Cette thèse est consacrée à l’analyse numérique en particulier aux estimations a posteriori de l’erreur dans la méthode de décomposition asymptotique partielle de domaine. Il s’agit de problèmes au dérivées partielles elliptiques linéaires et semi- linéaires avec une source qui ne dépend que d’une seule variable dans une partie du domaine. La MAPDD - Méthod of Asymptotic Partial Domain Decomposition - est une méthode inventée par Grigori . Panasenko et développée dans les références [G.P98, G.P99]. L’aidée principale est de remplacer un problème 3D ou 2D par un problème hybride combinée 3D−1D, 3D−2D ou 2D−1D, ou la dimension du problème diminue dans une partie du domaine. Des méthodes de calcul efficaces de solution pour le problème hybride en résultant sont récemment devenues disponibles pour plusieurs systèmes (linéaires/non linéaires, fluide/solide, etc.) ainsi chaque sous-problème est calcul ́ avec un code indépendant de type boîte noire [PBB10, JLB09, JLB11]. La position de la jonction entre les problèmes hétérogènes est asymptotiquement estimée dans les travaux de G. Panasenko [G.P98]. La méthode MAPDD a été conçu pour traiter des problèmes ou un petit paramètre apparaître, et fournit un développement en série de la solution avec des solutions de problèmes simplifiées à l’égard de ce petit paramètre. Dans le problème considéré dans les chapitres 3 et 4, aucun petit paramètre n’existe, mais en raison de considérations géométriques concernant le domaine on suppose que la solution ne diffère pas significativement d’une fonction qui dépend seulement d’une variable dans une partie du domaine Ω. La théorie de MAPDD n’est pas adaptée pour une telle situation, et si cette théorie est appliquée formellement elle ne fournit pas d’estimation d’erreur. / This thesis is devoted to numerical analysis in particular a postoriori estimates of the error in the method of asymptotic partial domain decomposition. There are problems in linear elliptic partial and semi-linear with a source which depends only of one variable in a portion of domain. Method of Asymptotic Partial Decomposition of a Domain (MAPDD) originates from the works of Grigori.Panasonko [12, 13]. The idea is to replace an original 3D or 2D problem by a hybrid one 3D − 1D; or 2D − 1D, where the dimension of the problem decreases in part of domain. Effective solution methods for the resulting hybrid problem have recently become available for several systems (linear/nonlinear, fluid/solid, etc.) which allow for each subproblem to be computed with an independent black-box code [21, 17, 18]. The location of the junction between the heterogeneous problems is asymptotically estimated in the works of Panasenko [12]. MAPDD has been designed for handling problems where a small parameter appears, and provides a series expansion of the solution with solutions of simplified problems with respect to this small parameter. In the problem considered in chapter 3 and 4, no small parameter exists, but due to geometrical considerations concerning the domain Ω it is assumed that the solution does not differ very much from a function which depends only on one variable in a part of the domain. The MAPDD theory is not suited for such a context, but if this theory is applied formally it does not provide any error estimate. The a posteriori error estimate proved in this chapter 3 and 4, is able to measure the discrepancy between the exact solution and the hybrid solution which corresponds to the zero-order term in the series expansion with respect to a small parameter when it exists. Numerically, independently of the existence of an asymptotical estimate of the location of the junction, it is essential to detect with accuracy the location of the junction. Let us also mention the interest of locating with accuracy the position of the junction in blood flows simulations [23]. Here in this chapter 3,4 the method proposed is to determine the location of the junction (i.e. the location of the boundary Γ in the example treated) by using optimization techniques. First it is shown that MAPDD can be expressed with a mixed domain decomposition formulation (as in [22]) in two different ways. Then it is proposed to use an a posteriori error estimate for locating the best position of the junction. A posteriori error estimates have been extensively used in optimization problems, the reader is referred to, e.g. [1, 11].

Orientador: Marcelo Muniz Silva Alves / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T16:23:24Z (GMT). No. of bitstreams: 1 Ritter_Donizete_M.pdf: 621556 bytes, checksum: 2bf0368b784f3a2be59ca3c2552f4908 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho abordamos a teoria dos Códigos Corretores de Erros clássica e também os códigos sobre ordens parciais, com algumas comparações entre os dois casos. Enfocamos, particularmente, a definição de Alfabeto, a distância de Hamming, os códigos lineares e a definição de matriz geradora de um código; o estudo dos limitantes de Singleton e de Hamming, além de tratar dos Códigos de Hamming. Em relação aos Códigos em Conjuntos Parcialmente Ordenados, apresentamos a definição de ordens parciais, métricas sobre conjuntos ordenados, contagem dos elementos da "bola", resultados sobre Ideais e o Código de Hamming Estendido; estudamos o caso da ordem cadeia ("chain poset"), analisando os códigos de uma cadeia e os códigos de duas cadeias de mesmo comprimento e, por fim, nos dedicamos ao estudo das "Métricas POSET", que admitem códigos binários perfeitos de codi-mensão m, caracterizando assim os Códigos Posets m-corretores de erros. Nosso objetivo é apresentar um texto, acessível a alunos de graduação, que contemple a teoria básica dos Códigos Corretores de Erros, no entanto, forneça uma noção sobre os códigos sobre ordens parciais. / Abstract: In this work, we address the classical theory of error-correcting codes and the theory of codes over poset spaces, also known as poset codes, establishing comparisons between these two cases. In particular, we present the definition of alphabet, the Hamming distance, linear codes and the definition of a generating matrix for a linear code; we also present the Singleton and Hamming bounds, alongside with the Hamming codes. With respect to poset codes, we present the definitions of partial orders and of the poset metric, the counting of the number of elements in a ball in a poset space, some results on ideals in posets and the extended Hamming code; we study the chain poset case, analysing the cases of codes over a chain poset and codes over a union of two chains of the same length and, finally, we study the poset metrics that allow m-perfect binary codes of codimension m, thus characterizing these codes. Our aim is to present a text, accessible for undergraduates, that encompasses the basic theory of error-correcting codes and, nonetheless, also provides some notions on poset codes. / Mestrado / Teoria dos Erros / Mestre em Matemática