We consider the problem of analytic continuation of the solution of the multidimensional Lame system in infinite domains through known values of the solution and the corresponding strain tensor on a part of the boundary, i.e,the Cauchy problem.

In this paper we consider the regularization of the Cauchy problem for a system of second order differential equations with constant coefficients.

We use Darcy's law and conservation of mass to model the flow of a fluid through a porous medium. It is a second order elliptic system with a heterogeneous coefficient. We consider the equations written in mixed form. In the heterogeneous case, we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom. In the case of a locally periodic heterogeneous coefficient of period epsilon, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when epsilon is small. Moreover, we present numerical examples to assess its performance when the coefficient is not obviously locally periodic. We show that the new mortar method works well, and better than polynomial mortar spaces. On the other hand, we also propose to use multiscale mortars as a coarse component to construct a two-level preconditioner for the saddle point linear system arising from the fine scale discretization of the mixed finite element system. The two-level preconditioners are constructed based on the interfaces. We propose a framework to define the interpolation operators for the face based two-level preconditioners for different combination of coarse and fine scale mortar spaces for matching and nonmatching grids. In this dissertation, we show that for quasi-homogeneous problems and matching grids, the condition number of the preconditioned interface operator is bounded by (log(H/h))², which is the same as the traditional two-level preconditioners, for quasi-homogeneous problems. We show several numerical examples to demonstrate that for the strongly heterogeneous porous media, it is often desirable and even necessary to use a higher dimensional coarse mortar space to construct the coarse preconditioner to achieve convergence. We apply our ideas to study slightly compressible single phase and two-phase flow in a porous medium. We find that for the nonlinear single phase problem, the two-level preconditioners could be successfully applied to the symmetrized linear system. For the two-phase problem, using the fine scale, instead of multiscale, velocity solutions from the flow problem can greatly benefit the transport problem. / text

Dans cette thèse, nous étudions des problèmes aux limites pour les systèmes elliptiques sous forme divergence avec coefficients complexes dans L^{infty}. Nous prouvons des estimations a priori, discutons de la solvabilité et d'extrapolation de la solvabilité. Nous utilisons une transformation via des équations Cauchy-Riemann généralisées due à P. Auscher, A. Axelsson et A. McIntosh. On peut résoudre les équations Cauchy-Riemann généralisées via la semi-groupe engendré par un opérateur différentiel perturbé d'ordre un de type Dirac. A l'aide du semi-groupe, nous étudions la théorie L^{p} avec une discussion sur la bisectorialité, le calcul fonctionnel holomorphe et les estimations hors-diagonales pour des opérateurs dans le calcul fonctionnel. En particulier, nous développons une théorie L^{p}-L^{q} pour des opérateurs dans le calcul fonctionnel d'opérateur de type Dirac perturbé. Les problèmes de Neumann, Régularité et Dirichlet se formulent avec des estimations quadratiques et des estimations pour la fonction maximale nontangentielle. Cela conduit à à démontrer de telles estimations pour le semi-groupe d'opérateur de Dirac Pour cela, nous utilisons les espaces Hardy associés et les identifions dans certains cas avec des sous-espaces des espaces de Hardy et Lebesgue classiques. Nous obtenons enfin des estimations a priori pour les problème aux limites via une extension utilisant des espaces de Sobolev associés. Nous utilisons les estimations a priori pour une discussion sur la solvabilité des problèmes aux limites et montrer un théorème d'extrapolation de la solvabilité. / In this this thesis we study boundary value problems for elliptic systems in divergence form with complex coefficients in L^{\infty}. We prove a priori estimates, discuss solvability and extrapolation of solvability. We use a transformation to generalized Cauchy-Riemann equations due to P. Auscher, A. Axelsson, and A. McIntosh. The generalized Cauchy-Riemann equations can be solved by the semi-group generated by a perturbed first order Dirac/differential operator. In relation to semi-group theory we setup the L^p theory by a discussion of bisectoriality, holomorphic functional calculus and off-diagonal estimates for operators in the functional calculus. In particular, we develop an L^p-L^q theory for operators in the functional calculus of the first order perturbed Dirac/differential operators. The formulation of Neumann, Regularity and Dirichlet problems involve square function estimates and nontangential maximal function estimates. This leads us to discuss square function estimates and nontangential maximal function estimates involving operators in the functional calculus of the perturbed first order Dirac/differential operator. We discuss the related Hardy spaces associated to operators and prove identifications by subspaces of classical Hardy and Lebesgue spaces. We obtain the a priori estimates by an extension of the square function estimates and nontangential maximal function estimates to Sobolev spaces associated to operators. We use the a priori estimates for a discussion of solvability and extrapolation of solvability.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-06T12:56:02Z No. of bitstreams: 1 TATIANA ROCHA DE SOUZA - DISSERTAÇÃO PPGMAT 2005..pdf: 635145 bytes, checksum: 84477cb78515ac5241ef5b362a0ff141 (MD5) / Made available in DSpace on 2018-07-06T12:56:02Z (GMT). No. of bitstreams: 1 TATIANA ROCHA DE SOUZA - DISSERTAÇÃO PPGMAT 2005..pdf: 635145 bytes, checksum: 84477cb78515ac5241ef5b362a0ff141 (MD5) Previous issue date: 2005-08 / Capes / Neste trabalho estudamos a eficiência do Método de Galerkin na resolução de problemas e sistemas Elípticos lineares, não-lineares, variacionias e não-variacionais. / In this work we study the Galerkin Method efficiency in solving of linear, nonlinear, variational and nonvariational Elliptic problems and Elliptic systems.

The theory of homogenization allows to find for a given system of partial differential equations governing a model with a very complicated internal struc- ture a system governing a model without this structure, whose solution is in a certain sense an approximation of the solution of the original problem. In this thesis, methods of the theory of homogenization are applied to three sys- tems of partial differential equations. The first one governs a flow of a class of non-Newtonian fluid through a porous medium. The second system is utilized for modeling of a flow of a fluid through an electric field wherein the viscosity depends significantly on the intensity of the electric field. For the third system is considered an elliptic operator having growth and coercivity indicated by a general anisotropic inhomogeneous N-function. 1