Functional approach to a posteriori error estimation for elliptic optimal control problems with distributed control

A new approach to the a posteriori analysis of distributed optimal control problems is presented. The approach is based on functional type a posteriori estimates that provide computable and guaranteed bounds of errors for any conforming approximations of a boundary value problem. Computable two-sided a posteriori estimates for the cost functional and estimates for approximations of the state and control functions are derived. Numerical results illustrate the efficiency of the approach. Bibliography: 35 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 3–14     

On a class of two-dimensional adjoint integral equations of Volterra type

In a rectangular domain, we consider a two-dimensional integral equation of Volterra type with fixed singular kernels. For various values of the parameters occurring in this equation, we prove its unique solvability and establish the asymptotic behavior of the obtained solution.     

Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems

We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.     

Estimates of deviations from exact solutions of initial boundary value problems for the wave equation

We derive computable upper bounds of the difference between an exact solution of the initial boundary value problem for a linear hyperbolic equation and any function in a space-time cylinder that belongs to the respective energy class. We prove that the bounds vanish if and only if the approximate solution coincides with the exact one. Bibliography: 13 titles. Dedicated to the jubilee of dear Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 93–106.     

Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary‐value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn‐ and Friedrichs‐type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L² norm of a vector‐valued function from H¹(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L² norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Description of a gas discharge as a kinetic transition

The ignition of a gas glow discharge is described as a kinetic transition, for which the charge-carrier concentrations in the plasma region play the role of order parameter, while the external parameter is the field strength in the cathode space.     

Structure of the glow of a nanosecond diffuse discharge in a strongly nonuniform electric field

The structure of the glow of a diffuse discharge in air under atmospheric pressure is studied in detail in the “rod (cathode)-plane” geometry for an electrode gap of 10 cm and a cathode tip radius from 3 cm to 2.4 μm. The amplitude of voltage across the gap was ～ 220 kV for voltage growth rate of ～10¹³ V/s and a pulse duration of 180 ns. It is found that the shape of the discharge glow strongly depends on the radius of the cathode tip. For large values of the radius, the multichannel form of the glow prevails, which is statistically transformed into a volume glow as the radius decreases from 5 mm to 60 μm. This transition is accompanied by a decrease in the amplitude of the discharge current from 440 to 140 A on the average. For ultrasmall radii of the cathode (2.4–7.7 μm), the multichannel form of the glow prevails again and the amplitude of the discharge current increases up to ～300 A.     

Heat transfer associated with viscous two-dimensional channel flow in a transverse acoustic field

An analytic solution is proposed for the problem of the effect of a transverse resonant acoustic field on the heat transfer process in laminar two-dimensional channel flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 41–49, May–June, 1985.The author wishes to thank V. E. Nakoryakov for formulating the problem and discussing the results.     

Problem with generalized fractional differentiation operators for a hyperbolic equation degenerating in the interior of the domain

We study the unique solvability of a nonlocal problem with generalized fractional differentiation operators in the boundary condition for a hyperbolic equation degenerating in the interior of the domain.