An algorithm for evolutionary surfaces.(English)Zbl 0714.65092

Using the finite element method in space coordinates and then an implicit time discretization in time t, the author presents a numerical algorithm to evaluate the mean curvature flow of surfaces with or without boundary which are described in time by the equation $$\partial u/\partial t- \Delta_{S(t)}u=2 H(u)n,\quad u(,0)=u_ 0,$$ where $$\Delta_{S(t)}$$ is the Beltrami operator on S(t), H being the mean curvature of S(t). As a limit case as $$t\to \infty$$, this method can give a solution to Plateau’s problem for H-surfaces. Some numerical results and their graphic pictures are also presented.
Reviewer: L.P.Lebedev

MSC:

 65Z05 Applications to the sciences 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35Q80 Applications of PDE in areas other than physics (MSC2000) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text:

References:

 [1] [B] Brakke, K.A.: the motion of a surface by its mean curvature. Math. Notes, Princeton Univ. Press, Princeton N.J., 1978 · Zbl 0386.53047 [2] [D] Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: S. Hildebrandt, R. Leis (ed.) Partial differential equations and calculus of variations. Lect. Notes Math Vol. 1357, pp. 142-155, Berlin Heidelberg New York: Springer 1988 [3] [H1] Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom.20, 237-266 (1984) · Zbl 0556.53001 [4] [H2] Huisken, G.: Non-parametric mean curvature evolution with boundary conditions. CMA Research Report 44 (1987) [5] [H3] Huisken, G.: Asymptotic behaviour for singularities of the mean curvature flow. CMA Research Report 49 (1987) [6] [W] Wohlrab, O.: Zur numerischen Behandlung von parametrischen Minimalfl?chen mit halbfreien R?ndern. Dissertation Bonn 1989
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.