EFFICIENT RIGOROUS NUMERICS FOR HIGHER-DIMENSIONAL PDEs VIA ONE-DIMENSIONAL ESTIMATES

Autor(es)

Gameiro, Marcio ^[1] ; Lessard, Jean-Philippe ^{[2, 3]}

Afiliação do(s) autor(es)

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

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

^[3] BCAM, Derio 48160, Bizkaia - Spain

• Tipo de documento: Artigo Científico
• Fonte: SIAM JOURNAL ON NUMERICAL ANALYSIS; v. 51, n. 4, p. 2063-2087, 2013.
• Citações Web of Science: 19

Resumo

We present an efficient rigorous computational method which is an extension of the work Analytic Estimates and Rigorous Continuation for Equilibria of Higher-Dimensional PDEs (M. Gameiro and J.-P. Lessard, J. Differential Equations, 249 (2010), pp. 2237 2268). The idea is to generate sharp one-dimensional estimates using interval arithmetic which are then used to produce high-dimensional estimates. These estimates are used to construct the radii polynomials which provide an efficient way of determining a domain on which the contraction mapping theorem is applicable. Computing the equilibria using a finite-dimensional projection, the method verifies that the numerically produced equilibrium for the projection can be used to explicitly define a set which contains a unique equilibrium for the PDE. A new construction of the polynomials is presented where the nonlinearities are bounded by products of one-dimensional estimates as opposed to using FFT with large inputs. It is demonstrated that with this approach it is much cheaper to prove that the numerical output is correct than to recompute at a finer resolution. We apply this method to PDEs defined on three-and four-dimensional spatial domains. (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