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Combined effect of explicit time-stepping and quadrature for curvature driven flows. (English) Zbl 0859.65066

The flow of a closed surface of codimension 1 in \(\mathbb{R}^n\) driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter. The discretization is made by conforming piecewise linear finite elements, with mass lumping, over a quasi-uniform and weakly acute mesh combined with forward differences with uniform time-step. An efficient algorithm, the so-called dynamic mesh algorithm, is obtained. It shows finite speed of propagation and discrete nondegeneracy. It is demonstrated that the fully discrete solution converges past singularities to the true interface and the rate of convergence is discussed. Smooth flows are also analyzed.
Reviewer: V.Arnăutu (Iaşi)

MSC:

65K10 Numerical optimization and variational techniques
35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
49Q05 Minimal surfaces and optimization
49M15 Newton-type methods
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35B50 Maximum principles in context of PDEs
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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