Sergey I. Repin

A posteriori error estimation for variational problems with uniformly convex functionals

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form inf {F(v) + G(Λv)}, where F: V → R is a convex lower semicontinuous functional, G: Y → R is a uniformly convex functional, V and Y are reflexive Banach spaces, and Λ: V → Y is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via the duality theory.

A posteriori error estimation for nonlinear variational problems by duality theory

In this paper, we use the duality theory of the calculus of variations to derive a posteriori error estimates. We obtain a general form of this (duality) error estimate and show that the known classes of a posteriori error estimates are its particular cases. Bibliography: 21 titles.

A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions

H1, independently of the discretization method chosen. In particular, our error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available. We will present different strategies for the evaluation of the error estimator. Only one constant appears in its definition which is the one from Friedrichs inequality; that constant depends solely on the domain geometry, and the estimator is quite non-sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.

A unified approach to a posteriori error estimation based on duality error majorants

In this paper, we present a unified approach to computing sharp error estimates for approximate solutions of elliptic type boundary value problems in variational form. This approach is based on the duality theory of the calculus of variations which analyses the original (primal) variational problem together with another (dual) one. In previous papers this theory was used to obtain a new tool of a posteriori error estimation – duality error majorant (DEM). The latter provides an upper bound for the energy norm of the error for any approximation that belongs to the set of admissible functions of the considered variational problem. For this reason, DEM can be easily used to estimate the accuracy of various post-processed finite element approximations as well as to those computed by boundary element or by finite difference methods. The objective of this paper is to present practically convenient forms of DEM and to discuss computational aspects of this error estimation strategy. The performance of the proposed error estimation method is demonstrated through several examples, where duality majorants are computed and compared with the results obtained by other methods.

A posteriori error estimation of goal-oriented quantities by the superconvergence patch recovery

The paper is concerned with a posteriori error estimation in terms of special problem-oriented quantities. In many practically interesting cases, such a quantity is represented as a linear functional that controls the behavior of a solution in certain subdomains, along some lines, or at especially interesting points. The method of estimating quantities of interest is usually based upon the analysis of the adjoint boundary-value problem, whose right hand side is formed by the considered linear functional. On this way, we propose a new effective modus operandi. It is based on two principles: (a) the original and adjoint problems are solved on non-coinciding meshes, and (b) the term presenting the product of gradients of errors of the primal and adjoint problems is estimated by using the gradient averaging technique. Numerical tests confirm high effectivity of this approach.

A posteriori error estimation for elasto-plastic problems based on duality theory

Abstract In this paper we introduce a new approach to a posteriori error estimation for elasto-plastic problems based on the duality theory of the calculus of variations. We show that, in spite of the prevailing view, duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem. Rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts: the error in the constitutive law and the error in the equilibrium equations. This representation is both physically meaningful and computationally important. The duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the Galerkin orthogonality condition. In particular, they encompass the familiar smoothening or gradient averaging techniques commonly used in practice. We prove that the duality error estimators are ‘equivalent to the error’ when the approximate solution is the exact solution of the discretised problem. Moreover, in the case of linear elasticity, the known residual type error estimators can be obtained from the duality error estimators. Numerical results for a model elasto-plastic problem show the accuracy of the duality error estimators.

A posteriori error estimates for approximate solutions to variational problems with strongly convex functionals

A method of obtaining a posteriori estimates for the difference between an exact solution and an approximate solution is suggested. The method is based on the duality theory of variational calculus. The general form of such an estimate is derived for a broad class of variational problems. The estimate converges to zero as the approximate solution converges to the exact one. The general estimates are considered in detail for some classes of variational problems. Bibliography: 25 titles.

A posteriori error estimates for boundary-value problems related to the biharmonic operator

Abstract This paper is concerned with boundary-value problems related to the biharmonic operator. The main goal of the paper is to derive a posteriori error estimates valid for any conforming approximations of the considered problems. For this purpose, the general approach that follows from the duality theory of the calculus of variations is used. The consistency of the derived a posteriori error estimates is proved and the corresponding computational strategies are discussed.

A Posteriori Estimation of Dimension Reduction Errors for Elliptic Problems on Thin Domains

A new a posteriori error estimator is presented for the verification of the dimensionally reduced models stemming from the elliptic problems on thin domains. The original problem is considered in a general setting, without any specific assumptions on the domain geometry, coefficients, and the right-hand sides. For the energy norm of the error of the zero-order dimension reduction method, the proposed estimator is shown to always provide a guaranteed upper bound. In the case when the original domain has constant thickness (but, possibly, nonplane upper and lower faces), the estimator demonstrates the optimal convergence rate as the thickness tends to zero. It is also flexible enough to successfully cope with infinitely growing right-hand sides in the equation when the domain thickness tends to zero. The numerical tests indicate the efficiency of the estimator and its ability to accurately represent the local error distribution needed for an adaptive improvement of the reduced model.

A reliable incremental method of computing the limit load in deformation plasticity based on compliance

The aim of this paper is to introduce an enhanced incremental procedure that can be used for the numerical evaluation and reliable estimation of the limit load. A conventional incremental method of limit analysis is based on parametrization of the respective variational formulation by the loading parameter ? ? ( 0 , ? l i m ) , where ? l i m is generally unknown. The enhanced incremental procedure is operated in terms of an inverse mapping ? : α ? ? where the parameter α belongs to ( 0 , + ∞ ) and its physical meaning is work of applied forces at the equilibrium state. The function ? is continuous, nondecreasing and its values tend to ? l i m as α ? + ∞ . Reduction of the problem to a finite element subspace associated with a mesh T h generates the discrete limit parameter ? l i m , h and the discrete counterpart ? h to the function ? . We prove pointwise convergence ? h ? ? and specify a class of yield functions for which ? l i m , h ? ? l i m . These convergence results enable to find reliable lower and upper bounds of ? l i m . Numerical tests confirm computational efficiency of the suggested method.The aim of this paper is to introduce a new incremental procedure that can be used for numerical evaluation of the limit load. Existing incremental type methods are based on parametrization of the energy by the loading parameter