We study approximation techniques for incompressible flows with heterogeneous properties. Speci cally, we study two types of phenomena. The first is the flow of a viscous incompressible fluid through a rigid porous medium, where the permeability of the medium depends on the pressure. The second is the ow of a viscous incompressible fluid with variable density. The heterogeneity is the permeability and the density, respectively. For the first problem, we propose a finite element discretization and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence is exponential, we propose a splitting scheme which involves solving only two linear systems. For the second problem, we introduce a fractional time-stepping scheme which, as opposed to other existing techniques, requires only the solution of a Poisson equation for the determination of the pressure. This simpli cation greatly reduces the computational cost. We prove the stability of first and second order schemes, and provide error estimates for first order schemes. For all the introduced discretization schemes we present numerical experiments, which illustrate their performance on model problems, as well as on realistic ones.

Fluorescence enhanced optical tomography is a promising molecular imaging modality which employs a near infrared fluorescent molecule as an imaging agent and time-dependent measurements of fluorescent light propagation and generation. In this dissertation a novel fluorescence tomography algorithm is proposed to reconstruct images of targets contrasted by fluorescence within the tissues from boundary fluorescence emission measurements. An adaptive finite element based reconstruction algorithm for high resolution, fluorescence tomography was developed and validated with non-contact, planewave frequency-domain fluorescence measurements on a tissue phantom. The image reconstruction problem was posed as an optimization problem in which the fluorescence optical property map which minimized the difference between the experimentally observed boundary fluorescence and that predicted from the diffusion model was sought. A regularized Gauss-Newton algorithm was derived and dual adaptive meshes were employed for solution of coupled photon diffusion equations and for updating the fluorescence optical property map in the tissue phantom. The algorithm was developed in a continuous function space setting in a mesh independent manner. This allowed the meshes to adapt during the tomography process to yield high resolution images of fluorescent targets and to accurately simulate the light propagation in tissue phantoms from area-illumination. Frequency-domain fluorescence data collected at the illumination surface was used for reconstructing the fluorescence yield distribution in a 512 cm3, tissue phantom filled with 1% Liposyn solution. Fluorescent targets containing 1 micro-molar Indocyanine Green solution in 1% Liposyn and were suspended at the depths of up to 2cm from the illumination surface. Fluorescence measurements at the illumination surface were acquired by a gain-modulated image intensified CCD camera system outfitted with holographic band rejection and optical band pass filters. Excitation light at the phantom surface source was quantified by utilizing cross polarizers. Rayleigh resolution studies to determine the minimum detectable sepatation of two embedded fluorescent targets was attempted and in the absence of measurement noise, resolution down to the transport limit of 1mm was attained. The results of this work demonstrate the feasibility of high-resolution, molecular tomography in clinic with rapid non-contact area measurements.

The finite element method since its development in the 1950’s has been used extensively in solving complex problems involving partial differential equations. The conventional finite element methods use piecewise Lagrange interpolation functions for approximating displacements. The aim of this research is to explore finite element analysis using B-spline interpolation. B-splines are piecewise defined polynomial curves which provide higher continuity of derivatives than piecewise Lagrange interpolation functions. This work focuses on the implementation and comparison of the B-spline finite elements in contrast with the conventional finite elements. This thesis observes that the use of B-spline interpolation functions can reduce the computational cost significantly. It is an efficient technique and can be conveniently implemented into the existing finite element programs.

Thesis (Ph.D.), Memorial University of Newfoundland, 2000. / Bibliography: leaves R-1-R-17. Also available online.

Thesis (M.A.)--University of Hawaii at Manoa, 2005. / Includes bibliographical references (leaves 146-156).

Thesis (M.A.)--University of Hawaii at Manoa, 2005. / Includes bibliographical references (leaves 146-156). Also available via World Wide Web.

Thesis (M.S.)--Rutgers University, 2009. / "Graduate Program in Chemistry." Includes bibliographical references (p.54-56).

Thesis (Ph.D.)--Memorial University of Newfoundland, 2002. / Bibliography: leaves R1-R38. Also available online.

Thesis (M. Phil.)--University of Hong Kong, 1980.

Thesis (M. Phil.)--University of Hong Kong, 1981.