An Introductory Review on A Posteriori Error Estimation in Finite Element Computations
From MaRDI portal
Publication:6071824
DOI10.1137/21m1464841zbMath1528.65104arXiv2110.02160MaRDI QIDQ6071824
Ludovic Chamoin, Frédéric Legoll
Publication date: 29 November 2023
Published in: SIAM Review (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/2110.02160
Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Extrapolation to the limit, deferred corrections (65B05) Research exposition (monographs, survey articles) pertaining to numerical analysis (65-02) Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs (65N50)
Cites Work
- Convergence of Adaptive Finite Element Methods
- Strategies for computing goal-orienteda posteriori error measures in non-linear elasticity
- Local problems on stars: A posteriori error estimators, convergence, and performance
- Error estimators for viscoplastic materials: application to forming processes
- Arbitrary Lagrangian–Eulerian finite element analysis of strain localization in transient problems
- ERROR ESTIMATION AND ADAPTIVITY IN ELASTOPLASTICITY
- A Posteriori Error Estimation for Hierarchic Models of Elliptic Boundary Value Problems on Thin Domains
- Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations
- A posteriori error estimators for nonconforming finite element methods
- A Convergent Adaptive Algorithm for Poisson’s Equation
- Mixed Finite Element Methods and Applications
- A “flux-free” nodal Neumann subproblem approach to output bounds for partial differential equations
- A residual a posteriori error estimate for the Virtual Element Method
- An MsFEM Approach Enriched Using Legendre Polynomials
- Residuala posteriorierror estimation for the Virtual Element Method for elliptic problems
- An Adaptive Multiscale Finite Element Method
- Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations
- Computing with hp-ADAPTIVE FINITE ELEMENTS
- A posteriorierror bounds for reduced-basis approximations of parametrized parabolic partial differential equations
- A New Finite Element Gradient Recovery Method: Superconvergence Property
- Generalized Green's Functions and the Effective Domain of Influence
- The Mathematical Theory of Finite Element Methods
- Error estimation of stress intensity factors for mixed‐mode cracks
- Equilibrated residual error estimator for edge elements
- Spectral Methods
- Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations
- Reduced-basis output bound methods for parabolic problems
- ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES
- Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation
- Superconvergent recovery of gradients on subdomains from piecewise linear finite‐element approximations
- Finite Elements
- Output bounds for reduced-order approximations of elliptic partial differential equations
- Goal-oriented error estimation and adaptivity for the finite element method
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients
- Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics problems
- An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress fields in finite element analyses
- Multiscale modeling using goal-oriented adaptivity and numerical homogenization. I: Mathematical formulation and numerical results
- Goal-adaptive isogeometric analysis with hierarchical splines
- A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods
- Strict upper bounds of the error in calculated outputs of interest for plasticity problems
- Computable bounds of functional outputs in linear visco-elastodynamics
- Convergence and optimality of \({\mathbf {hp}}\)-\textbf{AFEM}
- Fast estimation of discretization error for FE problems solved by domain decomposition
- A new non-intrusive technique for the construction of admissible stress fields in model verification
- Adaptive isogeometric analysis by local \(h\)-refinement with T-splines
- On the verification of model reduction methods based on the proper generalized decomposition
- An error estimator for separated representations of highly multidimensional models
- Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems
- Adaptive finite element methods in computational mechanics
- Error estimation and adaptivity in elastodynamics
- Finite element recovery techniques for local quantities of linear problems using fundamental solutions
- Computational error estimates and adaptive processes for some nonlinear structural problems
- A family of higher order mixed finite element methods for plane elasticity
- Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method
- A non-intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity problems
- A posteriori error analysis of the heterogeneous multiscale method for homogenization problems
- Goal-oriented adaptivity for GMsFEM
- A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization
- Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Application to transport and continuum mechanics.
- Strict upper error bounds on computed outputs of interest in computational structural mechanics
- Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems
- A review of some a posteriori error estimates for adaptive finite element methods
- Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems
- Adaptive finite elements for flow problems with moving boundaries. I. Variational principles and a posteriori estimates
- A feedback finite element method with a posteriori error estimation. I: The finite element method and some basic properties of the a posteriori error estimator
- A relationship between stabilized finite element methods and the Galerkin method with bubble functions
- Stabilized finite element methods. I.: Application to the advective- diffusive model
- Recent experiences with error estimation and adaptivity. I: Review of error estimators for scalar elliptic problems
- New advances on a posteriori error on constitutive relation in f. e. analysis
- A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations
- A unified approach to a posteriori error estimation using element residual methods
- A posteriori error estimation and adaptive mesh-refinement techniques
- A general method for recovering equilibrating element tractions
- A feed-back approach to error control in finite element methods: Application to linear elasticity
- Energy norm a posteriori error estimation for discontinuous Galerkin methods.
