O(t) of โบ(t), and let n be the unit normal vector to โบ(t) pointing towards I(t). The surface โบ(t) propagates in the A new finite element method is discussed for approximating evolving interfaces in R n whose normal velocity equals mean curva-normal direction n with velocity ture 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 nฯช1 without having the drawbacks of front-tracking strategies. A graded dynamic mesh around the propagating front is the sole partition present at where stands for the sum of the principal curvatures any time step and is significantly smaller than a full mesh. Time (positive if I(t) is locally mean convex) and g ฯญ g(x, t)
stepping is explicit, but stability constraints force small time steps only when singularities develop, whereas relatively large time steps is a forcing function. The evolution of โบ(t) may exhibit are allowed before or past singularities, when the evolution is singularities and topological changes, such as breaking, smooth. The explicit marching scheme also guarantees that at most merging and extinction. So the classical geometric apone layer of elements has to be added or deleted per time step, proach fails to describe the problem past singularities, and thereby making mesh updating simple and, thus, practical. Perforfront-tracking methods (FT) may also break down [16].
mance and potentials are fully documented via a number of numerical simulations in 2D, 3D, 4D, and 8D, with axial symmetries. They
They do not only have to compute explicitly, a delicate include tori and cones for the mean curvature flow, minimal and issue for large principal curvatures but small , but also prescribed mean curvature surfaces with given boundary, fattening rely on a catalog of singularities to replace (1.1) whenever for smooth driving force, and volume constraint. แฎ 1996 Academic breaking or merging occurs. It is not surprising then that FT Press, Inc.
are not proven to converge as the discretization parameters tend to zero, most notably for unsmooth flows. When applicable in 2D, however, FT are efficient due to their low 1. REACTION-DIFFUSION APPROACH WITH computational complexity.
DOUBLE OBSTACLE
We present a dynamic mesh algorithm (DMA) insensi-The ever increasing interest in the curvature dependent tive to singularity formation which retains the local strucmotion of fronts stems from its intrinsic mathematical ture of the geometric flow, and thus the computational beauty and difficulty, as well as its applications to phase complexity typical of R nฯช1 . Our method extends naturally transitions in materials science, flame propagation, comto higher dimensions, requiring mainly an efficient mesh bustion theory, crystal growth, etc. [13]. In its classical generator, whereas implementing FT for tracking interformulation, let โบ(t) ส R n be an oriented interface which faces in higher dimensions is a nontrivial matter. Prelimisplits R n into two disjoint regions, the inside I(t) and outside nary results were reported in [20], and the theoretical foundations in [21][22][23][24][25][26]29]. It is based on combining a singularly perturbed reaction-diffusion equation, the so-called Al-* Partially supported by NSF Grant DMS-9305935, MURST, and CNR Contract 94.00139.01. len-Cahn equation, with a double obstacle potential. 296