An algorithm for evolutionary surfaces. (English) Zbl 0714.65092

Dziuk, G.

Numer. Math. 58, No. 6, 603-611 (1991).

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

Keywords:

algorithm; evolutionary surfaces; finite element method; implicit time discretization; mean curvature flow of surfaces; Beltrami operator; Plateau’s problem; H-surfaces; numerical results

References:

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[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