Combined effect of explicit time stepping and quadrature for curvature driven flows

Numer. Math., 74 (1996), 105-136.

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA

Claudio Verdi
Dipartimento di Matematica
Universita di Milano
Milano, 20133, Italy


The flow of a closed surface of codimension $1$ in $\Rn$ driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter $\eps>0$. Conforming piecewise linear finite elements, with mass lumping, over a quasi-uniform and weakly acute mesh of size $h$ are further used for space discretization, and combined with forward differences for time discretization with uniform time-step $\tau$. The resulting explicit schemes are the basis for an efficient algorithm, the so-called dynamic mesh algorithm, and exhibit finite speed of propagation and discrete nondegeneracy. No iteration is required, not even to handle the obstacle constraints. The zero level set of the fully discrete solution is shown to converge past singularities to the true interface, provided $\tau,h^2\approx o(\eps^4)$ and no fattening occurs. If the more stringent relations $\tau,h^2\approx\O(\eps^6)$ are enforced, then an interface rate of convergence $\O(\eps)$ is derived in the vicinity of regular points, along with a companion $\O(\eps^{1/2})$ for type I singularities. For smooth flows, an interface rate of convergence of $\O(\epsq)$ is proven, provided $\tau,h^2 \approx\O(\eps^5)$ and exact integration is used for the potential term. The analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature, which bears some intrinsic interest, Lipschitz properties of viscosity solutions of the level set approach, and discrete nondegeneracy. These basic ingredients are also discussed.