by Li Jingzhi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 56-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Parameter identification problems --- p.1 / Chapter 1.2 --- Overview of existing numerical methods --- p.2 / Chapter 1.3 --- Outline of the thesis --- p.4 / Chapter 2 --- General Framework --- p.6 / Chapter 2.1 --- Abstract inverse problem --- p.6 / Chapter 2.2 --- Abstract multilevel models --- p.7 / Chapter 2.3 --- Abstract MMC algorithm --- p.9 / Chapter 3 --- Dual Viewpoint and Convergence Condition --- p.15 / Chapter 3.1 --- Dual viewpoint of nonlinear multigrid method --- p.15 / Chapter 3.2 --- Convergence condition of MMC algorithm --- p.16 / Chapter 4 --- Applications of MMC Algorithm for Parameter Identification in Elliptic and Parabolic Systems --- p.20 / Chapter 4.1 --- Notations --- p.20 / Chapter 4.2 --- Parameter identification in elliptic systems I --- p.21 / Chapter 4.3 --- Parameter identification in elliptic systems II --- p.23 / Chapter 4.4 --- Parameter identification in parabolic systems I --- p.24 / Chapter 4.5 --- Parameter identification in parabolic systems II --- p.25 / Chapter 5 --- Numerical Experiments --- p.27 / Chapter 5.1 --- Test problems --- p.27 / Chapter 5.2 --- Smoothing property of gradient methods --- p.28 / Chapter 5.3 --- Numerical examples --- p.29 / Chapter 6 --- Conclusion Remarks --- p.55 / Bibliography --- p.56

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Motivated by the reservoir engineering concept of the productivity index of a producing oil well in an isolated reservoir, we analyze a time dependent functional, diffusive capacity, on the solutions to initial boundary value problems for a parabolic equation. Sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions. The dependence of the constant diffusive capacity on the type of the boundary condition (Dirichlet, Neumann or third-type boundary condition) is investigated using a known variational principle and confirmed numerically for various geometrical settings. An important comparison between two principal constant values of a diffusive capacity is made, leading to the establishment of criteria when the so-called pseudo-steady-state and boundary-dominated productivity indices of a well significantly differ from each other. The third type boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well. The questions of stabilization and uniqueness of the time independent values of the diffusive capacity are addressed. The derived formulas are used in numerical study of evaluating the productivity index of a well in a general three-dimensional reservoir for a variety of well configurations.

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<p>We prove maximum norm regularity properties of L-stable finite difference</p><p>methods for linear-second order parabolic equations with coefficients</p><p>independent of time, valid for large time steps. These results are almost</p><p>sharp; the regularity property for first differences of the numerical solution</p><p>is of the same form as that of the continuous problem, and the regularity</p><p>property for second differences is the same as the continuous problem except for</p><p>logarithmic factors. </p><p>This generalizes a result proved by Beale valid for the constant-coefficient</p><p>diffusion equation, and is in the spirit of work by Aronson, Widlund and</p><p>Thome&eacute.</p><p>To prove maximum norm regularity properties for the homogeneous problem, </p><p> we introduce a semi-discrete problem (discrete in space, continuous in time).</p><p>We estimate the semi-discrete evolution operator and its spatial differences on</p><p>a sector of the complex plan by constructing a fundamental solution.</p><p>The semi-discrete fundamental solution is obtained from the fundamental solution to the frozen coefficient problem by adding a correction term found through an iterative process.</p><p>From the bounds obtained on the evolution operator and its spatial differences,</p><p>we find bounds</p><p>on the resolvent of the discrete elliptic operator and its differences through</p><p>the Laplace transform</p><p>representation of the resolvent. Using the resolvent estimates and the</p><p>assumed stability properties of the time-stepping method in the Cauchy integral</p><p>representation of the fully discrete solution operator</p><p>yields the homogeneous regularity result.</p><p>Maximum norm regularity results for the inhomogeneous</p><p>problem follow from the homogeneous results using Duhamel's principle. The results for the inhomogeneous</p><p>problem</p><p>imply that when the time step is taken proportional to the grid width, the rate of convergence of the numerical solution and its first</p><p>differences is second-order in space, and the rate of convergence for second</p><p>differences</p><p>is second-order except for logarithmic factors .</p><p>As an application of the theory, we prove almost sharp maximum norm resolvent estimates for divergence</p><p>form elliptic operators on spatially periodic grid functions. Such operators are invertible, with inverses and their first differences bounded in maximum norm, uniformly in the grid width. Second differences of the inverse operator are bounded except for logarithmic factors.</p> / Dissertation

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 93-99).

We study three different problems in non linear PDE. The first problem relates to finding equilibrium configurations in periodic media, more precisely, given an Area-Dirichlet functional J, which is periodic under integer translations and given three planes in R[superscript d], we proof there exists at least one minimizer such that it’s positive part, negative part and zero set remain at a uniform bounded distance of each plane. The second and third problem are related to non local diffusion, in the elliptic non symmetric case and parabolic case. In both cases we are interested in proving interior regularity for solutions of the aforementioned equations. / text

Thesis (DTech. degree in the Dept. of Mathematics and Statistics.)-Tshwane University of Technology, 2008.

Motivated by the reservoir engineering concept of the productivity index of a producing oil well in an isolated reservoir, we analyze a time dependent functional, diffusive capacity, on the solutions to initial boundary value problems for a parabolic equation. Sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions. The dependence of the constant diffusive capacity on the type of the boundary condition (Dirichlet, Neumann or third-type boundary condition) is investigated using a known variational principle and confirmed numerically for various geometrical settings. An important comparison between two principal constant values of a diffusive capacity is made, leading to the establishment of criteria when the so-called pseudo-steady-state and boundary-dominated productivity indices of a well significantly differ from each other. The third type boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well. The questions of stabilization and uniqueness of the time independent values of the diffusive capacity are addressed. The derived formulas are used in numerical study of evaluating the productivity index of a well in a general three-dimensional reservoir for a variety of well configurations.