Standard Subspaces Research Articles

First-Order System Least Squares (FOSLS) for Convection-Diffusion Problems: Numerical Results

The focus of this paper is on planar linear convection-diffusion problems, to which we apply a special form of first-order system least squares (FOSLS [Cai et al., SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799; Cai, Manteuffel, and McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 425--454]). This we do by introducing the gradient of the primary variable, scaled by certain exponential functions. The convection-diffusion equation is then recast as a minimization principle for a functional corresponding to a sum of weighted L2 norms of the resulting first-order system. Discretization is accomplished by a Rayleigh--Ritz method based on standard finite element subspaces, and the resulting linear systems are solved by basic multigrid algorithms. The main goal here is to obtain optimal discretization accuracy and solver speed that is essentially uniform in the size of convection. Our results show that the FOSLS approach achieves this goal in general when the performance is measured either by the functional or by an equivalent weighted H1 norm. Included in our study is a multilevel adaptive refinement method based on locally uniform composite grids and local error estimates based on the functional itself.

First-Order System Least Squares for Second-Order Partial Differential Equations: Part II

This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in $n=2$ or 3 dimensions as a system of first-order equations. In part I [Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799] a similar functional was developed and shown to be elliptic in the $H(\divv) \times H^1$ norm and to yield optimal convergence for finite element subspaces of $H(\divv) \times H^1$. In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the $(H^1)^{n+1}$ norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of $(H^1)^{n+1}$. Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation-based approaches, the least-squares approach developed here applies directly to convection--diffusion--reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.

Abstract Twisted Duality for Quantum Free Fermi Fields

Using the properties of standard subspaces of the 1-particle space for the free Fermi field we prove the twisted duality for the von Neumann algebras associated with real closed subspaces. This is done by application of the Tomita-Takesaki theory.

Chained Aggregation: A Geometric Analysis

Chained aggregation has been introduced as a procedure for transforming a linear time-invariant system into a special form called the Generalized Hessenberg Representation (GHR). Here, we give a geometric analysis which shows that the GHR exhibits explicitly the system's observability structure. We also explore the details of using state feedback to produce an exact aggregated lower order model. The procedure developed is then given a geometric interpretation which reveals two standard subspaces, V*and R*. The importance of this geometric structure in problems connected with this procedure is described

Chebyshev subspaces of finite codimension in spaces of continuous functions

AbstractA. L. Garkavi in 1967 characterized those compact metric spaces X with the property that the space C(X) of real-valued continuous functions possesses Chebyshev subspaces of fine codimension ≥ 2. Here compact Hausdorif spaces with the same property are characterized in terms of certain standard subspaces of the space [0, 1] × {0, 1} equipped with a lexicographic order topology. Garkavi's result for metric spaces is exhibited as a corollary. The proof depends upon a simplification of a characterization by Garkavi of the Chebyshev subspaces of finite codimension in C(X).Subject classification (Amer. Math. Soc. (MOS) 1970): primary 41 A 65, 46 E 15; secondary 54 G 99.