- A posteriori error estimation for unilateral contact with matching and non-matching meshes
- Adaptive finite element methods for elliptic equations with non-smooth coefficients
- Constitutive relation errror estimators for time-dependent nonlinear FE analysis
- A numerical study of a posteriori error estimators for convection-diffusion equations
- A fully automatic \(hp\)-adaptivity
- Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations
- A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows
- A posteriori error estimates for the virtual element method
- Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems
- Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty
- A note on the design of \(hp\)-adaptive finite element methods for elliptic partial differential equations
- The \(h,p\) and \(h\)-\(p\) version of the finite element method; basis theory and applications
- A posteriori error estimates for nonconforming finite element schemes
- A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems
- A tutorial in elementary finite element error analysis: A systematic presentation of a priori and a posteriori error estimates
- A new a posteriori error estimation for nonlinear time-dependent finite element analysis
- A numerical method with a posteriori error estimation for determining the path taken by a propagating crack
- Adaptive strategy for transient/coupled problems. Applications to thermoelasticity and elastodynamics
- Constitutive relation error estimator for transient finite element analysis
- A new strategy for improving a finite element method based on explicit error estimates
- A unified approach to remeshing strategies for finite element \(h\)-adaptivity
- Local error estimators for finite element linear analysis
- Constitutive relation error estimators for (visco)plastic finite element analysis with softening
- On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors
- A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity
- Pointwise and local error estimates for the quantities of interest in two-dimensional elasticity
- A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems
- A posteriori error estimates for nonconforming finite element methods
- Evaluation of the local quality of stresses in 3D finite element analysis.
- Estimation of modeling error in computational mechanics.
- Variational discretization of elliptic boundary value problems.
- Theory and practice of finite elements.
- Adaptive finite element methods with convergence rates
- Dual analysis with general boundary conditions
- Error estimation and adaptive meshing in strongly nonlinear dynamic problems
- Error estimation including pollution assessment for nonlinear finite element analysis
- A posteriori estimation and adaptive control of the error in the quantity of interest. I: A posteriori estimation of the error in the von Mises stress and the stress intensity factor
- Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method. II
- A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem
- \textit{A posteriori} error estimation and adaptive strategy for the control of MsFEM computations
- \textit{A posteriori} error estimation for isogeometric analysis using the concept of constitutive relation error
- Guaranteed and sharp a posteriori error estimates in isogeometric analysis
- A posteriori estimates distinguishing the error components and adaptive stopping criteria for numerical approximations of parabolic variational inequalities
- A posteriori error estimation and adaptivity in non-intrusive couplings between concurrent models
- Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods
- Goal-oriented error estimation and adaptivity in MsFEM computations
- Improved ZZ a posteriori error estimators for diffusion problems: conforming linear elements
- A new equilibrated residual method improving accuracy and efficiency of flux-free error estimates
- Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis
- Numerical model reduction with error control in computational homogenization of transient heat flow
- \textit{A posteriori} error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems
- Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers
- Residual-based a posteriori error estimation for stochastic magnetostatic problems
- A posteriori error estimation for generalized finite element methods
- Certified reduced basis approximation for parametrized partial differential equations and applications
- Computable upper and lower bounds on eigenfrequencies
- Adaptive reduced basis finite element heterogeneous multiscale method
- Verification of stochastic models in uncertain environments using the constitutive relation error method
- Optimality of a standard adaptive finite element method
- An accurate \(\mathbf H\)(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems
- Adaptive computations of a posteriori finite element output bounds: a comparison of the hybrid-flux approach and the flux-free approach
- Error indicators for mixed finite elements in 2-dimensional linear elasticity
- Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods
- A posteriori error estimates for finite element discretizations of the heat equation
- Goal-oriented a posteriori error estimates in linear elastic fracture mechanics
- Subdomain-based flux-free a posteriori error estimators
- The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations
- Computing upper and lower bounds for the \(J\)-integral in two-dimensional linear elasticity
- Convergence of an adaptive \( hp\) finite element strategy in one space dimension
- Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver
- Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages 145–197 of Stress Analysis, Edited by O. C. Zienkiewicz and G. S. Holister, Published by John Wiley & Sons, 1965
