Numerical solution of the nonstationary Stokes system by methods of adjoint-equation theory and optimal control theory

Methods in optimal control and the adjoint-equation theory are applied to the design of iterative algorithms for the numerical solution of the nonstationary Stokes system perturbed by a skew-symmetric operator. A general scheme is presented for constructing algorithms of this kind as applied to a broad class of problems. The scheme is applied to the nonstationary Stokes equations, and the convergence rate of the corresponding iterative algorithm is examined. Some numerical results are given.     

Numerical Integration of Differential Algebraic Systems and Invariant Manifolds

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.     

The critical Neumann problem for semilinear elliptic equations with concave perturbations

We consider the semilinear Neumann problem involving the critical Sobolev exponent with an indefinite weight function and a concave purturbation. We prove the existence of two distinct solutions.      

Singular limits for a 4-dimensional semilinear elliptic problem with exponential nonlinearity

Using some nonlinear domain decomposition method, we prove the existence of branches of solutions having singular limits for some 4-dimensional semilinear elliptic problem with exponential nonlinearity.     

Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis

We consider hysteresis perturbations of a system of ODEs which possesses an asymptotically stable periodic solution z_∗. Where the oscillation of an appropriate projection of this periodic solution is smaller than some threshold number defined by the hysteresis nonlinearity, it is shown that the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of z_∗. The main result is a theorem on the stability of this continuum.     

On Local Perturbations of a Three-Dimensional Boundary Layer

A high-Reynolds steady-state viscous incompressible fluid flow is investigated in the neighborhood of a small three-dimensional irregularity located on the smooth surface of a body and oriented almost along the skin-friction lines. The regime in which quasi-two-dimensional flow with a given pressure gradient is realized on the irregularity scale is studied in detail. The numerical solution of the corresponding boundary value problem for the boundary layer equations is obtained. It is shown that, as distinct from the plane case, this solution is unique.     

Variation in the wave resistance of an amphibian air-cushion vehicle moving over a broken-ice field

The nonstationary rectilinear motion of an amphibian air-cushion vehicle (AACV) on a water surface covered with finely broken ice is considered for various modes of velocity variation. The influence of the water depth, flotation parameters, and mode of motion on the wave resistance of the vehicle is analyzed. Maneuvering methods for increasing or decreasing the wave resistance of AACVs are proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 97–102, January–February, 2007.     

Discrete maximum principles for FE solutions of nonstationary diffusion‐reaction problems with mixed boundary conditions

In this article, we derive and discuss sufficient conditions for providing validity of the discrete maximum principle for nonstationary diffusion‐reaction problems with mixed boundary conditions, solved by means of simplicial finite elements and the θ time discretization method. The theoretical analysis is supported by numerical experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Vibrations of a piezoceramic cylinder subject to nonstationary electric excitation

The paper proposes a method to solve the problem of vibrations of a radially polarized piezoelectric cylinder subject to nonstationary electric excitation. The dynamic electromechanical state of the cylinder is analyzed. The time-dependences of electric and mechanical characteristics are plotted. The distribution of these characteristics over the cross section of a short cylinder is examined. The region of end disturbances in a long cylinder is identified __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 73–79, March 2007.     

Small random perturbations of one-dimensional diffusion processes

In the present paper we consider the small random perturbations of one-dimensional diffusion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero. Partially supported by the Young Teachers Foundation of Beijing Institute of Technology