Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

Autor(es)

Gameiro, Marcio ^[1] ; Lessard, Jean-Philippe ^[2] ; Pugliese, Alessandro ^[3]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

^[2] Univ Laval, Dept Math & Stat, 1045 Ave Med, Quebec City, PQ G1V 0A6 - Canada

^[3] Univ Bari Aldo Moro, Dipartimento Matemat, Via E Orabona 4, I-70125 Bari - Italy

• Tipo de documento: Artigo Científico
• Fonte: FOUNDATIONS OF COMPUTATIONAL MATHEMATICS; v. 16, n. 2, p. 531-575, APR 2016.
• Citações Web of Science: 8

Resumo

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn-Hilliard equation. (AU)

• Processo FAPESP: 10/00875-9 - Métodos topológicos e numéricos rigorosos para bifurcações em sistemas dinâmicos
• Beneficiário: Marcio Fuzeto Gameiro
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 13/07460-7 - Computações rigorosas para EDPs
• Beneficiário: Marcio Fuzeto Gameiro
• Modalidade de apoio: Auxílio à Pesquisa - Regular