The estimation of the solution to the matrix equation AX-XB=C is primarily dependent on the quantity sep(A,B) introduced by Stewart. Varah has given some examples to show that $sep_F(A,B)$ can be very small even though the eigenvalues of A and B are well separated. In this paper we give some lower bounds of $sep_F(A,B)$.

In this paper we consider the nonlinear operator equation $\lambda x=Lx+G(\lambda,x)$ where $L$ is a closed linear operator of $X-›X, X$ is a real Banach Space, with a simple eigenvalue $\lambda_0\neq 0$. We discretize its Liapunov-Schmidt bifurcation equation instead of the original nonlinear operator equation and estimate the approximating order of our approximate solution to the genuine solution. Our method is more convenient and more accurate. Meanwhile we put forward several abstract Newton-type iterative schemes, which are more efficient for practical computation, and get the result of their super-linear convergence.

In this paper, we deal with the discrete-discontinuous finite element method for solving the time-dependent neutron transport equation in two-dimensional planar geometry. Its stability and convergence are proved. The numerical results are given. Compared with SN method it is of higher accuracy and superconvergence.

In this paper, an abstract error estimate of mixed finite element methods using nonconforming elements is presented. In addition, a class of nonconforming rectangular elements is proposed, and applied to Stokes equations. The optimal error estimate is given.

For a missing-value problem in linear models, all the iterative procedures employed up to now are entirely within the framework of the EM algorithm. This paper proposes a new iterative procedure which is not the EM algorithm in the most general case. In the procedure nothing is assumed about the error distribution in the proof of its convergence. The convergence rate is also obtained.

In this paper, we introduce a method of numerical integration on a polytope of multiple dimensions. Its basic idea is to cut a polytope into some simplices and sum the integral values on the simplices. We give the integral formulae, the expressions to estimate the error and the method for cutting a polytope into simplices. Some examples are provided to explain the adaptability of the method.

In this paper we suggest a second-order fluid in cell (FLIC) method for the one-dimensional unsteady compressible flow problems. The numerical result obtained by the present method is compared with the one obtained with the original FLIC method and the exact solution for a shock tube problem.

The finite difference scheme is constructed for a nonlinear elliptic equation; its convergence is proved.

It is proved that Zienkiewicz's triangles for plate bending problems pass Stummel's generalized patch test — a necessary and sufficient condition for convergence of nonconforming finite elements — for mesh (a) generated by three sets of parallel lines, but do not pass it when "union jack" mesh (b) or when another mesh (c) is used. In the latter two cases the approximations are divergent.