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May/June 2013 Convergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds
Benoît Merlet, Thanh Nhan Nguyen
Differential Integral Equations 26(5/6): 571-602 (May/June 2013).

Abstract

In this paper, we consider discretizations of systems of differential equations on manifolds that admit a strict Lyapunov function. We study the long-time behavior of the discrete solutions. In the continuous case, if a solution admits an accumulation point for which a Lojasiewicz inequality holds then its trajectory converges. Here we continue the work started in [18] by showing that discrete solutions have the same behavior under mild hypotheses. In particular, we consider the $\theta$-scheme for systems with solutions in $\mathbf{R}^d$ and a projected $\theta$-scheme for systems defined on an embedded manifold. As illustrations, we show that our results apply to existing algorithms: 1) Alouges' algorithm for computing minimizing discrete harmonic maps with values in the sphere, and 2) a discretization of the Landau--Lifshitz equations of micromagnetism.

Citation

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Benoît Merlet. Thanh Nhan Nguyen. "Convergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds." Differential Integral Equations 26 (5/6) 571 - 602, May/June 2013.

Information

Published: May/June 2013
First available in Project Euclid: 14 March 2013

zbMATH: 1299.65162
MathSciNet: MR3086400

Subjects:
Primary: 65L06 , 65L20 , 65P05

Rights: Copyright © 2013 Khayyam Publishing, Inc.

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Vol.26 • No. 5/6 • May/June 2013
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