Adaptive refinement based on asymptotic expansions of finite element solutions for node insertion in 1d

We consider refinement of finite element discretizations by splitting nodes along edges. For this process, we derive asymptotic expansions of Galerkin solutions of linear second‐order elliptic equations. Thereby, we calculate a topological derivative w.r.t. node insertion for functionals such as the...

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Published inMitteilungen der Gesellschaft für Angewandte Mathematik und Mechanik Vol. 35; no. 2; pp. 175 - 190
Main Authors Friederich, Jan, Leugering, Günther, Steinmann, Paul
Format Journal Article
LanguageEnglish
Published Berlin WILEY-VCH Verlag 01.11.2012
WILEY‐VCH Verlag
Wiley Subscription Services, Inc
Online AccessGet full text
ISSN0936-7195
1522-2608
DOI10.1002/gamm.201210012

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Summary:We consider refinement of finite element discretizations by splitting nodes along edges. For this process, we derive asymptotic expansions of Galerkin solutions of linear second‐order elliptic equations. Thereby, we calculate a topological derivative w.r.t. node insertion for functionals such as the total potential energy, minimization of which decreases the approximation error in the energy norm. Hence, these sensitivities can be used to define indicators for local h ‐refinement. Our results suggest that this procedure leads to an efficient adaptive refinement method. This presentation is concerned with a model problem in 1d. The extension of this concept to higher dimensions will be the subject of forthcoming publications (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Bibliography:ark:/67375/WNG-P6VGKSXH-J
ArticleID:GAMM201210012
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ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0936-7195
1522-2608
DOI:10.1002/gamm.201210012