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2005-09-18 - Colloque/Présentation - communication orale - Anglais - 40 page(s)

Troestler Christophe , "Convergence of a mountain pass algorithm with projector." in International Conference of Numerical Analysis and Applied Mathematics, Rhodes, Grèce, 2005

  • Codes CREF : Calcul des variations (DI1128), Analyse numérique (DI1123), Equations différentielles et aux dérivées partielles (DI1127)
  • Unités de recherche UMONS : Analyse numérique (S835)
  • Centres UMONS : Modélisation mathématique et informatique (CREMMI)

Abstract(s) :

(Anglais) Since the seminal work of Ambrosetti and Rabinowitz [1], variational prin- ciples such as the mountain pass lemma have been in widespread use to prove the existence of solutions of nonlinear PDEs. However, it is only twenty years later that a numerical scheme based on the mountain pass lemma was pro- posed by Choi and McKenna [2]. Other authors have subsequently designed algorithms to find solutions with higher Morse indexes. Despite the fact that these algorithms seem to work well in practice, very little is known about their convergence; the only result being by Chen, Englert, and Zhou, about the convergence of the mountain pass algorithm [3]. Often solutions with special properties (such as non-negativity) are de- sired. This can be expressed by saying that the solution belongs to the image of a (often nonlinear) projector. In this talk, we will present a mountain pass type algorithm with projector and establish its convergence. Applications to semilinear elliptic BVPs will be given. [1] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. [2] Y. S. Choi, P. J. McKenna, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Analysis TMA 20 (1993), 417–437. [3] G. Chen, B. G. Englert, J. Zhou, Convergence analysis of an optimal scaling algorithm for semilinear elliptic boundary value problems, Contemporary Math. 357 (2004), American Mathematical Society, 69–84