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     

低场脉冲核磁共振测定二醋酸纤维素丝束中油剂

研究了采用低场脉冲核磁共振(NMR)测定二醋酸纤维素丝束油剂的新方法，给出NMR法测定二醋酸纤维素丝束油荆含量的标准曲线。同时对NMR法测量二醋酸纤维素丝束油荆不确定度进行了评估。结果表明，NMR法具有分析速度快、无污染、无损伤和准确度高的优点，适合于二醋酸纤维素丝束油荆的分析。     

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.     

On the asymptotic stability ofθ-methods for delay differential equations

Stability regions of -methods for the linear delay differential test equations

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

Comparison theorems forL-monosplines of minimal norm

Summary LetLM _N be the set of allL-monosplines withN free knots, prescribed by a pair (x;E) of pointsx = {x _i } ₁ ⁿ ,a <x ₁ < ... <x _n <b and an incidence matrixE = (e _ij ) _i=1 ⁿ , _r-1 ^j=0 with Denote byLM _N ^O the subset ofLM _N consisting of theL-monosplines withN simple knots (n=N). We prove that theL-monosplines of minimalL _p-norms inLM _N belong toLM _N ^O .The results are reformulated as comparison theorems for quadrature formulae.