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Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied. This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid. Moreover, the scaling between these two grid sizes is super-linear. Approximation, stability and convergence aspects of a fully discrete scheme are analyzed. At last a numrical example is given whose results show that the algorithm proposed in this paper is effcient.

Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.

An information system is one of the very important mathematical models in the field of artificial intelligence. The notion of an information system homomorphism as a kind of tool to study the relation between two information systems was introduced in J.W. Grzymala-Busse (Algebraic properties of knowledge representation systems, in: Proceedings of the ACM SIGART International Symposium on Methodologies for Intelligent Systems, Knoxville, 1986, pp. 432–440). In this paper, some invariant characters of an information system under a homomorphism are discussed.

In this paper two modern numerical methods for computational physics are introduced. One of them is the nonlinear Galerkin method for nonlinear partial differential equations; the approximation properties are discussed and the errors are estimated. For the other one the Schwarz algorithm for the neutron transport equation is constructed, and the convergence properties are proved.

An information system is one of the very important mathematical models in the field of artificial intelligence. The notion of information system's homomorphism as a kind of tools to study the relation between two information systems was introduced by Grzymala-Busse (1986). In this paper, some invariant characteristics of an information system under a homomorphism are discussed.

An optimal control finite element approximation of the penalization type for the solution of incompressible viscous Navier–Stokes equations is presented. This paper has proved the convergence of its minimizing sequence and the existence of the solution for the optimal control problem corresponding to the penalty variational problem of the Navier–Stokes equations. Based on this method and the conjugate gradient algorithm, the program for solving the three-dimensional Navier–Stokes problem has been developed, and some numerical results have also been provided.