Comoving mesh method for multi-dimensional moving boundary problems: Mean-curvature flow and Stefan problems

The “comoving mesh method” or CMM is a Lagrangian-type numerical scheme recently developed for numerically solving classes of moving boundary problems. The scheme is well-suited for solving, for example, the Hele-Shaw flow problem, the curve-shortening problem, and the well-known Bernoulli free boundary problem. This finite element method exploits the idea that the normal velocity field of a moving boundary can be extended smoothly throughout the entire domain of the problem’s definition using, for instance, the Laplace operator. By doing so, the finite element mesh of the domain is easily updated at every time step by moving the nodal points along this velocity field. As a result, one avoids the need to generate a new computational mesh at every time step. In this exposition, we further develop and demonstrate the practicality of the method by solving moving boundary problems set in higher dimensions and its application to solving the Stefan problem in two dimensions. Numerical examples are provided for illustration purposes.