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.

We study the integral equation related to the three and higher dimensional superradiance problem. Collective radiation phenomena has attracted the attention of many physicists and chemists since the pioneering work of R. H. Dicke in 1954. We first consider the three-dimensional superradiance problem and find a differential operator that commutes with the integral operator related to the problem. We find all the eigenfunctions of the differential operator and obtain a complete set of eigensolutions for the three-dimensional superradiance problem. Generalization of the three-dimensional superradiance integral equation is provided. A commuting differential operator is found for this generalized problem. For the three dimensional superradiance problem, an alternative set of complete eigenfunctions is also provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak. The uniqueness of the differential operator commuting with that kernel is indicated. Finally, a concentration problem for the signals which are bandlimited in disjoint frequency-intervals is considered. The problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted in a given time interval. A numerical algorithm for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L2(−∞,∞) are also proved. Numerical computations are carried out in support of the theory.

The modeling of electromagnetic phenomena described by the Maxwell's equations is of critical importance in many practical applications. The numerical simulation of these equations is challenging and much more involved than initially believed. Consequently, many discretization techniques, most of them quite complicated, have been proposed. In this dissertation, we present and analyze a new methodology for approximation of the time-harmonic Maxwell's equations. It is an extension of the negative-norm least-squares finite element approach which has been applied successfully to a variety of other problems. The main advantages of our method are that it uses simple, piecewise polynomial, finite element spaces, while giving quasi-optimal approximation, even for solutions with low regularity (such as the ones found in practical applications). The numerical solution can be efficiently computed using standard and well-known tools, such as iterative methods and eigensolvers for symmetric and positive definite systems (e.g. PCG and LOBPCG) and reconditioners for second-order problems (e.g. Multigrid). Additionally, approximation of varying polynomial degrees is allowed and spurious eigenmodes are provably avoided. We consider the following problems related to the Maxwell's equations in the frequency domain: the magnetostatic problem, the electrostatic problem, the eigenvalue problem and the full time-harmonic system. For each of these problems, we present a natural (very) weak variational formulation assuming minimal regularity of the solution. In each case, we prove error estimates for the approximation with two different discrete least-squares methods. We also show how to deal with problems posed on domains that are multiply connected or have multiple boundary components. Besides the theoretical analysis of the methods, the dissertation provides various numerical results in two and three dimensions that illustrate and support the theory.

Thesis (Ph. D.)--University of Oregon, 2000. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-128). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 129-131). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 129-131). Also available on the Internet.

Let x be a real reflexive separable locally uniformly convex banach space with locally uniformly convex dual spacex *. Let t:x\supset d(t)\rightarrow 2 {x *} be maximal monotone with 0\in t(0), 0\in intd(t) and c:x\supset d(c)\rightarrow x *. Assume that $l\subset d(c)$ is a dense linear subspace of x, c is of class (s_+)_l and \langle cx,x\rangle\geq-\psi(\lx\l), x\in d(c), where \psi:\mathbb{r} +\rightarrow\mathbb{r} + is nondecreasing. a new topological degree is developed for the sum t+c in chapter one. This theory extends the recent degree theory for the operators c of type (s_+)_{0,l} in [15]. unlike such a recent extension to multivalued (s_+)_{0,l}-type operators, the current approach utilizes the approximate degree d(t_t+c,g,0), t\downarrow 0, where t_t = (T {-1}+tJ {-1}) {-1}and G is an open bounded subset of X and is such that $0\in G$, for the single-valued mapping $T_t+C$. The subdifferential\partial\varphi, for \varphi belonging to a large class of proper c onvex lower semicontinuous functions, gives rise to operators T to which this degree theory applies. Theoretical applications to problems of Nonlinear Analysis are included, as well as applications from the field of partial differential equations. Let T:X\supset D(T)\rightarrow 2 {X *} be maximal monotone with compact resolvents, i.e, the operator $(T+\epsilonJ) {-1}:X *\rightarrow X is compact for every \in 0. We present a relevant result in chapter 2 that says there exists an open ball around zero in the image of a relatively open set by a continuous and bounded perturbation of a maximal monotone operator with compact resolvents. The generalized degree function for compact perturbations of m-accretive operators established by Y. -Z Chen in [7] isextended to the case of a multivalued compact perturbations of maximal monotone maps by appealing to the topological degree forset-valued compact fields in locally convex spaces introduced by Tsoy Wo-Ma in [25]. Such is the content of the thi rd chapter. A unified eigen value theory is developed for the pair(T,S), where T:X\supset D(T)\rightarrow 2 {X *} is aquasimonotone-type operator which belong to the so-called A_G(QM) class introduced by Arto Kittila in [23] and S is abounded demicontinuous mapping of class (S)_+. Conditions are given for the existence of a pair (x,\lambda)\in (0,\infty)\times(D(T+S)\cap\partial G)$ such that Tx+\lambda Sx\ni 0$. This is the content of Chapter 4.

I consider Dirichlet eigenvalues for an elliptic system in a region that consists of two domains joined by a thin tube. Under quite general conditions, I am able to give a rate on the convergence of the eigenvalues as the tube shrinks away. I make no assumption on the smoothness of the coefficients and only mild assumptions on the boundary of the domain. Also, I consider the Green function associated with the mixed problem on a Lipschitz domain with a general decomposition of the boundary. I show that the Green function is Hölder continuous, which shows how a solution to the mixed problem behaves.

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.

Thesis (Ph. D.)--Ohio University, November, 1995. / Title from PDF t.p.