Ph.D. Recipients and their Thesis Abstracts

PDE / Numerical Analysis

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Estimates for and Properties of Mixed Finite Element Methods for Elliptic Problems

Abstract: We consider two mixed finite element methods for a general second-order linear elliptic problem on a domain $\Omega\subset R^n$. These methods originate from two different forms of the same partial differential equation, and they sometimes behave in substantially different ways. The choice of element space used in these methods also has an effect on their behavior. In this thesis we consider the effects of different choices of elements and methods on the optimality and localization properties of the resulting approximations.

In the "divergence'' form method, the vector variable in the mixed method approximates $– A \nabla u$, while in the "conservation'' form method, the vector variable approximates $–(A \nabla u – \vec{b} u)$. Here u solves the given elliptic scalar problem, A is a matrix of coeffients, and $\vec{b}$ is a vector of coefficients. We demonstrate via general L_2 error estimates that, in the divergence form method, the errors in the vector and scalar variables are weakly coupled, while in the conservation form method they are strongly coupled. The strong coupling in the latter case leads to suboptimal convergence when members of the Brezzi-Douglas-Marini (BDM) family of elements for simplicial spaces are used, a fact which we demonstrate computationally. The well-known Raviart-Thomas-Nedelec family of spaces, in contrast, always give optimal convergence.

Using pointwise error estimates which generalize previously known almost-best-approximation maximum-norm estimates, Schatz has recently shown that standard finite element methods for elliptic problems yield errors which are in a certain sense mostly local in character, except in the lowest-order piecewise linear case. We carry out a similar pointwise error analysis for the mixed methods described above. Our estimates indicate that localization occurs in both of these mixed methods except when the lowest order BDM simplicial elements are used, and we again confirm the sharpness of our theoretical results via computational examples.

Asymptotic Behavior of Solutions of One Dimensional Parabolic SPDE

Abstract: We investigate the weak convergence as time t\to\infty of the solutions of SPDE in one dimension with Dirichlet boundary conditions. We first prove a result for the linear case, which is easy since the solution is Gaussian. Then we get a result for nonlinear drift case by proving a new comparison theorem for SPDE. To investigate an elliptic SPDE with Dirichlet boundary condition, we set up an equivalence between an elliptic SPDE and a system of SODE. From there, we construct a counter example that shows non-uniqueness of the solution for SPDE, and look at how the Lipschitz constant of the drift term affects the uniqueness.

Superconvergence Points in Locally Uniform Finite Element Meshes for Second Order Two-Point Boundary Value Problems

Abstract: A superconvergence point for a finite element approximation is a family of points \xi(h) such that the pointwise error at the point has a higher degree of accuracy than the global error. Such points can be sought for either function values or derivatives; we shall restrict ourselves to function values and first derivatives. Recent results have shown that such points can be found at points where the mesh is locally symmetric (i.e. symmetric in a Ch\ln 1/h neighborhood of the point); in particular, if the mesh is locally uniform, superconvergence points occur at the meshpoints and midpoints of intervals.

We find the location of all interior superconvergence points for function values and derivatives occurring in the finite element discretization of second order two point boundary value problems. We further note that a complete understanding of this problem leads to a knowledge of the location of superconvergence points in certain second order problems on multidimensional tensor product spaces and in the L_2 projection.

The problem can be reduced to one of finding the roots of even or odd polynomials on an interval satisfying certain orthogonality conditions and having certain roots at the endpoints. The resulting pattern of superconvergence points is found to have several remarkable features.

Essays in Mathematical Economics and Economic Theory

Abstract: This dissertation consists of two parts, one on the effect of open market operations and one on the existence of demand function in L^1 space, respectively. The first part of the thesis (Chapters 1–6) is to study how prices respond to open-market operations based on a transactions-based model of money demand with the use of homothetic utility function. It is shown that a monetary injection will lead to a gradual rise in prices. Of special interest is the determination of a stabilizing monetary injection policy corresponding to which the equilibrium prices are most stable.

The equilibrium price in this model is governed by a second order, nonlinear difference equation which can be reduced to a functional equation of the form f(f\circ f(x), f(x), x)= 0. The stable manifold theorem from dynamical systems theory is employed to establish the existence of f by relating it to the graph of a stable manifold of an auxiliary two-dimensional functional. Furthermore the contraction mapping theorem is established to prove the existence and design numerical algorithms for equilibrium prices based on some properly constructed nonlinear functionals. With some efficient numerical algorithms proposed and analyzed in this theses, the equilibria are computed very accurately and the behavior of the equilibria can be clearly observed through computer graphics.

The second part of the thesis (Chapter 7) is concerned with an optimization problem motivated by a generalized Cass-Shell sunspots model that allows for an infinite state of nature. The space for the demand function is L^1 which lacks the useful properties such as the reflexivity and weak compactness for closed bounded sets that are used in the conventional optimization theories. A new theory is developed to deal with these kinds of optimization problems. The proof of the main theorem depends on various results from real analysis and functional analysis, and particularly a new theorem from modern probability theory.