Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/oxford-university-press/quasioptimal-cardinality-of-afem-driven-by-nonresidual-estimators-uu363rm0LV
References (48)
-
P. Ciarlet (2002)
The finite element method for elliptic problems, 40
-
Zhiming Chen, Jianchao Feng (2004)
An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problemsMath. Comput., 73
-
R. Nochetto, K. Siebert, A. Veeser (2009)
Theory of adaptive finite element methods: An introduction -
A. Bonito, R. Nochetto (2010)
Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin MethodSIAM J. Numer. Anal., 48
-
P. Bassanini, A. Elcrat (1997)
Elliptic Partial Differential Equations of Second Order -
L. Diening, C. Kreuzer (2008)
Linear Convergence of an Adaptive Finite Element Method for the p-Laplacian EquationSIAM J. Numer. Anal., 46
-
O. Zienkiewicz, J. Zhu (1987)
A simple error estimator and adaptive procedure for practical engineerng analysisInternational Journal for Numerical Methods in Engineering, 24
-
Mirosław Krzyżański (1971)
Partial differential equations of second order -
M. Ainsworth, J. Oden (2000)
A Posteriori Error Estimation in Finite Element Analysis -
L. Scott, Shangyou Zhang (2010)
FINITE ELEMENT INTERPOLATION OF NONSMOOTH FUNCTIONS -
C. Carstensen, R. Verfürth (1999)
Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element MethodsSIAM Journal on Numerical Analysis, 36
-
C. Traxler (1997)
An algorithm for adaptive mesh refinement inn dimensionsComputing, 59
-
R. Verfürth (1994)
A posteriori error estimation and adaptive mesh-refinement techniquesJournal of Computational and Applied Mathematics, 50
-
I. Babuska, A. Miller (1987)
A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimatorApplied Mechanics and Engineering, 61
-
D. Braess, J. Schöberl (2007)
Equilibrated residual error estimator for edge elementsMath. Comput., 77
-
L. Ridgway, Scott And, Shangyou Zhang (1990)
Finite element interpolation of nonsmooth functions satisfying boundary conditionsMathematics of Computation, 54
-
R. Stevenson (2007)
Optimality of a Standard Adaptive Finite Element MethodFoundations of Computational Mathematics, 7
-
P. Binev, W. Dahmen, R. DeVore, P. Petrushev (2002)
Approximation Classes for Adaptive MethodsSerdica. Mathematical Journal, 28
-
S. Prudhomme, F. Nobile, L. Chamoin, J. Oden (2004)
Analysis of a subdomain‐based error estimator for finite element approximations of elliptic problemsNumerical Methods for Partial Differential Equations, 20
-
T. Strouboulis, Lin Zhang, D. Wang, I. Babuska (2006)
A posteriori error estimation for generalized finite element methodsComputer Methods in Applied Mechanics and Engineering, 195
-
E. Bänsch (1991)
Local mesh refinement in 2 and 3 dimensionsIMPACT Comput. Sci. Eng., 3
-
C. Kreuzer, K. Siebert (2011)
Decay rates of adaptive finite elements with Dörfler markingNumerische Mathematik, 117
-
A. Veeser (2002)
Convergent adaptive finite elements for the nonlinear LaplacianNumerische Mathematik, 92
-
C. Carstensen (2003)
All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliableMath. Comput., 73
-
P. Morin, R. Nochetto, K. Siebert (2002)
Convergence of Adaptive Finite Element MethodsSIAM Rev., 44
-
Igor Kossaczký (1994)
A recursive approach to local mesh refinement in two and three dimensionsJournal of Computational and Applied Mathematics, 55
-
N. Parés, P. Díez, A. Huerta (2006)
Subdomain-based flux-free a posteriori error estimatorsComputer Methods in Applied Mechanics and Engineering, 195
-
W. Dörfler (1996)
A convergent adaptive algorithm for Poisson's equationSIAM Journal on Numerical Analysis, 33
-
K. Mekchay, R. Nochetto (2005)
Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEsSIAM J. Numer. Anal., 43
-
P. Morin, K. Siebert, A. Veeser (2008)
A BASIC CONVERGENCE RESULT FOR CONFORMING ADAPTIVE FINITE ELEMENTSMathematical Models and Methods in Applied Sciences, 18
-
P. Morin, R. Nochetto, K. Siebert (2000)
Data Oscillation and Convergence of Adaptive FEMSIAM J. Numer. Anal., 38
-
R. Bank, Jinchao Xu (2003)
Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with SuperconvergenceSIAM J. Numer. Anal., 41
-
I. Babuska, T. Strouboulis (2001)
The finite element method and its reliability -
C. Carstensen, S. Funken (1999)
Fully Reliable Localized Error Control in the FEMSIAM J. Sci. Comput., 21
-
Alan WEISERAbstract (1985)
Some a posteriori error estimators for elliptic partial differential equationsMathematics of Computation, 44
-
A. Cohen, R. DeVore, R. Nochetto (2012)
Convergence Rates of AFEM with H−1 DataFoundations of Computational Mathematics, 12
-
R. Bank, Jinchao Xu (2003)
Asymptotically Exact A Posteriori Error Estimators, Part II: General Unstructured GridsSIAM J. Numer. Anal., 41
-
G. Kunert, R. Verfürth (2000)
Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshesNumerische Mathematik, 86
-
R. Rodríguez (1994)
Some remarks on Zienkiewicz‐Zhu estimatorNumerical Methods for Partial Differential Equations, 10
-
P. Binev, W. Dahmen, R. DeVore (2004)
Adaptive Finite Element Methods with convergence ratesNumerische Mathematik, 97
-
P. Morin, R. Nochetto, K. Siebert (2003)
Local problems on stars: A posteriori error estimators, convergence, and performanceMath. Comput., 72
-
R. Stevenson (2008)
The completion of locally refined simplicial partitions created by bisectionMath. Comput., 77
-
A. Schmidt, K. Siebert (2005)
Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA, 42
-
F. Bornemann, B. Erdmann, R. Kornhuber (1996)
A posteriori error estimates for elliptic problems in two and three space dimensionsSIAM Journal on Numerical Analysis, 33
-
W. Mitchell (1989)
A comparison of adaptive refinement techniques for elliptic problemsACM Trans. Math. Softw., 15
-
J. Maubach, Finite Barker, Orlando, R. Bank, Siam Welfert (2017)
Local bisection refinement for $n$-simplicial grids generated by reflection -
K. Siebert (2011)
A convergence proof for adaptive finite elements without lower boundIma Journal of Numerical Analysis, 31
-
J. Cascón, C. Kreuzer, R. Nochetto, K. Siebert (2008)
Quasi-Optimal Convergence Rate for an Adaptive Finite Element MethodSIAM J. Numer. Anal., 46
- Publisher
- Oxford University Press
- Copyright
- The author 2011. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0272-4979
- eISSN
- 1464-3642
- DOI
- 10.1093/imanum/drr014
- Publisher site
- See Article on Publisher Site
Abstract
We examine adaptive finite element methods (AFEMs) with any polynomial degree satisfying rather general assumptions on the a posteriori error estimators. We show that several nonresidual estimators satisfy these assumptions. We design an AFEM with single Drfler marking for the sum of error estimator and oscillation, prove a contraction property for the so-called total error, namely the scaled sum of energy error and oscillation, and derive quasioptimal decay rates for the total error. We also re-examine the definition and role of oscillation in the approximation class.
Journal
IMA Journal of Numerical Analysis – Oxford University Press
Published: Jan 28, 2012
Keywords: error reduction convergence optimal cardinality adaptive algorithm