A dynamic mesh algorithm for curvature dependent evolving interfaces
A dynamic mesh algorithm is considered for approximating evolving interfaces in \(\mathbb{R}^n\) which propagate in the normal direction with velocity equal mean curvature plus a forcing function. The method is insensitive to singularity formation which retains the local structure of the geometric flow, and thus the computational complexity typical of \(\mathbb{R}^{n- 1}\). The approach extends naturally to higher dimensions, requiring only an efficient mesh generator, whereas implementing a front-tracking method in three dimensions is a difficult task. The dynamic mesh algorithm uses piecewise linear finite elements, with mass lumping, over a dynamic grid for space digitization and forward differences in time, with adaptive control of the time step. Time stepping is explicit, but stability constraints force small time steps only when singularities develop, relatively large time steps are allowed at beginning or past singularities, when the evolution is smooth. Explicit time-stepping guarantees that at most one layer element has to be added or deleted per time step, thereby making the grid updating simple and, thus, practical. The authors present a number of numerical experiments for the geometric movement in 2, 3, 4, and 8 dimensions, with axial symmetries. Motions by mean curvature (plus forcing), volume constraints and minimal surfaces are discussed.
scientific article; zbMATH DE number 881855