Arbitrary-order structure-preserving discretizations for geometric curvature flows

Abstract: Geometric flows, where an immersed manifold evolves in time according to its own geometry, exhibit important structural properties. For example, surface diffusion dissipates surface area while conserving volume; it is desirable to preserve these properties on discretization. This has motivated a substantial body of research on structure-preserving discretizations for these flows, albeit at low order in time. In this work, we present the first discretization of geometric curvature flows (curve shortening/mean curvature flow and curve/surface diffusion) that preserves the evolution of area and volume at arbitrary order in space and time. The key idea is to introduce auxiliary variables in a particular way so that the derivation of the area dissipation law can be replicated after discretization with continuous Petrov--Galerkin in time. These auxiliary variables are indicated by a general strategy for structure-preservation in time that applies to many other problems. The proposed scheme also preserves mesh quality in the same manner as the minimal deformation rate strategy. We demonstrate its structure-preserving properties and high-order convergence on several benchmark examples.