Remarks on the accuracy of algorithms for motion by mean curvature in bounded domains

TL;DR

This paper evaluates the accuracy of algorithms simulating mean curvature motion in bounded domains, introduces new identities for translation solutions, and proposes modifications to improve computational precision.

Contribution

It derives new identities for translation solutions and proposes a simple algorithm modification to enhance accuracy in simulations.

Findings

Error increases with boundary velocity V, especially near boundaries

Proposed algorithm modification improves computational accuracy

Benchmark solutions are used to verify algorithm performance

Abstract

Simulations of motion by mean curvature in bounded domains, with applications to bubble motion and grain growth, rely upon boundary conditions that are only approximately compatible with the equation of motion. Three closed form solutions for the problem exist, governing translation, rotation and expansion of a single interface [1], providing the only benchmarks for algorithm verification. We derive new identities for the translation solution. Then we estimate the accuracy of a straightforward algorithm to recover the analytical solution for different values of the velocity V given along the boundary. As expected, for large V the error can reach unacceptable levels especially near the boundary. We discuss factors influencing the accuracy and propose a simple modification of the algorithm which improves the computational accuracy.