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Primer of Adaptive Finite Element Methods

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Book cover Multiscale and Adaptivity: Modeling, Numerics and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2040))

Abstract

Adaptive finite element methods (AFEM) are a fundamental numerical instrument in science and engineering to approximate partial differential equations. In the 1980s and 1990s a great deal of effortwas devoted to the design of a posteriori error estimators, following the pioneering work of Babu?ska. These are computable quantities, depending on the discrete solution(s) and data, that can be used to assess the approximation quality and improve it adaptively. Despite their practical success, adaptive processes have been shown to converge, and to exhibit optimal cardinality, only recently for dimension d > 1 and for linear elliptic PDE. These series of lectures presents an up-to-date discussion of AFEM encompassing the derivation of upper and lower a posteriori error bounds for residual-type estimators, including a critical look at the role of oscillation, the design of AFEM and its basic properties, as well as a complete discussion of convergence, contraction property and quasi- optimal cardinality of AFEM

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Authors and Affiliations

  1. Department of Mathematics and Institute of Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  2. Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, I-20133, Milano, Italy

    Andreas Veeser

Authors
  1. Ricardo H. Nochetto
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  2. Andreas Veeser
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Correspondence to Ricardo H. Nochetto .

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Nochetto, R.H., Veeser, A. (2011). Primer of Adaptive Finite Element Methods. In: Multiscale and Adaptivity: Modeling, Numerics and Applications. Lecture Notes in Mathematics(), vol 2040. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24079-9_3

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