Convergence Past Singularities for a Fully Discrete Approximation of Curvature-Driven Interfaces
Ricardo H. Nochetto, C. Verdi · SIAM Journal on Numerical Analysis · 1997
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 two dimensions of the parabolic projection in $L^\infty$ with respect to both order and regularity requirements for functions in $W^{2,1}_p$.