A posteriori error estimators for the first-order least-squares finite element method
type: Item
instance of: scholarly article
title: A posteriori error estimators for the first-order least-squares finite element method
zbMATH Open document ID: 1203.65231
author: Jaeun Ku
author: Eun-Jae Park
published in: Journal of Computational and Applied Mathematics
publication date: 2010-10-25T00:00:00Z
review text: The aim of the authors is to construct an a posteriori error estimator based on the computation of adequate norms of the residual for the model problem written in a first-order least-squares finite element setting. Their analysis is based on the argument given in the previously obtained and published study on the weak coupling of solutions of least-square methods. In the first-order least-squares method, the built-in functional provides a natural error estimator. However, when someone is interested in certain quantities of interest, then the built-in energy error estimator may not be optimal and may overestimate the actual error. This phenomenon is observed in the accomplished numerical experiments. The authors are constructing a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method for second-order elliptic problems. In the beginning of the paper, they describe the first-order least-squares method and state known a priori error estimates. Then, they are deriving an a posteriori error estimator based on the residuals of the least-squares finite element solution. It is proven that an a posteriori error estimator yields the global upper bounds and the local lower bounds on the error of interest. Finally, some numerical results are presented to show the actual performance of the obtained estimator. The effectivity indices computed from numerical examples show that constructed error estimator is asymptotically exact. According to the author's knowledge, this is the first result providing an a posteriori error estimator different from the usual built-in energy error indicators.
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65N50
zbMATH DE Number: 5804885
zbMATH Keywords: least square method
zbMATH Keywords: finite element methods
zbMATH Keywords: a posteriori error estimators
zbMATH Keywords: adaptive mesh refinement
zbMATH Keywords: numerical experiments
reviewed by: Tzvetan Semerdjiev
MaRDI profile type: Publication
full work available at URL: j.cam.2010.06.004
OpenAlex ID: W1988190368
cites work: A posteriori error estimation in finite element analysis
cites work: Error Estimates for Adaptive Finite Element Computations
cites work: Some A Posteriori Error Estimators for Elliptic Partial Differential Equations
cites work: Adaptive finite element methods with convergence rates
cites work: A posteriori error estimate for the mixed finite element method
cites work: Convergence of Adaptive Finite Element Methods
cites work: Q4886148
cites work: Local error estimates and adaptive refinement for first-order system least squares (FOSLS)
cites work: Local error estimation and adaptive remeshing scheme for least-squares mixed finite elements
cites work: An Adaptive Least‐Squares Mixed Finite Element Method for Elasto‐Plasticity
cites work: A Posteriori Least-Squares Finite Element Error Analysis for the Navier–Stokes Equations
cites work: Weak coupling of solutions of first-order least-squares method
cites work: First-Order System Least Squares for Second-Order Partial Differential Equations: Part I
cites work: Least-Squares Mixed Finite Elements for Second-Order Elliptic Problems
cites work: Least Squares Methods for Elliptic Systems
cites work: A least-squares approach based on a discrete minus one inner product for first order systems
cites work: Least-quares for second-order elliptic problems
cites work: First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity
cites work: Least-squares mixed finite element methods for non-selfadjoint elliptic problems. I: Error estimates
cites work: Q4300595
cites work: Q3890207
cites work: Q4136469
cites work: The $L^2$ Norm Error Estimates for the Div Least‐Squares Method
cites work: Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary Conditions
cites work: A Convergent Adaptive Algorithm for Poisson’s Equation
DOI: 10.1016/J.CAM.2010.06.004
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description: scientific article; zbMATH DE number 5804885