A converging finite element scheme for motion by curvature of a network with a triple junction

TL;DR

This paper introduces a new finite element scheme for simulating the curvature-driven evolution of a network of three curves meeting at a triple junction, ensuring stability, accuracy, and proper junction motion.

Contribution

It proposes a novel semi-discrete finite element method with a variational formulation and error analysis for networks with triple junctions, including a regularization approach.

Findings

Proved convergence and optimal error estimates for the scheme

Numerical tests confirm the scheme's accuracy and stability

Analyzed the effect of regularization parameter on results

Abstract

A new semi-discrete finite element scheme for the evolution of three parametrized curves by curvature flow that are connected by a triple junction is presented and analyzed. In this triple junction, conditions are imposed on the angles at which the curves meet. One of the key problems in analyzing motion of networks by curvature law is the choice of a tangential velocity that allows for motion of the triple junction, does not lead to mesh degeneration, and is amenable to an error analysis. Our approach consists in considering a perturbation of a classical smooth formulation. The problem we propose admits a natural variational formulation that can be discretized with finite elements. The perturbation can be made arbitrarily small when a regularization parameter shrinks to zero. Convergence of the new scheme including optimal error estimates are proved. These results are supported by some numerical tests. Finally, the influence of the small regularization parameter on the properties of scheme and the accuracy of the results is numerically investigated.