# Convergence past singularities for a fully discrete approximation
of curvature driven interfaces

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SIAM J. Numer. Anal., 34 (1997), 490-512.
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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
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Abstract

Consider a closed surface in $\Rn$ of codimension $1$
which propagates in the normal direction with
velocity proportional to its mean curvature plus a forcing term.
This geometric problem is first approximated by a singularly
perturbed parabolic double obstacle problem with small parameter
$\eps>0$. Conforming piecewise linear finite elements over a
quasi-uniform and strongly acute mesh of size $h$ are further
used for space discretization, and combined with backward
differences for time discretization with uniform time-step $\tau$.
It is shown that the zero level set of the fully discrete solution
converges past singularities to the true interface, provided
$\tau,h^2\approx o(\eps^3)$ and no fattening occurs.
If the more stringent relations
$\tau,h^2\approx\O(\eps^4)$ are enforced, then a linear rate of
convergence $\O(\eps)$ for interfaces is derived in the vicinity
of regular points, namely those for which the underlying
viscosity solution is nondegenerate. Singularities and their smearing
effect are also studied. The analysis is based on
constructing discrete barriers via a parabolic projection,
Lipschitz dependence
of viscosity solutions with respect to perturbations
of data, and discrete nondegeneracy. These issues are proven, along with
quasi-optimality in 2D of the parabolic projection
in $L^\infty$ with respect to both order and
regularity requirements for functions in $W^{2,1}_p$.