Comoving mesh method for certain classes of moving boundary problems

TL;DR

The paper introduces the comoving mesh method, a finite element scheme for efficiently solving certain moving boundary problems like Hele-Shaw flow and mean curvature flow, demonstrated through numerical experiments.

Contribution

It develops a novel Lagrangian-type numerical scheme that extends boundary velocities into the domain for improved boundary tracking in moving boundary problems.

Findings

Method effectively updates boundary and mesh at each time step.

Numerical experiments confirm the practicality and convergence of the scheme.

The scheme accurately approximates solutions for classical moving boundary problems.

Abstract

A Lagrangian-type numerical scheme called the "comoving mesh method" or CMM is developed for numerically solving certain classes of moving boundary problems which include, for example, the classical Hele-Shaw flow problem and the well-known mean curvature flow problem. This finite element scheme exploits the idea that the normal velocity field of the moving boundary can be extended throughout the entire domain of definition of the problem using, for instance, the Laplace operator. Then, the boundary as well as the finite element mesh of the domain are easily updated at every time step by moving the nodal points along this velocity field. The feasibility of the method, highlighting its practicality, is illustrated through various numerical experiments. Also, in order to examine the accuracy of the proposed scheme, the experimental order of convergences between the numerical and manufactured solutions for these examples are also calculated.