A completely exponentially fitted difference scheme is considered for the singular perturbation problem: $\epsilon U^{''}+a(x) U^{'}-b(x) U=f(x) \ {\rm for}  \ 0 \lt x \lt 1$, with U(0), and U(1) given, $\epsilon \in (0,1]$ and $a(x) \gt α \gt 0, b(x)\geq 0$. It is proven that the scheme is uniformly second-order accurate.

This paper presents a method of finding a strictly feasible solution for linear constraints. We prove, under certain assumption, that the method is convergent in a finite number of iterations, and give the sufficient and necessary conditions for the infeasibility of the problem. Actually, it can be considered as a constructive proof for the Farkas lemma.

The Drazin inverse of a lower Hessenberg matrix is considered. If $A$ is a singular lower Hessenberg matrix and $a_{i,i+1}=\neq 0,i=1,2,\cdots,n-1$, then $A^D$ can be given, and expressed explicitly by elements of $A$. The structure of the Drazin inverse of a lower Hessenberg matrix is also studied.

In this paper we present a posteriori error estimator in a suitable norm of mixed finite element solution for two-dimensional stationary Stokes problem. The estimator is optimal in the sense that, up to multiplicative constant, the upper and lower bounds of the error are the same. The constants are independent of the mesh and the true solution of the problem.

Very recently, there is a growing interest in studying parallel and distributed stochastic approximation algorithms. Previously, we suggest such an algorithm to find zeros or locate maximum values of a regression function with large state space dimension in [1], and derived the strong consistency property for that algorithm. In the present work, we concern ourselves with the problem of asymptotic properties of such an algorithm. We will study the limit behavior of the algorithm and obtain the rate of convergence and asymptotic normality results.

In order to obtain the numerical solution for a one-dimensional parabolic system, an unconditionally stable difference method is investigated in [1]. If the number of unknown functions is M, for each time step only M times of calculation are needed. The rate of convergence is $O(\tau+h^2)$. On the basis of [1], an alternating calculation difference scheme is presented in [2]; the rate of the convergence is $O(\tau^2+h^2)$. The difference schemes in [1] and [2] are economic ones. For the $\alpha$-$th$ equation, only $U_{\alpha}$ is an unknown function; the others $U_{\beta}$ are given evaluated either in the last step or in the present step. So the practical calculation is quite convenient.
The purpose of this paper is to derive a trilayer difference scheme for one-dimensional parabolic systems. It is known that the scheme is also unconditionally stable and the rate of convergence is $O(\tau^2+h^2)$.

For the problems of the left and right matrix Padé approximations, we give the necessary and sufficient conditions for the existence of their solutions. If the left Padé approximant exists, then we prove that its uniqueness is equivalent to the existence of right Padé approximants, and we further give the exact results about the dimension of the linear space $^LR^{*}(m,n)$ formed from the left Padé approximants.  

The paper describes an interpolation procedure of formulating shape functions for a new energy-orthogonal plate element. Sample problems using the new element show satisfactory numerical results.  

We present a semi-coarsening procedure, i.e., coarsening in one space direction, to improve the convergence rate of the multigrid solver presented in [5] for solving the 2D steady Navier-Stokes in primitive variables when the aspect ratio of grid cells is not equal to 1, i.e., when $h_x/h_y \gg 1 \ {\rm or} \ll 1$, where $h_x$ is the grid step in $x$ direction and $h_y$ is the grid step in $y$ direction, $x$ and $y$ represent the Cartesian coordinates.