Neste trabalho desenvolvemos métodos numéricos para aproximação de solução da equação do calor, baseados nos princípios extremos dúais de Noble e Sewell, onde usamos o método dos Elementos Finitos para a discretização. Exibimos um espaço de Hilbert X, uma forma bilinear\'s a ele associada e verificamos todas as condições do lema de Max-Milgram com as quais temos prova de existência e unicidade de solução da nossa formulação. Além disso provamos um teorema de convergência. Nós usamos funções lineares por partes no tempo e no espaço. Os problemas de minimização e maximização resultantes, são resolvidos por um método de Gradientes Conjugado matricial. Para uma precisão de 10-5, são necessárias cerca de n/20 iterações para n grande, onde n é o tamanho da discretização. / In this work we develop numerical methods for approximate solutions of the heat equation, based on the dual extremum principies of Noble and Sewell, where we use the Finite Element Method for discretization. We exhibit a Hilbert Space X, bilinear form S associated to it and we verify ali the condi tions of Lax-Milgram\'s lemma with Which we get proof of existence and uniqueness of solution of our formulation.Flurtilemore we prove a convergence theorem. We use piecewise linear functions both in time- and in space. The resulting minimization and maximization problents are solved by a matricial form of the Conjugate Gradient method . For n large enough it was needed about n/20 iterations to achiev the precision of 10-5, n is the size of the discretization.

Não disponível / In this work, we study the problem of separating the range of a given map, by its image; we also study the topology of certain stable maps. By working with immersions with normal crossings, Mm → Nm-1, we firstly obtain interesting results which guarantee the separation of N by f(M), under certain conditions. Then, by using the Stein Factorization of a given stable map and some results on the classification of 1-connected 4-manifolds, we obtain interesting information on the topologies of the singular set and of the domain of such a map, particularly in the special generic case. Finally, by working with stable maps whose only singularities are fold points and whose domain has low dimension, we simplify its singular set, by cancelling components by local deformations, in order to get topological information of the source.

Não disponível / In this work, we study the problem of separating the range of a given map, by its image; we also study the topology of certain stable maps. By working with immersions with normal crossings, Mm → Nm-1, we firstly obtain interesting results which guarantee the separation of N by f(M), under certain conditions. Then, by using the Stein Factorization of a given stable map and some results on the classification of 1-connected 4-manifolds, we obtain interesting information on the topologies of the singular set and of the domain of such a map, particularly in the special generic case. Finally, by working with stable maps whose only singularities are fold points and whose domain has low dimension, we simplify its singular set, by cancelling components by local deformations, in order to get topological information of the source.