Convergence of a semi-discrete scheme for the curve shortening flow. (English) Zbl 0811.65112

Dziuk, Gerhard

Math. Models Methods Appl. Sci. 4, No. 4, 589-606 (1994).

Summary: Convergence for a spatial discretization of the curvature flow for curves in possibly higher codimension is proved in \(L^ \infty((0,T), L^ 2(\mathbb{R}/2\pi))\cap L^ 2((0,T), H^ 1(\mathbb{R}/2\pi))\). Asymptotic convergence in these norms is achieved for the position vector and its time derivative which is proportional to curvature. The underlying algorithm rests on a formulation of mean curvature flow which uses the Laplace-Beltrami operator and leads to tridiagonal linear systems which can easily be solved.

MSC:

65Z05	Applications to the sciences
65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
37C80	Symmetries, equivariant dynamical systems (MSC2010)
35K15	Initial value problems for second-order parabolic equations

Keywords:

semidiscretization; finite difference method; convergence; curvature flow