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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. Mathematics Subject Classification (2000). 35Q30, 35D05, 76D05.

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. Introduction.Even with the use of good mesh generators, discretization schemes, and solution algorithms, the computational simulation of complex, turbulent, or chaotic systems still remains a formidable endeavor. For example, typical finite element codes may require many thousands of degrees of freedom for the accurate simulation of fluid flows. The situation is even worse for optimization problems for which multiple solutions of the complex state system are usually required or in feedback control problems for which real-time solutions of the complex state system are needed. The need for so many degrees of freedom results from two causes. First, phenomena occurring over very small spatial and temporal scales have to be resolved. For example, flows at even moderate values of the Reynolds number are usually turbulent. In a direct computational simulation, very small eddies have to be accurately resolved, even if one is only interested in the large-scale features of the flow. Second, the functions or grid stencils used to effect approximation have no direct relation to the solution of the complex state system. For example, the standard, locally supported, piecewise polynomial basis functions used to define a finite element approximation of the flow solution are not specially designed for a particular flow or for flows in general. Thus, in a typical finite element calculation for a turbulent flow, many thousands of degrees of freedom are needed in order to properly resolve the small eddies.The natural question to ask is whether or not it is really necessary to have thousands of degrees of freedom in order to produce accurate approximations to solutions of complex systems. For example, for turbulent flows, one would like to identify highly persistent spatio-temporal structures using relatively inexpensive, low-dimensional approaches instead of using expensive standard techniques in the numerical solution

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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 H 1 (Ω) 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.