Nonlinear functional analysis and its applications
Nonlinear functional analysis and its applications
Adaptive finite element methods in computational mechanics
Computer Methods in Applied Mechanics and Engineering - Special issue on reliability in computational mechanics
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods
SIAM Journal on Numerical Analysis
Computational Differential Equations
Computational Differential Equations
Convergence of Adaptive Finite Element Methods
SIAM Review
Adaptive Finite Element Methods with convergence rates
Numerische Mathematik
Optimality of a Standard Adaptive Finite Element Method
Foundations of Computational Mathematics
Framework for the A Posteriori Error Analysis of Nonconforming Finite Elements
SIAM Journal on Numerical Analysis
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
SIAM Journal on Numerical Analysis
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The a priori convergence of finite element methods is based on the density property of the sequence of finite element spaces which essentially assumes a quasi-uniform mesh-refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. Adaptive finite element methods (AFEMs) automatically refine exclusively wherever their refinement indication suggests to do so and consequently leave out refinements at other locations. In other words, the density property is violated on purpose and the a priori convergence is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh in many practical examples accompanied by smaller computational costs; the disadvantage is that the desirable convergence property is not guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not theoretically justified from the start that the adaptive mesh-refinement will generate an accurate solution at all. In order to foster the development of a convergence theory and improved design of AFEMs in computational engineering and sciences, this paper describes a particular version of an AFEM and analyses convergence results for three model problems in computational mechanics: linear elastic material (A), nonlinear monotone elastic material (B), and Hencky elastoplastic material (C). It establishes conditions sufficient for error-reduction in (A), for energy-reduction in (B), and eventually for strong convergence of the stress field in (C) in the presence of small hardening.