Orientador: German Jesus Lozada Cruz / Banca: Luiz Augusto Fernandes de Oliveira / Banca: José Marcio Machado / Resumo: O objetivo deste trabalho é estudar problemas de valor de contorno do tipo {ÿ + f(t) =0 y(0)=0˙ y(1)= ky(η), (1) onde η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). Para antingirmos nosso objetivo usamosas funções de Green G(t,s)que nos permitem escrever a solução do problema(1)na seguinte forma: w(t)= ∫ 1 0 G(t,s)f(s)ds. Usando esta solução, investigamos através do ponto fixo de Schauder a solvabilidade do problema não linear { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / Abstract: The main goal of this work is study the following boundary value problems {ÿ + f(t) = 0 =0 y(0)=0˙ y(1)= ky(η), (1), where η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). To achieve our goal we use the Green's function G(t,s) which allow us to write the solution of the problem (2) in the form: w(t)= ∫ 1 0 G(t,s)f(s)ds. Using this solution and the Schauder point theory, also we study the solvability of a nonlinear problem { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / Mestre

No description available.

The paper deals with tau-extrapolation - a modification of the multigrid method, which leads to solutions with an improved con- vergence order. The number of numerical operations depends linearly on the problem size and is not much higher than for a multigrid method without this modification. The paper starts with a short mathematical foundation of the tau-extrapolation. Then follows a careful tuning of some multigrid components necessary for a successful application of tau-extrapolation. The next part of the paper presents numerical illustrations to the theoretical investigations for one- dimensional test problems. Finally some experience with the use of tau-extrapolation for the Navier-Stokes equations is given.

La résolution numérique de n’importe quelle discrétisation d’équations aux dérivées partielles non linéaires requiert le plus souvent un algorithme itératif. En général, la discrétisation des équations aux dérivées partielles donne lieu à des systèmes de grandes dimensions. Comme la résolution des grands systèmes est très coûteuse en terme de temps de calcul, une question importante se pose: afin d’obtenir une solution approchée de bonne qualité, quand est-ce qu’il faut arrêter l’itération afin d’éviter les itérations inutiles ? L’objectif de cette thèse est alors d’appliquer, à différentes équations, une méthode qui nous permet de diminuer le nombre d’itérations de la résolution des systèmes en gardant toujours une bonne précision de la méthode numérique. En d’autres termes, notre but est d’appliquer une nouvelle méthode qui fournira un gain remarquable en terme de temps de calcul. Tout d’abord, nous appliquons cette méthode pour un problème non linéaire modèle. Nous effectuons l’analyse a priori et a posteriori de la discrétisation par éléments finis de ce problème et nous proposons par la suite deux algorithmes de résolution itérative correspondants. Nous calculons les estimations d’erreur a posteriori de nos algorithmes itératifs proposés et nous présentons ensuite quelques résultats d’expérience numériques afin de comparer ces deux algorithmes. Nous appliquerons de même cette approche pour les équations de Navier-Stokes. Nous proposons un schéma itératif et nous étudions la convergence et l’analyse a priori et a posteriori correspondantes. Finalement, nous présentons des simulations numériques montrant l’efficacité de notre méthode. / The numerical resolution of any discretization of nonlinear PDEs most often requires an iterative algorithm. In general, the discretization of partial differential equations leads to large systems. As the resolution of large systems is very costly in terms of computation time, an important question arises. To obtain an approximate solution of good quality, when is it necessary to stop the iteration in order to avoid unnecessary iterations? A posteriori error indicators have been studied in recent years owing to their remarkable capacity to enhance both speed and accuracy in computing. This thesis deals with a posteriori error estimation for the finite element discretization of nonlinear problems. Our purpose is to apply a new method that allows us to reduce the number of iterations of the resolution system while keeping a good accuracy of the numerical method. In other words, our goal is to apply a new method that provides a remarkable gain in computation time. For a given nonlinear equation we propose a finite element discretization relying on the Galerkin method. We solve the discrete problem using two iterative methods involving some kind of linearization. For each of them, there are actually two sources of error, namely discretization and linearization. Balancing these two errors can be very important, since it avoids performing an excessive number of iterations. Our results lead to the construction of computable upper indicators for the full error. Similarly, we apply this approach to the Navier-Stokes equations. Several numerical tests are provided to evaluate the efficiency of our indicators.

Cette thèse propose trois nouveaux développements de la méthode des bases réduites (RB) et de la méthode d'interpolation empirique (EIM) pour des problèmes non-linéaires. La première contribution est une nouvelle méthodologie, la méthode progressive RB-EIM (PREIM) dont l'objectif est de réduire le coût de la phase de construction du modèle réduit tout en maintenant une bonne approximation RB finale. L'idée est d'enrichir progressivement l'approximation EIM et l'espace RB, contrairement à l'approche standard où leurs constructions sont disjointes. La deuxième contribution concerne la RB pour les inéquations variationnelles avec contraintes non-linéaires. Nous proposons une combinaison RB-EIM pour traiter la contrainte. En outre, nous construisons une base réduite pour les multiplicateurs de Lagrange via un algorithme hiérarchique qui conserve la positivité des vecteurs cette base. Nous appliquons cette stratégie aux problèmes de contact élastique sans frottement pour les maillages non-coïncidents. La troisième contribution concerne la réduction de modèles avec assimilation de données. Une méthode dédiée a été introduite dans la littérature pour combiner un modèle numérique avec des mesures expérimentales. Nous élargissons son cadre d'application aux problèmes instationnaires en exploitant la méthode POD-greedy afin de construire des espaces réduits pour tout le transitoire temporel. Enfin, nous proposons un nouvel algorithme qui produit des espaces réduits plus représentatifs de la solution recherchée tout en minimisant le nombre de mesures nécessaires pour le problème réduit final / This thesis introduces three new developments of the reduced basis method (RB) and the empirical interpolation method (EIM) for nonlinear problems. The first contribution is a new methodology, the Progressive RB-EIM (PREIM) which aims at reducing the cost of the phase during which the reduced model is constructed without compromising the accuracy of the final RB approximation. The idea is to gradually enrich the EIM approximation and the RB space, in contrast to the standard approach where both constructions are separate. The second contribution is related to the RB for variational inequalities with nonlinear constraints. We employ an RB-EIM combination to treat the nonlinear constraint. Also, we build a reduced basis for the Lagrange multipliers via a hierarchical algorithm that preserves the non-negativity of the basis vectors. We apply this strategy to elastic frictionless contact for non-matching meshes. Finally, the third contribution focuses on model reduction with data assimilation. A dedicated method has been introduced in the literature so as to combine numerical models with experimental measurements. We extend the method to a time-dependent framework using a POD-greedy algorithm in order to build accurate reduced spaces for all the time steps. Besides, we devise a new algorithm that produces better reduced spaces while minimizing the number of measurements required for the final reduced problem

The paper deals with tau-extrapolation - a modification of the multigrid method, which leads to solutions with an improved con- vergence order. The number of numerical operations depends linearly on the problem size and is not much higher than for a multigrid method without this modification. The paper starts with a short mathematical foundation of the tau-extrapolation. Then follows a careful tuning of some multigrid components necessary for a successful application of tau-extrapolation. The next part of the paper presents numerical illustrations to the theoretical investigations for one- dimensional test problems. Finally some experience with the use of tau-extrapolation for the Navier-Stokes equations is given.

No description available.

Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann. / In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.

Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann. / In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.