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Convergence of an algorithm for mean curvature motion. (English) Zbl 0802.65098
In a recent paper of J. Bence, B. Merriman and S. Osher [Diffusion generated motion by mean curvature (Preprint 1992)] a new computational algorithm was proposed for tracking the time evolution of a set in \(n\)-dimensional space whose boundary moves with the velocity equaling its mean curvature. The algorithm computes the mean curvature flow, in terms of solutions of the usual heat equation, continually reinitialized after short time steps.
The present paper employs nonlinear semigroup theory to reconcile this algorithm with the “level set” approach to mean curvature flow used in the paper of S. Osher and J. A. Sethian [J. Comput. Phys. 79, No. 1, 12-49 (1988; Zbl 0659.65132)], the author and J. Spruck [J. Diff. Geom. 33, No. 3, 635-681 (1991; Zbl 0726.53029) and Y.-G. Chen, Y. Giga and S. Goto [ibid. 33, No. 3, 749-786 (1991; Zbl 0715.35037)].

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
47H20 Semigroups of nonlinear operators
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