It is known that selection of a well-conditioned set of vectors from given subspaces is the key step for solving the robust pole assignment problem. In this paper we suggest two numerical methods for selecting such set of vectors. The numerical methods, Method (I) and Method (II), are described, and some numerical results are presented.

In this paper, a new finite element scheme for Navier-Stokes equations is proposed, in which three different partitions (in the two dimensional case) are used to construct finite element subspaces of the velocity field and the pressure. The error estimate of the finite element than approximation is given. The precision of this new scheme has the same order as the scheme $Q_2/P_0$, but it is more economical that the scheme $Q_2/P_0$.

A sharper upperbound of the difference between a diagonal element and the corresponding eigenvalue of a symmetric tridiagonal matrix is given. The bound can be used in QL and QR algorithms and Rayleigh quotient approximation. The change of eigenvalues is estimated when the first off-diagonal element $\beta_1$ is replaced by zero and when two neighboring off-diagonal elements $ \beta_{i-1},\beta_i$ are replaced by zero.

The paper is devoted to the study and analysis of the mixed stiffness finite element method for the Navier-Stokes equations, based on a formulation of velocity-pressure-stress deviatorics. The method used low order Lagrange elements and leads to optimal error order of convergence for velocity, pressure, and stress deviatorics by means of mesh-dependent norms defined in this paper. The main advantage of the MSFEM is that the stream function can not only be employed to satisfy the divergence constraint but stress deviatorics can also be eliminated at the element level so that it is unnecessary to solve a larger algebraic system containing stress multipliers, or to develop a special code for computing the MSFE solutions of Navier-Stokes equations because we can use the computing codes used in solving the Navier-Stokes equations with the velocity-pressure formulation, or even the computing codes used in solving the problems of solid mechanics.  

Two types of combination methods for accelerating the convergence of the finite difference method are presented. The first is based on an interpolation principle (correction method) and the second one on extrapolation principle. They improve the convergence form $O(h^2)$ to $O(h^4)$. The main advantage when compared with standard methods, is that the computational work can be split into independent parts, which can be carried out in parallel.