Functional A Posteriori Error Estimates for Time-Periodic Parabolic Optimal Control Problems

This article is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of the functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds.     

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     

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 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     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.     

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 viscous flow problems with rotation

A new functional type a posteriori error estimates for the Stokes problem with rotating term are presented. The estimates give guaranteed upper bounds for the energy norm of the error and provide reliable error indication. Computational properties of the estimates are demonstrated by a number of numerical examples. Bibliography: 37 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 23–34.     

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Summary. In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when . We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Received January 25, 1999 / Revised version received June 5, 2000 Published online March 20, 2001     

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

A posteriori error estimates for the Crank-Nicolson method for parabolic equations

We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

     

A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations

In this paper, we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations. The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L ²-norm and for state y and costate p measured in energy norm. Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.      

A posteriori error estimates for the generalized Stokes problem

The paper concerns a posteriori estimates of functional type for the difference between exact and approximate solutions to a generalized Stokes problem. The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem using the method suggested by the first author. The estimates obtained can be classified into two types. Estimates of the first type are valid only for solenoidal functions, while estimates of the second type are applicable for any functions that belong to the energy space of the respective problem and satisfy the boundary conditions. In the second case, the estimates include an additional penalty term with a multiplier defined by the constant in the Ladyzhenskaya-Babuška-Brezzi condition. It is proved that a posteriori estimates for the velocity field yield computable estimates of the difference between exact and approximate pressure functions in the L₂-norm. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires to solve only finite-dimensional problems. Bibliography: 34 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 89–101.     

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.     

Discontinuous Galerkin finite element method for a nonlinear boundary value problem

In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm （stronger than the H1 norm） and the L2 norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.     

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.     

Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

The paper is devoted to 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 consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow. This work was supported by the Academy Research Fellowship No. 208628 from the Academy of Finland.     

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.     

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.     

双线性非协调有限元逼近的梯度恢复后验误差估计

给出了二阶椭圆方程的双线性非协调有限元逼近的梯度恢复后验误差估计.该误差估计是在Q_1非协调元上得到的,并给出了误差的上下界.进一步证明该误差估计在拟一致网格上是渐进精确地.证明依赖于clement插值和Helmholtz分解,数值结果验证了理论的正确性.