Pointwise a posteriori error control for elliptic obstacle problems

We consider a finite element method for the elliptic obstacle problem over polyhedral domains in ^d, which enforces the unilateral constraint solely at the nodes. We derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle, which is just assumed to be Hölder continuous. They exhibit optimal order and localization to the non-contact set. We illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi-optimal meshes. Partially supported by NSF Grant DMS-9971450 and NSF/DAAD Grant INT-9910086.Partially suported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF'.Partially supported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF', and by the TMR network ``Viscosity solutions and their Applications', Italian M.I.U.R. projects ``Scientific Computing: Innovative Models and Numerical Methods' and ``Symmetries, Geometric Structures, Evolution and Memory in PDEs'.Mathematics Subject Classification (1991):65N15, 65N30, 35J85     

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     

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)     

POD a-posteriori error estimates for linear-quadratic optimal control problems

The main focus of this paper is on an a-posteriori analysis for the method of proper orthogonal decomposition (POD) applied to optimal control problems governed by parabolic and elliptic PDEs. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the POD model, is from the (unknown) exact one. Numerical examples illustrate the realization of the proposed approach for linear-quadratic problems governed by parabolic and elliptic partial differential equations.     

A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints

In this paper optimal control problems governed by elliptic semilinear equations and subject to pointwise state constraints are considered. These problems are discretized using finite element methods and a posteriori error estimates are derived assessing the error with respect to the cost functional. These estimates are used to obtain quantitative information on the discretization error as well as for guiding an adaptive algorithm for local mesh refinement. Numerical examples illustrate the behavior of the method.     

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.     

A note on functional a posteriori estimates for elliptic optimal control problems

In this work, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed, and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two‐sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 403–424, 2017     

Elliptic reconstruction and a posteriori error estimates for parabolic optimal control problems

In this article, a semidiscrete finite element method for parabolic optimal control problems is investigate. By using elliptic reconstruction, a posteriori error estimates for finite element discretizations of optimal control problem governed by parabolic equations with integral constraints are derived.     

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     

A global a posteriori error estimate for quasilinear elliptic problems

A global a posteriori error estimate, valid even if uniqueness fails, is obtained for a class of quasilinear elliptic partial differential equations. It is applied to the analysis of finite element methods for such problems, and to a primitive Fourier series method for the stationary Navier-Stokes problem in three dimensions at arbitrary Reynolds number.This research was supported by the National Science Foundation, Grant No. NSF-GP-37069.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Residual-based a posteriori error control for Space-time finite element approximations of parabolic optimal control problems

In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori finite element error estimators for parametrized nonlinear boundary value problems

A posteriori error estimators for finite element solutions of multi—parameter nonlinear partial differential equations are based on an element—by—element solution of local linearizations of the nonlinear equation. In general, the associated bilinear form of the linearized Problems satisfies a Gårding—type inequality. Under appropriate assumption it is shown that the error estimators are bounded by constant multiples of the true error in a suitable norm. Computational experiments indicate that the estimators are effective, inexpensive, and insensitive to the choice of the local coordinate system on the solution manifold.     

Reliable a posteriori error estimation for state-constrained optimal control

We derive a reliable a posteriori error estimator for a state-constrained elliptic optimal control problem taking into account both regularisation and discretisation. The estimator is applicable to finite element discretisations of the problem with both discretised and non-discretised control. The performance of our estimator is illustrated by several numerical examples for which we also introduce an adaptation strategy for the regularisation parameter.     

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.     

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babu?ka’s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi’s saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babu?ka’s inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

We propose a characteristic finite element discretization of evolutionary type convection-diffusion optimal control problems. Nondivergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.     

A priori and a posteriori error analysis for virtual element discretization of elliptic optimal control problem

Numerical Algorithms - In this paper, a virtual element method (VEM) discretization of elliptic optimal control problem with pointwise control constraint is investigated. Virtual element discrete...