Estimation of Deviations from the Exact Solution for the Reissner-Mindlin Plate Problem

In this paper, we obtain a majorant of the difference between the exact solution and any conforming approximate solution of the Reissner-Mindlin plate problem. This majorant is explicitly computable and involves constants that depend only on given data of the problem. The majorant allows us to compute guaranteed upper bounds of errors with any desired accuracy and vanishes if and only if the approximate solution coincides with the exact one. Bibliography: 12 titles. To N. N. Uraltseva with gratitude __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 145–157.     

A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem

In this article, we derive a posteriori error estimates for the Hencky plasticity problem. These estimates are formulated in terms of the stresses and present guaranteed and computable bounds of the difference between the exact stress field and any approximation of it from the energy space of the dual variational problem. They consist of quantities that can be considered as penalties for the violations of the equilibrium equations, the yield condition and the constitutive relations that must hold for the exact stresses and strains. It is proved that the upper bound tends to zero for any sequence of stresses that tends to the exact solution of the Haar–Karman variational problem. An important ingredient of our analysis is a collection of Poincaré type inequalities involving the L ¹ norms of the tensors of small deformation. Estimates of this form are not new, however we will present computable upper bounds for the constants being involved even for rather complicated domains.     

A posteriori error estimates for approximations of evolutionary convection–diffusion problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed. Bibliography: 7 titles.     

The Elastic–Plastic Torsion Problem: A Posteriori Error Estimates for Approximate Solutions

We consider the variational inequality that describes the torsion problem for a long elasto-plastic bar. Using duality methods of the variational calculus, we derive a posteriori estimates of functional type that provide computable and guaranteed upper bounds of the energy norm of the difference between the exact solution and any function from the corresponding energy space that satisfies the Dirichlet boundary condition.     

A new method to build confidence regions for solutions of stochastic variational inequalities

Stochastic variational inequalities model a large class of equilibrium problems subject to data uncertainty. The true solution to such a problem is usually estimated by a solution to its sample average approximation (SAA) problem. This article proposed a new method to build asymptotically exact confidence regions for the true solution that are computable from the SAA solution.     

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.     

Solitary wave solutions of the improved KdV equation by VIM

The variational iteration method (VIM) is applied to solve numerically the improved Korteweg-de Vries equation (IKdV). A correction function is constructed with a general Lagrange multiplier that can be identified optimally via the variational theory. This technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the IKdV equation. Propagation of single, interaction of two, and three solitary waves, and also birth of solitons have been discussed. Three invariants of motion have been evaluated to determine the conservation properties of the problem. This procedure is promising for solving other nonlinear equations.     

On the regularization of mixed complementarity problems

A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MOP). Monotonicity of the VIP implies that the MOP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP.     

Functional a posteriori error estimates for elliptic problems in exterior domains

This paper is concerned with the derivation of computable and guaranteed upper bounds of the difference between the exact and approximate solutions of an exterior domain boundary value problem for a linear elliptic equation. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of functional type) were derived earlier for problems in bounded domains of RN. Bibliography: 4 titles. Illustrations: 1 figure.     

Two-sided a posteriori error bounds for electro-magnetostatic problems

This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between exact and approximate solutions of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not invoke specific properties of the approximation method. For this reason, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for elliptic problems. Bibliography: 24 titles.     

Estimates of deviations from exact solutions of variational inequalities based upon Payne–Weinberger inequality

A new method for obtaining computable estimates for the difference between exact solutions of elliptic variational inequalities and arbitrary functions in the respective energy space is suggested. The estimates are obtained by transforming the corresponding variational inequality without the use of variational duality arguments. These estimates are valid for any function in the energy class and contain no constants depending on the mesh used to find an approximate solution. This method for linear elliptic and parabolic problems was earlier suggested by the author. The guaranteed error bounds we derive can be of two types. Estimates of the first type contain only one global constant, which is a constant in the Friedrichs type inequality. Estimates of the second type are based on the decomposition of Ω into convex subdomains and the Payne–Weinberger inequalities for these subdomains. Bibliography: 20 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 81–90.     

On a posteriori error estimates for the stationary Navier-Stokes problem

We obtain a computable upper bound for the difference between a solution to the stationary Navier-Stokes problem and any solenoidal vector-valued function satisfying the boundary condition and possessing necessary differentiability properties. For sufficiently small velocities this estimate implies an estimate of the deviation from exact solution in the energy norm and the uniqueness of a weak solution. Bibliography: 3 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 89–92.     

Multilevel a posteriori error analysis for reaction–convection–diffusion problems

We present a new approach to the a posteriori error analysis of stable Galerkin approximations of reaction–convection–diffusion problems. It relies upon a non-standard variational formulation of the exact problem, based on the anisotropic wavelet decomposition of the equation residual into convection-dominated scales and diffusion-dominated scales. The associated norm, which is stronger than the standard energy norm, provides a robust (i.e., uniform in the convection limit) control over the streamline derivative of the solution. We propose an upper estimator and a lower estimator of the error, in this norm, between the exact solution and any finite dimensional approximation of it. We investigate the behaviour of such estimators, both theoretically and through numerical experiments. As an output of our analysis, we find that the lower estimator is quantitatively accurate and robust.     

Numerical verification of solutions for variational inequalities

In this paper, we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities. This technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates for obstacle problems, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Further, a numerical example for an obstacle problem is presented. Received October 28,1996 / Revised version received December 29,1997     

On the Order of Decrease of Uniform Moduli of Smoothness for the Classes of Functions E p,m [ɛ]

In this paper, we solve the problem of the exact order of decrease of uniform moduli of smoothness for the classes of 2π-periodic functions of several variables with a given majorant of the sequence of total best approximations in the metric of L _p , 1 ≤ p < ∞.     

Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie‐modified Riemann–Liouville derivative

In this article, the fractional variational iteration method is employed for computing the approximate analytical solutions of degenerate parabolic equations with fractional time derivative. The time‐fractional derivatives are described by the use of a new approach, the so‐called Jumarie modified Riemann–Liouville derivative, instead in the sense of Caputo. The approximate solutions of our model problem are calculated in the form of convergent series with easily computable components. Moreover, the numerical solution is compared with the exact solution and the quantitative estimate of accuracy is obtained. The results of the study reveal that the proposed method with modified fractional Riemann–Liouville derivatives is efficient, accurate, and convenient for solving the fractional partial differential equations in multi‐dimensional spaces without using any linearization, perturbation or restrictive assumptions. Copyright © 2011 John Wiley & Sons, Ltd.     

General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.     

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.     

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.     

Lower convergence of approximate solutions to vector quasi-variational problems

In this article, various types of approximate solutions for vector quasi-variational problems in Banach spaces are introduced. Motivated by [M.B. Lignola, J. Morgan, On convergence results for weak efficiency in vector optimization problems with equilibrium constraints, J. Optim. Theor. Appl. 133 (2007), pp. 117–121] and in line with the results obtained in optimization, game theory and scalar variational inequalities, our aim is to investigate lower convergence properties (in the sense of Painlevé–Kuratowski) for such approximate solution sets in the presence of perturbations on the data. Sufficient conditions are obtained for the lower convergence of ‘strict approximate’ solution sets but counterexamples show that, in general, the other types of solutions do not lower converge. Moreover, we prove that any exact solution to the limit problem can be obtained as the limit of a sequence of approximate solutions to the perturbed problems.