Approximation and application of minimizing movements for surface PDE

TL;DR

This paper extends minimizing movements to surface PDEs using the closest point method, developing approximation algorithms that enable simulation of complex surface-constrained curvature flows.

Contribution

It introduces new approximation methods for minimizing movements on surfaces and demonstrates their application to multiphase, volume-preserving curvature flows.

Findings

Convergence of the proposed approximation methods is observed.

Developed threshold dynamics algorithms for surface interfacial motions.

Enabled simulation of multiphase, volume-preserving curvature flows on surfaces.

Abstract

By employing the closest point method, we extend the applicability of minimizing movements to the surface PDE setting. The corresponding approximation methods are created, and their convergence is observed. The numerical methods are then used to develop threshold dynamics algorithms for surface-constrained interfacial motions. In particular, show how the minimizing movements enable one to approximate multiphase, volume-preserving, curvature flows on surfaces via generalized MBO and HMBO algorithms.