Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes

We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.     

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

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 posteriori analysis of finite element discretizations of stochastic partial differential delay equations

In this paper, we study a posteriori error estimates for finite element approximation of stochastic partial differential delay equations containing a noise. We derive an energy norm a posteriori bounds for an Euler time-stepping method combined with a standard Galerkin schemes for the problems. For accessibility, we first address the spatially semidiscrete case and then move to the fully discrete scheme.     

A posteriori error estimates for a compressible Stokes system

A linearized compressible viscous Stokes system is considered. The a posteriori error estimates are defined and compared with the true error. They are shown to be globally upper and locally lower bounds for the true error of the finite element solution. Some numerical examples are given, showing an efficiency of the estimator. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 412–431, 2004.     

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.     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

A Posteriori Error Estimates for Distributed Convex Optimal Control Problems

In this paper, we present an a posteriori error analysis for finite element approximation of distributed convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximation schemes for control problems. Explicit estimates are obtained for some model problems which frequently appear in real-life applications.     

Robustness in a posteriori error analysis for FEM flow models

Summary. We derive a residual-based a posteriori error estimator for a stabilized finite element discretization of certain incompressible Oseen-like equations. We focus our attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem and mesh parameters. We also consider a scalar reaction-convection-diffusion problem and a divergence-free projection problem in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction-convection-diffusion phenomena and, separately, of the incompressibility constraint. Received March 21, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001     

A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

In this paper, we conduct a goal-oriented a posteriori analysis for the error in a quantity of interest computed from a cell-centered finite volume scheme for a semilinear elliptic problem. The a posteriori error analysis is based on variational analysis, residual errors and the adjoint problem. To carry out the analysis, we use an equivalence between the cell-centered finite volume scheme and a mixed finite element method with special choice of quadrature.     

A residual-type a posteriori error estimate of finite volume element method for a quasi-linear elliptic problem

In this paper, we analyze a residual-type a posteriori error estimator of the finite volume element method for a quasi-linear elliptic problem of nonmonotone type and derive computable upper and lower bounds on the error in the H ¹-norm. Numerical experiments are provided to illustrate the performance of the proposed estimator.     

Implicit residual error estimators for the coupling of finite elements and boundary elements

We consider a symmetric Galerkin method for the coupling of finite elements and boundary elements for elliptic problems with a monotone operator in the finite element domain. We derive an a posteriori error estimator which involves the solution of equilibrated local Neumann problems in the finite element domain and requires computation of a residual term on the coupling interface. Finally, we discuss a similar approach for a coupling with Signorini contact conditions on the interface. Copyright © 1999 John Wiley & Sons, Ltd.     

A Posteriori Error Estimators and Adaptivity for Finite Element Approximation of the Non-Homogeneous Dirichlet Problem

Techniques are developed for a posteriori error analysis of the non-homogeneous Dirichlet problem for the Laplacian giving computable error bounds for the error measured in the energy norm. The techniques are based on the equilibrated residual method that has proved to be reliable and accurate for the treatment of problems with homogeneous Dirichlet data. It is shown how the equilibrated residual method must be modified to include the practically important case of non-homogeneous Dirichlet data. Explicit and implicit a posteriori error estimators are derived and shown to be efficient and reliable. Numerical examples are provided illustrating the theory.     

A posteriori error estimate for the Stokes–Darcy system

We consider the a posteriori error estimates for finite element approximations of the Stokes–Darcy system. The finite element spaces adopted are the Hood–Taylor element for the velocity and the pressure in fluid region and conforming piecewise quadratic element for the pressure in porous media region. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. Copyright © 2011 John Wiley & Sons, Ltd.     

Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems

A procedure for the construction of robust, upper bounds for the error in the finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method.     

An experimental survey of a posteriori Courant finite element error control for the Poisson equation

This comparison of some a posteriori error estimators aims at empirical evidence for a ranking of their performance for a Poisson model problem with conforming lowest order finite element discretizations. Modified residual-based error estimates compete with averaging techniques and two estimators based on local problem solving. Multiplicative constants are involved to achieve guaranteed upper and lower energy error bounds up to higher order terms. The optimal strategy combines various estimators.     

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.     

Reliable,goal‐oriented postprocessing for FE‐discretizations

In this article we present strategies to improve the quality of adaptive FE‐approximations measured in terms of linear functionals. The ideas are based on the so called dual‐weighted‐residual (DWR) approach to a posteriori error control for FE‐schemes. In more details, we exploit those parts of an underlying error representation, which are completely computable, to improve the FE‐solution. Furthermore, the remaining parts of the error identity can be estimated by well‐established a posteriori energy estimates yielding reliable error bounds for the postprocessed values. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Finite element convergence for the Joule heating problem with mixed boundary conditions

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes.     

A posteriori error estimates of functional type for variational problems related to generalized Newtonian fluids

The paper is focused on functional type a posteriori estimates of the difference between the exact solution of a variational problem modelling certain types of generalized Newtonian fluids and any function from the admissible energy class. In contrast to the a posteriori estimates obtained for example by the finite element method our estimates do not contain any local (mesh dependent) constants, and therefore they can be used regardless of the way in which an approximation has been constructed. Copyright © 2006 John Wiley & Sons, Ltd.