A dynamic mesh algorithm for curvature dependent evolving interfaces

A new finite element method is discussed for approximating evolving interfaces in $\Rn$ whose normal velocity equals mean curvature plus a forcing function. The method is insensitive to singularity formation and retains the local structure of the limit problem, and thus exhibits a computational complexity typical of $\R^{n-1}$ without having the drawbacks of front-tracking strategies. A graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh. Time stepping is explicit, but stability constraints force small time steps only when singularities develop, whereas relatively large time steps are allowed before or past singularities, when the evolution is smooth. The explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step, thereby making mesh updating simple, and thus practical. Performance and potentials are fully documented via a number of numerical simulations in 2D, 3D, 4D, and 8D, with axial symmetries. They include tori and cones for the mean curvature flow, minimal and prescribed mean curvature surfaces with given boundary, fattening for smooth driving force, and volume constraint.