Hyung-Chun Lee

Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions

Abstract. We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.

Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems

\noindent A reduced-order modeling methodology based on centroidal Voronoi tessellations (CVTs) is introduced. CVTs are special Voronoi tessellations for which the generators of the Voronoi diagram are also the centers of mass (means) of the corresponding Voronoi cells. For discrete data sets, CVTs are closely related to the h-means and k-means clustering techniques. A discussion of reduced-order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced-order bases. Then, detailed descriptions of CVT-based reduced-order bases and how they can be constructed from snapshot sets and how they can be applied to the low-cost simulation of complex systems are given. Subsequently, some concrete incompressible flow examples are used to illustrate the construction and use of CVT-based reduced-order bases. The CVT-based reduced-order modeling methodology is shown to be effective for these examples.

Error Estimates of Stochastic Optimal Neumann Boundary Control Problems

We study mathematically and computationally optimal control problems for stochastic partial differential equations with Neumann boundary conditions. The control objective is to minimize the expectation of a cost functional, and the control is of the deterministic, boundary-value type. Mathematically, we prove the existence of an optimal solution and of a Lagrange multiplier; we represent the input data in terms of their Karhunen-Loeve expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces.

Analysis of Neumann Boundary Optimal Control Problems for the Stationary Boussinesq Equations Including Solid Media

This article deals with Neumann boundary optimal control problems associated with the Boussinesq equations including solid media. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived.

Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions

We examine analytical and numerical aspects of optimal control problems for first-order elliptic systems in three dimensions. The particular setting we use is that of divcurl systems. After formulating some optimization problems, we prove the existence and uniqueness of the optimal solution. We then demonstrate the existence of Lagrange multipliers and derive an optimality system of partial differential equations from which optimal controls and states may be deduced. We then define least-squares finite element approximations of the solution of the optimality system and derive optimal estimates for the error in these approximations.

A Stochastic Galerkin Method for Stochastic Control Problems

In an interdisciplinary field on mathematics and physics, we examine a physical problem, fluid flow in porous media, which is represented by a stochastic partial differential equation (SPDE). We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature. We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem. Finally, we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates. AMS subject classifications: 65M55, 65N30

Pseudospectral Least-Squares Method for the Second-Order Elliptic Boundary Value Problem

The least-squares Legendre and Chebyshev pseudospectral methods are presented for a first-order system equivalent to a second-order elliptic partial differential equation. Continuous and discrete homogeneous least-squares functionals using Legendre and Chebyshev weights are shown to be equivalent to the H1(\Omega)

Analysis of Optimal Control Problems for the Two-Dimensional Thermistor System

norm and Chebyshev-weighted Div-Curl norm over appropriate polynomial spaces, respectively. The spectral error estimates are derived. The block diagonal finite element preconditioner is developed for the both cases. Several numerical tests are demonstrated on the spectral discretization errors and on performances of the finite element preconditioner.

FINITE ELEMENT APPROXIMATION AND COMPUTATIONS OF OPTIMAL DIRICHLET BOUNDARY CONTROL PROBLEMS FOR THE BOUSSINESQ EQUATIONS

An optimal control problem for the thermistor system is considered. First, the precise mathematical problem is established and the proof of existence of the optimal solution is given with appropriate function spaces. Then, Gâteaux differentiability is shown for the thermistor system, with respect to control, and the optimality system is obtained.

Analysis and Computational Methods of Dirichlet Boundary Optimal Control Problems for 2D Boussinesq Equations

Mathematical formulation and numerical solutions of an optimal Dirichlet boundary control problem for the Boussinesq equations are considered. The solution of the optimal control prob- lem is obtained by adjusting of the temperature on the boundary. We analyze flnite element approximations. A gradient method for the solution of the discrete optimal control problem is presented and analyzed. Finally, the results of some computational experiments are presented.