Many problems in the natural world have high contrast properties, like transport in composites, fluid in porous media and so on. These problems have huge numerical difficulties because of the singularities of their solutions. It may be really expensive to solve these problems directly by traditional numerical methods. It is necessary and important to understand these problems more in mathematical aspect first, and then using the mathematical results to simplify the original problems or develop more efficient numerical methods. In this thesis we are going to approximate the Dirichlet to Neumann map for the high contrast two phase composites. The mathematical formulation of our problem is to approximate the energy for an elliptic equation with arbitrary boundary conditions. The boundary conditions may have highly oscillations, which makes our problems very interesting and difficult. We developed a method to divide the domain into two different subdomains, one is close to and the other one is far from the boundary, and we can approximate the energy in these two subdomains separately. In the subdomain far from the boundary, the energy is not influenced that much by the boundary conditions. Methods for approximation of the energy in this subdomain are studied before. In the subdomain near the boundary, the energy depends on the boundary conditions a lot. We used a new method to approximate the energy there such that it works for any kind of boundary conditions. By this way, we can have the approximation for the total energy of high contrast problems with any boundary conditions. In other words, we can have a matrix up to any dimension to approximate the continuous Dirichlet to Neumann map of the high contrast composites. Then we will use this matrix as a preconditioner in domain decomposition methods, such that our numerical methods are very efficient to solve the problems in high contrast composites.

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Made available in DSpace on 2015-05-15T11:46:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 947145 bytes, checksum: 7294f81daf663930a60afca1aecbee7b (MD5) Previous issue date: 2010-07-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we talk about properties of the Dirichlet-to-Neumann map for the conductivity equation in a smooth manifold with boundary of R2. We use several times the Maximum Principle to conclude a Alternating Property of the Dirichlet-to- Neumann map. Using this property, we and that the Kernel satises a given set of inequalities. Finally, we note that these inequalities imply the Alternating Property of the Kernel of the Dirichlet-to-Neumann map. / Neste trabalho dissertamos sobre Propriedades do Funcional de Dirichlet-Neumann para uma equação de condutividade numa variedade diferenciavel bidimensional com bordo. Utilizamos varias vezes o Principio do Maximo para concluir que esse Funcional tem uma Propriedade Alternada. A partir dessa propriedade, verificamos que o Nucleo do Funcional satisfaz um conjunto especifico de desigualdades. Porém, verificamos que essas desigualdades implicam na Propriedade Alternada do Nucleo do Funcional.

This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.

This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.