Convergence of a crystalline algorithmfor the heat equation in one dimensionand for the motion of a graphby weighted curvature

Motion by (weighted) mean curvature is a geometric evolution law forsurfaces, representing steepest descent with respect to (an)isotropicsurface energy. It has been proposed that this motion couldbe computed by solving the analogous evolution law using a``crystalline'' approximation to the surface energy. We present thefirst convergence analysis for a numerical scheme of this type. Ourtreatment is restricted to one dimensional surfaces (curves in theplane) which are graphs. In this context, the scheme amounts to a newalgorithm for solving quasilinear parabolic equations in one spacedimension.