Efficient and reliable hierarchical error estimates for an elliptic obstacle problem

We present and analyze novel hierarchical a posteriori error estimates for a self-adjoint elliptic obstacle problem. Under a suitable saturation assumption, we prove the efficiency and reliability of our hierarchical estimates. The proof is based upon some new observations on the efficiency of some hierarchical error indicators. These new observations allow us to remove an additional regularity condition on the underlying grid required in the previous analysis. Numerical computations confirm our theoretical findings.     

Local problems on stars: A posteriori error estimators, convergence, and performance

A new computable a posteriori error estimator is introduced, which relies on the solution of small discrete problems on stars. It exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption. A simple adaptive strategy is designed, which simultaneously reduces error and data oscillation, and is shown to converge without mesh pre-adaptation nor explicit knowledge of constants. Numerical experiments reveal a competitive performance, show extremely good effectivity indices, and yield quasi-optimal meshes.

     

Robust hierarchical a posteriori error estimators for stabilized convection–diffusion problems

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection‐diffusion equations with height Péclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

An adaptive finite element method with asymptotic saturation for eigenvalue problems

This paper discusses adaptive finite element methods for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.     

Small data oscillation implies the saturation assumption

Summary. The saturation assumption asserts that the best approximation error in with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements. We establish a link between this assumption and the oscillation of , and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption. We also show that this condition is necessary, and asymptotically valid provided . Received November 17, 2000 / Published online July 25, 2001     

Estimateur hiérarchique robuste pour un problème de perturbation singulière

In this Note, we show that a modified and simplified version of the estimator of Bank–Weiser can be used to define a robust a posteriori error estimator for singularly perturbed problem. We prove without comparison with a residual estimator or saturation assumption, the equivalence of the estimator with the error in the energy norm and the robusteness with respect to the diffusion coefficient. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

双线形元的各向异性后验误差估计

The main aim of this paper is to give an anisotropic posteriori error estimator. We firstly study the convergence of bilinear finite element for the second order problem under anisotropic meshes.By using some novel approaches and techniques,the optimal error estimates and some superconvergence results are obtained without the regularity assumption and quasi-uniform assumption requirements on the meshes.Then,based on these results, we give an anisotropic posteriori error estimate for the second problem.     

A new residual posteriori error estimates of mixed finite element methods for convection‐diffusion‐reaction equations

We propose and analyze a new technique for developing residual‐based a posteriori error estimates over the stress and scalar displacement error for the lowest‐order Raviart–Thomas mixed finite element discretizations of convection‐diffusion‐reaction equations in two‐dimension space. The new technique is based on the abstract error estimates, the postprocessed approximation of the scalar displacement, and on the construction of an auxiliary problem. We consider the centered and upwind‐weighted mixed schemes, and concentrate the attention on the presence of an inhomogeneous and an anisotropic diffusion‐dispersion tensor and on a possible convection dominance. Global upper bounds can be directly computed on the base of the solution of the mixed schemes without any additional cost. Local lower bounds without any saturation assumption, hold from the case where convection or reaction are not present to convection‐ or reaction‐dominated equations, and their local efficiency depends on local or global variations in coefficients similar to Péclect number. Numerical experiments are reported to show the competitive behavior of the proposed posteriori error estimates, and to confirm the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 593–624, 2014     

Robust a posteriori error estimation for the nonconforming Fortin-Soulie finite element approximation

We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin-Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error.

     

Estimateur a posteriori en norme L∞ pour les équations elliptiquesL∞-a posteriori error estimator for elliptic equations

In this Note, we show that modification of Bank–Wieser estimator introduce an L^∞-a posteriori error estimator for conforming and nonconforming methods. We prove, without saturation assumption nor comparison with residual estimators, the equivalence with the L^∞ error. To cite this article: A. Agouzal, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 411–415.     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

Robust error estimates for the finite element approximation of elliptic optimal control problems

In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang-bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results.     

Robust error estimates for the finite element approximation of elliptic optimal control problems

In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang–bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results.     

Energy norm a posteriori error estimates for mixed finite element methods

This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields.

     

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.     

Gradient recovery type a posteriori error estimate for finite element approximation on non-uniform meshes

In this paper, we derive gradient recovery type a posteriori error estimate for the finite element approximation of elliptic equations. We show that a posteriori error estimate provide both upper and lower bounds for the discretization error on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough. The numerical results demonstrating the theoretical results are also presented in this paper.     

A Posteriori Error Estimators of Gradient Recovery Type for Fem of a Model Optimal Control Problem

In this paper, we derive an a posteriori error estimator of gradient recovery type for a model optimal control problem. We show that the a posteriori error estimator is equivalent to the discretization error in a norm of energy type on general meshes. Furthermore, when the solution of the control problem is smooth and the meshes are uniform, it is shown to be asymptotically exact.     

A Posteriori Error Estimates for Axisymmetric and Nonlinear Problems

We propose and examine the primal and dual finite element method for solving an axially symmetric elliptic problem with mixed boundary conditions. We derive an a posteriori error estimate and generalize the method used for a nonlinear elliptic problem. Finally, an a posteriori error estimate for a nonlinear parabolic problem based on the concept of hierarchical finite element basis functions is introduced.     

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.     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015