contexts [36,38,29], here being combined and applied for the first time in this context. Conservation of the global surfactant mass is attained in all cases with local conservation being achieved to machine precision under certain circumstances.…”