# 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

rhn@math.umd.edu
Claudio Verdi

Dipartimento di Matematica

Universita di Milano

Milano, 20133, Italy

verdi@paola.mat.unimi.it
##
Abstract

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.