- Constitutive relation errors for F.E. analysis considering (visco‐) plasticity and damage
- Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems
- Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence
- Local Flux Reconstructions for Standard Finite Element Methods on Triangular Meshes
- $hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems
- Adaptive Inexact Newton Methods with A Posteriori Stopping Criteria for Nonlinear Diffusion PDEs
- A Review of Unified A Posteriori Finite Element Error Control
- On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples
- Guaranteed computable bounds on quantities of interest in finite element computations
- An explicit residual-type error estimator for Q1-quadrilateral extended finite element method in two-dimensional linear elastic fracture mechanics
- On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments
- Computable exact bounds for linear outputs from stabilized solutions of the advection-diffusion-reaction equation
- New bounding techniques for goal-oriented error estimation applied to linear problems
- Certification of projection-based reduced order modelling in computational homogenisation by the constitutive relation error
- Equilibrium Finite Element Formulations
- ERROR ESTIMATION OF EIGENFREQUENCIES FOR ELASTICITY AND SHELL PROBLEMS
- A posteriori error estimator and error control for contact problems
- Calculation of strict error bounds for finite element approximations of non-linear pointwise quantities of interest
- A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation
- A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions
- A Posteriori Error Estimates Based on Hierarchical Bases
- Error indicators for the mortar finite element discretization of the Laplace equation
- A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers
- Analysis of the Zienkiewicz-Zhua-posteriori error estimator in the finite element method
- A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis
- The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements
- Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality
- An optimal control approach to a posteriori error estimation in finite element methods
- Steady-state convection-diffusion problems
- Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary Conditions
- Accuracy in finite element computation for eigenfrequencies
- An a posteriori error estimator for the Lame equation based on equilibrated fluxes
- A Posteriori Error Estimates for Lowest-Order Mixed Finite Element Discretizations of Convection-Diffusion-Reaction Equations
- A Posteriori Error Estimation for Discontinuous Galerkin Finite Element Approximation
- A constitutive relation error estimator based on traction-free recovery of the equilibrated stress
- Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis
- Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with XFEM
- A Goal-Oriented Adaptive Finite Element Method with Convergence Rates
- Theory of adaptive finite element methods: An introduction
- Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements
- Exact Bounds for Linear Outputs of the Advection-Diffusion-Reaction Equation Using Flux-Free Error Estimates
- Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error
- Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR-C technique
- A recovery-type error estimator for the extended finite element method based onsingular+smoothstress field splitting
- Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
- Guaranteed energy error bounds for the Poisson equation using a flux-free approach: Solving the local problems in subdomains
- Recovery of equilibrium on star patches using a partition of unity technique
- Some A Posteriori Error Estimators for Elliptic Partial Differential Equations
- Error Estimate Procedure in the Finite Element Method and Applications
- Estimation of the Error in the Reduced Basis Method Solution of Nonlinear Equations
- A simple error estimator and adaptive procedure for practical engineerng analysis
- The superconvergent patch recovery anda posteriori error estimates. Part 1: The recovery technique
- The superconvergent patch recovery anda posteriori error estimates. Part 2: Error estimates and adaptivity
- Error Estimates for Adaptive Finite Element Computations
- Explicit Error Bounds for a Nonconforming Finite Element Method
- Error estimation with postprocessed finite element solutions
- Explicit error bounds in a conforming finite element method
- A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem
- Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements
- Validation of a posteriori error estimators by numerical approach
- On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua
- Enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions
- CRITERIA TO ACHIEVE NEARLY OPTIMAL MESHES IN THEh-ADAPTIVE FINITE ELEMENT METHOD
- Recovering lower bounds of the error by postprocessing implicit residuala posteriorierror estimates
- Time finite elements and error computation for (visco)plasticity with hardening or softening
- Analysis of a subdomain‐based error estimator for finite element approximations of elliptic problems
- Fully Reliable Localized Error Control in the FEM
- Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
- Computing Bounds for Linear Functionals of Exact Weak Solutions to the Advection-Diffusion-Reaction Equation
- A Local A Posteriori Error Estimator Based on Equilibrated Fluxes
- Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation
- A posteriori analysis of the finite element discretization of some parabolic equations
- Patch recovery based on complementary energy
This page was built for publication: An Introductory Review on A Posteriori Error Estimation in Finite Element Computations
