A posteriori boundary element error estimation

An a posteriori error estimator is presented for the boundary element method in a general framework. It is obtained by solving local residual problems for which a local concept is introduced to accommodate the fact that integral operators are nonlocal operators. The estimator is shown to have an upper and a lower bound by the constant multiples of the exact error in the energy norm for Symm's and hypersingular integral equations. Numerical results are also given to demonstrate the effectiveness of the estimator for these equations. It can be used for adaptive h,p, and hp methods.     

A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.     

Adaptive mesh for algebraic orthogonal subscale stabilization of convective dispersive transport

We derive a residual a posteriori error estimator for the algebraic orthogonal subscales stabilization of convective dispersive transport equation. The estimator yields upper bound on the error which is global and lower bound that is local. Numerical studies show the behaviour of the error indicator and how it is robust to deal with singularities. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

We consider some (anisotropic and piecewise constant) diffusion problems in domains of R², approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.     

Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow

We derive computable a posteriori error estimates for the lowest order nonconforming Crouzeix-Raviart element applied to the approximation of incompressible Stokes flow. The estimator provides an explicit upper bound that is free of any unknown constants, provided that a reasonable lower bound for the inf-sup constant of the underlying problem is available. In addition, it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant.

     

Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.     

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms.One estimator is shown to be reliable and efficient,which yields global upper and lower bounds for the error in piecewise W1,p seminorm.The other one is proved to give a global upper bound of the error in Lp-norm.By taking the two estimators as refinement indicators,adaptive algorithms are suggested,which are experimentally shown to attain optimal convergence orders.     

A posteriori estimation of the linearization error for strongly monotone nonlinear operators

We investigate the a posteriori estimation of the modeling (or linearization) error which arises when a nonlinear problem is replaced by a linear model. Using the context of strongly monotone operators, we construct a computable upper estimator for this error, and also provide an estimator that gives a lower bound. Several numerical results illustrating our theory are provided.     

An adaptive stabilized finite element method for the generalized Stokes problem

In this work we present an adaptive strategy (based on an a posteriori error estimator) for a stabilized finite element method for the Stokes problem, with and without a reaction term. The hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. An equivalence result between the norm of the finite element error and the estimator is given, where the dependence of the constants on the physics of the problem is explicited. Several numerical results confirming both the theoretical results and the good performance of the estimator are given.     

A posteriori error estimation for convection dominated problems on anisotropic meshes

A singularly perturbed convection–diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Asymptotic bounds for the expected L error of a multivariate kernel density estimator

The kernel estimator of a multivariate probability density function is studied. An asymptotic upper bound for the expected L¹ error of the estimator is derived. An asymptotic lower bound result and a formula for the exact asymptotic error are also given. The goodness of the smoothing parameter value derived by minimizing an explicit upper bound is examined in numerical simulations that consist of two different experiments. First, the L¹ error is estimated using numerical integration and, second, the effect of the choice of the smoothing parameter in discrimination tasks is studied.     

Equilibrated error estimators for discontinuous Galerkin methods

We consider some diffusion problems in domains of ?^d, d = 2 or 3 approximated by a discontinuous Galerkin method with polynomials of any degree. We propose a new a posteriori error estimator based on H(div)‐conforming elements. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established with a constant depending on the aspect ratio of the mesh, the dependence with respect to the coefficients being also traced. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Robust a posteriori error estimates for subgrid stabilization of non-stationary convection dominated diffusive transport

We derive a robust residual a posteriori error estimator for time-dependent convection-diffusion-reaction problem, stabilized by subgrid viscosity in space and discretized by Crank-Nicolson scheme in time. The estimator yields upper bounds on the error which are global in space and time and lower bounds that are global in space and local in time. Numerical experiments illustrate the theoretical performance of the error estimator.     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.     

A Posteriori Control of Dimension Reduction Errors on Long Domains

In this work, we consider linear elliptic problems posed in long domains, i.e. the domains whose size in one coordinate direction is much greater than the size in the other directions. If the variation of the coefficients and right‐hand side along the emphasized direction is small, the original problem can be reduced to a lower‐dimensional one that is supposed to be much easier to solve. The a‐posteriori estimation of the error stemming from the model reduction constitutes the goal of the present work. For general coefficient matrix and right‐hand side of the equation, the reliable and efficient error estimator is derived that provides a guaranteed upper bound for the modelling error, exhibits the optimal asymptotics as the size of the domain tends to infinity and correctly indicates the local error distribution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori error estimators for the Stokes equations II non-conforming discretizations

Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.     

A Posteriori Error Estimates and a Local Refinement Strategy for a Finite Element Method to Solve Structural-Acoustic Vibration Problems

This paper deals with an adaptive technique to compute structural-acoustic vibration modes. It is based on an a posteriori error estimator for a finite element method free of spurious or circulation nonzero-frequency modes. The estimator is shown to be equivalent, up to higher order terms, to the approximate eigenfunction error, measured in a useful norm; moreover, the equivalence constants are independent of the corresponding eigenvalue, the physical parameters, and the mesh size. This a posteriori error estimator yields global upper and local lower bounds for the error and, thus, it may be used to design adaptive algorithms. We propose a local refinement strategy based on this estimator and present a numerical test to assess the efficiency of this technique.     

Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem

This article presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e., elements with very large aspect ratio. Our analysis covers two‐ and three‐dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh, which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

A posteriori error estimate for a modified weak Galerkin method solving elliptic problems

A residual‐type a posteriori error estimator is proposed and analyzed for a modified weak Galerkin finite element method solving second‐order elliptic problems. This estimator is proven to be both reliable and efficient because it provides computable upper and lower bounds on the actual error in a discrete H¹‐norm. Numerical experiments are given to illustrate the effectiveness of the this error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 381–398, 2017