Convergence of a crystalline approximation for an area-preserving motion

We consider an approximation of area-preserving motion in the plane by a generalized crystalline motion. The area-preserving motion is described by a parabolic partial differential equation with a nonlocal term, while the crystalline motion is governed by a system of ordinary differential equations. We show the convergence between these two motions. The convergence theorem is proved in two steps: first, an a priori estimate is established for a solution to the generalized crystalline motion; second, a discrete W1,p norms of the error is estimated for all 1⩽p<∞ and, passing p to infinity, a discrete W1,∞ error estimate is obtained. We also construct an implicit scheme which enjoys several nice properties such as the area-preserving and curve-shortening, and compare our scheme with a simple scheme.