Approximation of multiphase mean curvature flows with arbitrary nonnegative mobilities

TL;DR

This paper introduces a robust variational phase field method for approximating multiphase mean curvature flows with arbitrary nonnegative mobilities, including zero mobility cases, and validates it through analysis and numerical experiments.

Contribution

It generalizes existing approaches to handle arbitrary nonnegative mobilities by decomposing them into harmonically additive parts and establishes the method's second-order accuracy.

Findings

Method effectively approximates flows with zero mobility phases.

Numerical experiments confirm robustness and accuracy in 2D and 3D.

The approach is consistent with the sharp interface limit.

Abstract

This paper is devoted to the robust approximation with a variational phase field approach of multiphase mean curvature flows with possibly highly contrasted mobilities. The case of harmonically additive mobilities has been addressed recently using a suitable metric to define the gradient flow of the phase field approximate energy. We generalize this approach to arbitrary nonnegative mobilities using a decomposition as sums of harmonically additive mobilities. We establish the consistency of the resulting method by analyzing the sharp interface limit of the flow: a formal expansion of the phase field shows that the method is of second order. We propose a simple numerical scheme to approximate the solutions to our new model. Finally, we present some numerical experiments in dimensions 2 and 3 that illustrate the interest and effectiveness of our approach, in particular for approximating flows in which the mobility of some phases is zero.