We consider algorithms for preconditioning of two discontinuous Galerkin (DG) methods for second order elliptic problems, namely the symmetric interior penalty (SIPG) method and the method of Baumann and Oden. For the SIPG method we first consider two-level preconditioners using coarse spaces of either continuous piecewise polynomial functions or piecewise constant (discontinuous) functions. We show that both choices give rise to uniform, with respect to the mesh size, preconditioners. We also consider multilevel preconditioners based on the same two types of coarse spaces. In the case when continuous coarse spaces are used, we prove that a variable V-cycle multigrid algorithm is a uniform preconditioner. We present numerical experiments illustrating the behavior of the considered preconditioners when applied to various test problems in three spatial dimensions. The numerical results confirm our theoretical results and in the cases not covered by the theory show the efficiency of the proposed algorithms. Another approach for preconditioning the SIPG method that we consider is an algebraic multigrid algorithm using coarsening based on element agglomeration which is suitable for unstructured meshes. We also consider an improved version of the algorithm using a smoothed aggregation technique. We present numerical experiments using the proposed algorithms which show their efficiency as uniform preconditioners. For the method of Baumann and Oden we construct a preconditioner based on an orthogonal splitting of the discrete space into piecewise constant functions and functions with zero average over each element. We show that the preconditioner is uniformly spectrally equivalent to an appropriate symmetrization of the discrete equations when quadratic or higher order finite elements are used. In the case of linear elements we give a characterization of the kernel of the discrete system and present numerical evidence that the method has optimal convergence rates in both L2 and H1 norms. We present numerical experiments which show that the convergence of the proposed preconditioning technique is independent of the mesh size.

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1990.

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1991.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Not available / text

This thesis begins with a review on the topic of molecular electronics. The purpose of this review is to motivate the need for good theory to understand and predict molecular electronics behaviour. At present the most promising theoretical formalism for dealing with this problem is a combination of density functional theory and nonequilibrium Green's functions (NEGF-DFT). This formalism is especially attractive because it is an ab-initio technique, meaning that it is completely from first principles and does not require any empirical parameters. An implementation of this formalism has been developed by the research group of Hong Guo and is presented and explained here. A few other implementations which are similar but differ in some ways are also discussed briefly to highlight their various advantages and disadvantages. / One of the difficulties of ab-initio calculations is that they can be extremely costly in terms of the computing time and memory that they require. For this reason, in addition to using appropriate approximations, sophisticated numerical analysis tech niques need to be used. One of the bottlenecks in the NEGF-DFT method is solving the Poisson equation on a large real space grid. For studying systems incorporating a gate voltage it is required to be able to solve this problem with nonperiodic boundary conditions. In order to do this a technique called multigrid is used. This thesis examines the multigrid technique and develops an efficient implementation for the purpose of use in the NEGF-DFT formalism. For large systems, where it is necessary to use especially large real space grids, it is desirable to run simulations on parallel computing clusters to handle the memory requirements and make the code run faster. For this reason a parallel implementation of multigrid is developed and tested for performance. The multigrid tool is incorporated into the NEGF-DFT formalism and tested to ensure that it is properly implemented. A few calculations are made on a benzenedithiol system with gold leads to show the effect of an applied gate voltage.

Thesis (M.S.)--Brigham Young University. Dept. of Mechanical Engineering, 2006. / Includes bibliographical references (p. 81-84).

Various models are currently in existence for determining the deflection of lithospheric plates under an applied transverse load. The most popular models treat lithospheric plates as thin elastic or thin viscoelastic plates. The equations governing the deflection of such plates have been solved successfully in two dimensions using integral transform techniques. Three dimensional models have been solved using Fourier Series expansions assuming a sinusoidal variation for the load and deflection. In the engineering context, the finite element technique has also been employed. The current aim, however, is to develop an efficient solver for the three dimensional elastic and viscoelastic problems using finite difference techniques. A variety of loading functions may therefore be considered with minimum work involved in obtaining a solution for different forcing functions once the main program has been developed. The proposed method would therefore provide a valuable technique for assessing new models for the loading of lithospheric plates as well as a useful educational tool for use in geophysics laboratories. The multigrid method, which has proved to be a fast, efficient solver for elliptic partial differential equations, is examined as the basis for a solver of both the elastic and viscoelastic problems. The viscoelastic problem, being explicitly time-dependent, is the more challenging of the two and will receive particular attention. Multigrid proves to be a very effective method applicable to the solution of both the elastic and viscoelastic problems. / Science, Faculty of / Mathematics, Department of / Graduate