Previous article Next article Full AccessSIGESThttps://doi.org/10.1137/SIREAD000044000004000629000001BibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis issue's SIGEST paper brings together two prominent themes in numerical analysis---finite element methods and adaptivity.Some date the ideas of the finite element method to the Hurwitz--Courant book of 1922; the first mathematical application of the finite element method is often attributed to Richard Courant's 1943 work on solving a torsion problem using piecewise polynomials. Engineers independently began using finite element techniques during the 1950s, primarily for structural analysis, and the name "finite element method" was coined in 1960 by Ray Clough, a professor of structural engineering at the University of California, Berkeley. Since then, finite element methods and their variations have been widely studied and generalized, becoming in the process a solution technique that is almost ubiquitous in scienceand engineering.The concept of an adaptive numerical method---meaning, broadly speaking, one that uses information acquired while solving a problem to shape the solution procedure---is similarly woven into today's scientific computing. Adaptive methods are especially popular for problems containing singularities, sharp boundaries, or disparate scales. The buzzword "multiscale" frequently appears in recent characterizations of the most challenging problems, and is naturally linked with adaptivity.In 1978, the combination of these ideas---adaptivity and the finite element method---began with the seminal paper "Error Estimates for Adaptive Finite Element Computations," by Ivo Babuska and Werner Rheinboldt [SIAM Journal on Numerical Analysis, 15 (1978), pp.~736--754]. Thus it is particularly appropriate that this issue's SIGEST paper from that same journal, "Convergence of adaptive finite element methods," by Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, also involves adaptive finite element methods.As the authors observe, adaptive methods can succeed only with good a~posteriori error estimates to help guide changes in the mesh. This paper tackles the mostly open problem of analyzing convergence in more than one dimension for the adaptive method paradigm of repeated solution, error estimation, and refinement. Based on a measure called data oscillation, which captures information that can be missed in averaging, the authors present an algorithm and prove its fundamental error reduction property. The nonexpert reader will find a clear introduction to the a posteriori error estimates on which adaptive finite elements are founded, followed by presentation of a new algorithm and a proof of its error reduction property. Many illuminating numerical examples are presented throughout, and two of the many colorful figures appear on this issue's cover.We thank the authors for their contribution to SIAM Review. Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 44, Issue 4| 2002SIAM Review History Published online:04 August 2006 InformationCopyright © 2002 Society for Industrial and Applied Mathematics Article & Publication DataArticle DOI:10.1137/SIREAD000044000004000629000001Article page range:pp. 629-629ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics