Obstacle Mean Curvature Flow: Efficient Approximation and Convergence Analysis

TL;DR

This paper presents a simple, efficient numerical scheme for obstacle mean curvature flow that preserves key geometric properties and guarantees convergence to the viscosity solution, validated through numerical experiments.

Contribution

It introduces a naive yet structurally sound numerical method that combines the Merriman-Bence-Osher scheme with obstacle constraints, ensuring stability and convergence.

Findings

The scheme inherits a geometric comparison principle.

The scheme is unconditionally stable.

Numerical experiments validate the method's effectiveness.

Abstract

We introduce a simple and efficient numerical method to compute mean curvature flow with obstacles. The method augments the Merrimam-Bence-Osher scheme with a pointwise update that enforces the constraint and therefore retains the computational complexity of the original scheme. Remarkably, this naive scheme inherits both crucial structural properties of obstacle mean curvature flow: a geometric comparison principle and a minimizing movements interpretation. The latter immediately implies the unconditional stability of the scheme. Based on the comparison principle we prove the convergence of the scheme to the viscosity solution of obstacle mean curvature flow. Moreover, using the minimizing movements interpretation, we show convergence of a spatially discrete model. Finally, we present numerical experiments for a physical model that inspired this work.