Comparison theorems for weak splittings of bounded operators

Summary Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Wonicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.This work was supported by the National Science Foundation grants DMS-8807338 and INT-8918502     

On a problem proposed by H. Braß concerning the remainder term in quadrature for convex functions

Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n ^–2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw _, (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw _, is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday     

Parallel and superfast algorithms for Hankel systems of equations

Summary Utilizing kernel structure properties a unified construction of Hankel matrix inversion algorithms is presented. Three types of algorithms are obtained: 1)O(n ²) complexity Levinson type, 2)O (n) parallel complexity Schur-type, and 3)O(n log² n) complexity asymptotically fast ones. All algorithms work without additional assumption (like strong nonsingularity).     

Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation

Summary The good Boussinesq equationu _tt =–u _xxxx +u _xx +(u ²) _xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the good Boussinesq equation. Numerical experimentas are also reported.     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

Gaussian collocation via defect correction

Summary For the numerical integration of boundary value problems for first order ordinary differential systems, collocation on Gaussian points is known to provide a powerful method. In this paper we introduce a defect correction method for the iterative solution of such high order collocation equations. The method uses the trapezoidal scheme as the basic discretization and an adapted form of the collocation equations for defect evaluation. The error analysis is based on estimates of the contractive power of the defect correction iteration. It is shown that the iteration producesO(h ²), convergence rates for smooth starting vectors. A new result is that the iteration damps all kind of errors, so that it can also handle non-smooth starting vectors successfully.     

The Weierstrass mean

Theorem.Let the sequences {e _i ⁽ⁿ⁾ },i=1, 2, 3,n=0, 1, 2, ...be defined by where the e ⁽⁰⁾ s satisfy and where all square roots are taken positive. Then where the convergence is quadratic and monotone and where The discussions of convergence are entirely elementary. However, although the determination of the limits can be made in an elementary way, an acquaintance with elliptic objects is desirable for real understanding.     

Tracking poles and representing Hankel operators directly from data

Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL ² fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL ² which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH , analytic, approximation toG relative to theL norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

Spectral methods with sparse matrices

Summary Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N ⁴) for polynomials of degree N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N ²). Certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.