A class of k-block implicit hybrid methods for solving the initial value problem for ordinary differential equations are studied, which take a block of k new values at each step. These methods are examined for the property of A-stability. It is shown that the method of order 2k+2 exists uniquely, and these methods are A-stable for block sizes k=1,2,3,4,5.

In this paper, the concept of optimally scaled matrix and the estimate of $\|M^{-1}N\|_{\infty}$ in our previous paper are used to find the upper bounds of the spectral radii of the iterative matrices SOR, SSOR, AOR and SAOR. The sharpness of the upper bounds of the spectral radii of SOR and AOR is established. The proofs are very intuitive and may be considered as the geometrical interpretations of our theorems.

In this paper we give some sufficient conditions for the convergence of the AOR method, introduced by Hadjidimos [5], which include the ones from [1], [2], [5], [6], [7], [9], [10], [11], [12] and which show that the necessary condition given in [8] for the convergence of the AOR method is not valid. We give general conditions for the class of H-matrices, but they are not always easy to check in practice. Consequently, we give some more practical conditions concerning some subclasses of H-matrices.

By means of the theory of spline interpolation in Hilbert spaces, the bivariate polynomial natural spline interpolation to scattered data is constructed. The method can easily be carried out on a computer, and parallelly generalized to high dimensional cases as well. The results can be used for numerical integration in higher dimensions and numerical solution of partial differential equations, and so on.

A nonconforming finite element method of streamline diffusion type for solving the stationary and incompressible Navier-Stokes equation is considered. Velocity field and pressure field are approximated by piecewise linear and piecewise constant functions, respectively. The existence of solutions of the discrete problem and the strong convergence of a subsequence of discrete solutions are established. Error estimates are presented for the uniqueness case.

We study the numerical solution of high-order Volterra integro-differential equations by means of collocation techniques in certain polynomial spline spaces. The attainable order of global convergence and local superconvergence of these methods is analyzed.