This paper is intended to present wavelet Galerkin schemes for the boundary element method. Wavelet Galerkin schemes employ appropriate wavelet bases for the discretization of boundary integral operators. This yields quasisparse system matrices which can be compressed to O(N_J) relevant matrix entries without compromising the accuracy of the underlying Galerkin scheme. Herein, O(N_J) denotes the number of unknowns. The assembly of the compressed system matrix can be performed in O(N_J) operations. Therefore, we arrive at an algorithm which solves boundary integral equations within optimal complexity. By numerical experiments we provide results which corroborate the theory.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 214-220). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to the Department of Building and Construction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [170]-184)

Coastal regions are the most densely populated regions of the world. The populations of these regions continue to grow which has created a high demand for water that stresses existing water resources. Coastal aquifers provide a source of water for coastal populations and are generally part of a larger system where freshwater aquifers are hydraulically connected with a saline surface-water body. They are characterized by salinity variations in space and time, sharp freshwater/saltwater interfaces which can lead to dramatic density differences, and complex groundwater chemistry. Mismanagement of coastal aquifers can lead to saltwater intrusion, the displacement of fresh water by saline water in the freshwater regions of the aquifers, making them unusable as a freshwater source. Saltwater intrusion is of significant interest to water resource managers and efficient simulators are needed to assist them. Numerical simulation of saltwater intrusion requires solving a system of flow and transport equations coupled through a density equation of state. The scale of the problem domain, irregular geometry and heterogeneity can require significant computational resources. Also, modeling sharp transition zones and accurate flow velocities pose numerical challenges. Discontinuous Galerkin (DG) finite element methods (FEM) have been shown to be well suited for modeling flow and transport in porous media but a fully coupled DG formulation has not been applied to the variable density flow and transport model. DG methods have many desirable characteristics in the areas of numerical stability, mesh and polynomial approximation adaptivity and the use of non-conforming meshes. These properties are especially desirable when working with complex geometries over large scales and when coupling multi-physics models (e.g. surface water and groundwater flow models). In this dissertation, we investigate a new combined local discontinuous Galerkin (LDG) and non-symmetric, interior penalty Galerkin (NIPG) formulation for the non-linear coupled flow and solute transport equations that model saltwater intrusion. Our main goal is the formulation and numerical implementation of a robust, efficient, tightly-coupled combined LDG/NIPG formulation within the Department of Defense (DoD) Proteus Computational Mechanics Toolkit modeling framework. We conduct an extensive and systematic code and model verification (using established benchmark problems and proven convergence rates) and model validation (using experimental data) to verify accomplishment of this goal. Lastly, we analyze the accuracy and conservation properties of the numerical model. More specifically, we derive an a priori error estimate for the coupled system and conduct a flow/transport model compatibility analysis to prove conservation properties. / text

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Numerical methods have been used to predict the eigenperiods and eigenfunctions of the Earth’s Slichter modes, known as the Slichter triplets. In order to test the validity of our method, we have also computed the frequencies and displacement eigenfunctions of some of the inertial modes of the Earth’s fluid core. We use a Galerkin method to integrate the Three Potential Description (3PD) for a neutrally, stratified and rotating fluid core of a modified Preliminary Reference Earth Model (PREM). Moreover, the same mathematical tool is used for the computation of the frequencies and displacement amplitudes of the Slichter modes. In the Galerkin formulation of the 3PD, using the divergence theorem, we make use of the natural character of the boundary conditions to reduce the order of derivatives from second to first. To compute the frequencies of the Slichter modes, we solve simultaneously the equations of the inner core motion and the dynamics of the fluid core as described above. The results are compared to those in previous studies and it is shown that in the case of the inertial modes they agree well, which proves the validity of the approach. For the Slichter modes, however, it is shown that the results are significantly different from previous work for a similar Earth model. We have also plotted the displacement eigenfunctions for the motion of the fluid in the fluid core during the Slichter oscillations. It is shown that the pattern of motion is consistent with the motion of the inner core, which serves as a second test of the validity of our results. / x, 105 leaves : ill. ; 29 cm

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In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Pade approximation of exp(-z) when the p-th degree basis functions are used. Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids. Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cutting an embedded geometry out of the Cartesian grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.

Heidelberg, Univ., Diplomarbeit, 2008.