PUMA publications for /user/mhartmann/Henckyhttps://puma.ub.uni-stuttgart.de/user/mhartmann/HenckyPUMA RSS feed for /user/mhartmann/Hencky2019-06-17T10:59:50+02:00Convergence of Adaptive Finite Element Methodshttps://puma.ub.uni-stuttgart.de/bibtex/26577a49d6be6cb03cd99ed10f5af9860/mhartmannmhartmann2018-07-20T10:54:15+02:00Adaptive Convergence Convexity; FEM; Hencky Linear algorithm; elasticity; elastoplasticity; vorlaeufig <span class="authorEditorList"><span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Pedro Morin" itemprop="url" href="/person/182d27fed5937d15b2a4642665dea175f/author/0"><span itemprop="name">P. Morin</span></a></span>, <span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Ricardo H. Nochetto" itemprop="url" href="/person/182d27fed5937d15b2a4642665dea175f/author/1"><span itemprop="name">R. Nochetto</span></a></span>, und <span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Kunibert G. Siebert" itemprop="url" href="/person/182d27fed5937d15b2a4642665dea175f/author/2"><span itemprop="name">K. Siebert</span></a></span>. </span><span itemtype="http://schema.org/PublicationIssue" itemscope="itemscope" itemprop="isPartOf"><span itemtype="http://schema.org/Periodical" itemscope="itemscope" itemprop="isPartOf"><span itemprop="name"><em>SIAM Review</em></span></span> <em><span itemtype="http://schema.org/PublicationVolume" itemscope="itemscope" itemprop="isPartOf"><span itemprop="volumeNumber">44 </span></span>(<span itemprop="issueNumber">4</span>):
<span itemprop="pagination">631-658</span></em> </span>(<em><span>2002<meta content="2002" itemprop="datePublished"/></span></em>)Fri Jul 20 10:54:15 CEST 2018SIAM Review4631-658Convergence of Adaptive Finite Element Methods442002Adaptive Convergence Convexity; FEM; Hencky Linear algorithm; elasticity; elastoplasticity; vorlaeufig 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.