Functional a posteriori estimates for elliptic optimal control problems

In this note, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.     

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.     

Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems

In this article, we develop functional a posteriori error estimates for discontinuous Galerkin (DG) approximations of elliptic boundary‐value problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximations developed by S. Repin (see e.g., Math Comp 69 (2000) 481–500). On these grounds, we derive two‐sided guaranteed and computable bounds for the errors in “broken” energy norms. A series of numerical examples presented confirm the efficiency of the estimates. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Local a posteriori error estimators for variational inequalities

Local a posteriori error estimators for finite element approximation of variational inequalities are derived. These are shown to provide upper bounds on the discretization error. Numerical examples are given illustrating the theoretical results. © 1993 John Wiley & Sons, Inc.     

Residual type a posteriori error estimates for elliptic obstacle problems

Summary. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed. Received June 19, 1998 / Published online December 6, 1999     

Jumping problems for quasilinear elliptic variational inequalities

We investigate the existence and multiplicity of solutions for a certain class of quasilinear variational inequalities, whenever between the obstacle and the behavior at +￥+\infty of the nonlinearity there is a situation of jumping type.     

Eigenvalue problems for nonlinear elliptic variational inequalities on a cone

Eigenvalue problems for variational inequalities on a closed convex cone C in a real Banach space V, of the form 〈g′(v) ? λh′(v), w ? v〉 ? 0 for all w in C, are considered with the normalization g(v) = r, where g and h are real-valued C¹ functions on V. Theorems are proved on the existence of solutions λ(r) and v(r), and on their dependence upon the normalization constant r > 0. In particular, the relation, as r → 0, of λ(r), v(r) to solutions of the analogous problem with g″(0) and h″(0) in place of g′ and h′, is discussed. The theorems are applied to elliptic inequalities for Euler-Lagrange operators corresponding to multiple integral functionals on closed subspaces of Sobolev spaces, and to the inequality arising from the von Karman equations for the buckling of a thin elastic plate which is constrained to buckle in only one direction.     

Functional a posteriori error estimates for the reaction-convection-diffusion problem

In this paper, a general form of functional type a posteriori error estimates for linear reaction-convection-diffusion problems is presented. It is derived by purely functional arguments without attracting specific properties of the approximation method. The estimate provides a guaranteed upper bound of the difference between the exact solution and any conforming approximation from the energy functional class. It is also proved that the derived error majorants give computable quantities, which are equivalent to the error evaluated in the energy and combined primal-dual norms. Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 127–146.     

Some aspects of elliptic variational inequalities

In this paper we study an existence and the approximation of the solution of the elliptic variational inequality from an abstract axiomatic point of view. We discuss convergence results and give an error estimate for the difference of the two solutions in an appropriate norm. Also, we present some computational results by using fixed point method.     

Alternating iteration and elliptic variational inequalities

Summary Free boundary value problems, too complicated for formulation as a variational inequality, are broken up into two problems on overlapping regions. On one region the problem is treated as an ordinary boundary value problem; on the second region, the free boundary part of the problem is reduced to a variational inequality. By solving the two problems successively it is shown that under certain conditions the successive solutions converge to a single function that gives a solution of the original problem. Application to a filtration problem is given.     

Inverse coefficient problems for nonlinear elliptic variational inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.     

Monotone multigrid methods for elliptic variational inequalities II

Summary. We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches. Received March 26, 1994 / Revised version received September 22, 1994     

Reinforcement problems for elliptic equations and variational inequalities

Summary Consider the Dirichlet problem for an elliptic equation in a domain , with coefficients having discontinuity on a surface . Suppose divides into _{1 2}(₂ the inner core), the thickness of ₁ is of order of magnitude , and the modulus of ellipticity in ₁ is of order magnitude ₁. The asymptotic behavior of the solution is studied as 0, ₁ 0, provided lim (/₁) exists. Other questions of this type are studied both for elliptic equations and for elliptic variational inequalities.The second author is partially supported by National Science Foundation Grant 7406375 A01. The third author is partially supported by National Science Foundation Grant MC575-21416 A01.     

Gradient recovery type a posteriori error estimates of virtual element method for an elliptic variational inequality of the second kind

In this paper, gradient recovery type a posteriori error estimators of virtual element discretization are derived for a simplified friction problem, which is a typical elliptic variational inequality of the second kind. Both the reliability and the efficiency of the error estimators are proved. In addition, one numerical example is presented to show the efficiency of the adaptive VEM based on the derived error estimators.