Dziuk, Gerhard Convergence of a semi-discrete scheme for the curve shortening flow. (English) Zbl 0811.65112 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. Cited in 1 ReviewCited in 31 Documents 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 PDF BibTeX XML Cite \textit{G. Dziuk}, Math. Models Methods Appl. Sci. 4, No. 4, 589--606 (1994; Zbl 0811.65112) Full Text: DOI OpenURL