The flow of a closed surface of codimension 1 in R(R) driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter epsilon > 0. Conforming piecewise linear finite elements, with mass lumping, over a quasi-uniform and weakly acute mesh of size h are further used for space discretization, and combined with forward differences for time discretization with uniform time-step tau. The resulting explicit schemes are the basis for an efficient algorithm, the so-called dynamic mesh algorithm, and exhibit finite speed of propagation and discrete nondegeneracy. No iteration is required, not even to handle the obstacle constraints. The zero level set of the fully discrete solution is shown to converge past singularities to the true interface, provided tau, h(2) approximate to 0(epsilon(4)) and no fattening occurs. If the more stringent relations tau, h(2) approximate to 0(epsilon(6)) are enforced, then an interface rate of convergence O(epsilon) is derived in the vicinity of regular points, along with a companion O(epsilon(1/2)) for type I singularities. For smooth flows, an interface rate of convergence of O(epsilon(2)) is proven, provided tau, h(2) approximate to O(epsilon(5)) and exact integration is used for the potential term. The analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature, which bears some intrinsic interest, Lipschitz properties of viscosity solutions of the level set approach, and discrete nondegeneracy. These basic ingredients are also discussed.

Combined effect of explicit time-stepping and quadrature for curvature driven flows / R.H. Nochetto, C. Verdi. - In: NUMERISCHE MATHEMATIK. - ISSN 0029-599X. - 74:1(1996), pp. 105-136.

Combined effect of explicit time-stepping and quadrature for curvature driven flows

C. Verdi
Ultimo
1996

Abstract

The flow of a closed surface of codimension 1 in R(R) driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter epsilon > 0. Conforming piecewise linear finite elements, with mass lumping, over a quasi-uniform and weakly acute mesh of size h are further used for space discretization, and combined with forward differences for time discretization with uniform time-step tau. The resulting explicit schemes are the basis for an efficient algorithm, the so-called dynamic mesh algorithm, and exhibit finite speed of propagation and discrete nondegeneracy. No iteration is required, not even to handle the obstacle constraints. The zero level set of the fully discrete solution is shown to converge past singularities to the true interface, provided tau, h(2) approximate to 0(epsilon(4)) and no fattening occurs. If the more stringent relations tau, h(2) approximate to 0(epsilon(6)) are enforced, then an interface rate of convergence O(epsilon) is derived in the vicinity of regular points, along with a companion O(epsilon(1/2)) for type I singularities. For smooth flows, an interface rate of convergence of O(epsilon(2)) is proven, provided tau, h(2) approximate to O(epsilon(5)) and exact integration is used for the potential term. The analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature, which bears some intrinsic interest, Lipschitz properties of viscosity solutions of the level set approach, and discrete nondegeneracy. These basic ingredients are also discussed.
Settore MAT/08 - Analisi Numerica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/178607
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