Computer Methods in Applied Mechanics and Engineering
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A posteriori error estimation and adaptive mesh-refinement techniques
ICCAM'92 Proceedings of the fifth international conference on Computational and applied mathematics
A posteriori error estimates for boundary element methods
Mathematics of Computation
Error estimators for nonconforming finite element approximations of the Stokes problem
Mathematics of Computation
Mathematics of Computation
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
Adaptive Boundary Element Methods for Some First Kind Integral Equations
SIAM Journal on Numerical Analysis
A posteriori error estimate for the mixed finite element method
Mathematics of Computation
An a posteriori error estimate for a first-kind integral equation
Mathematics of Computation
SIAM Journal on Numerical Analysis
Data Oscillation and Convergence of Adaptive FEM
SIAM Journal on Numerical Analysis
An Adaptive Uzawa FEM for the Stokes Problem: Convergence without the Inf-Sup Condition
SIAM Journal on Numerical Analysis
Local problems on stars: a posteriori error estimators, convergence, and performance
Mathematics of Computation
Residual-based a posteriori error estimate for hypersingular equation on surfaces
Numerische Mathematik
Adaptive Finite Element Methods with convergence rates
Numerische Mathematik
A unifying theory of a posteriori finite element error control
Numerische Mathematik
Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs
SIAM Journal on Numerical Analysis
Optimality of a Standard Adaptive Finite Element Method
Foundations of Computational Mathematics
A unifying theory of a posteriori error control for nonconforming finite element methods
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Linear Convergence of an Adaptive Finite Element Method for the $p$-Laplacian Equation
SIAM Journal on Numerical Analysis
An Optimal Adaptive Finite Element Method for the Stokes Problem
SIAM Journal on Numerical Analysis
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
SIAM Journal on Numerical Analysis
Convergence and optimal complexity of adaptive finite element eigenvalue computations
Numerische Mathematik
Convergence of a standard adaptive nonconforming finite element method with optimal complexity
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A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity
SIAM Journal on Numerical Analysis
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SIAM Journal on Numerical Analysis
Decay rates of adaptive finite elements with Dörfler marking
Numerische Mathematik
An oscillation-free adaptive FEM for symmetric eigenvalue problems
Numerische Mathematik
Quasi-Optimality of Adaptive Nonconforming Finite Element Methods for the Stokes Equations
SIAM Journal on Numerical Analysis
Estimator reduction and convergence of adaptive BEM
Applied Numerical Mathematics
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Journal of Computational and Applied Mathematics
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This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators. Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.