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Adaptive finite element methods

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  04 September 2024

Andrea Bonito ,
Claudio Canuto ,
Ricardo H. Nochetto Open the ORCID record for Ricardo H. Nochetto [Opens in a new window]  and
Andreas Veeser
Andrea Bonito
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA E-mail: bonito@tamu.edu
Claudio Canuto
Affiliation:
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail: claudio.canuto@polito.it
Ricardo H. Nochetto
Affiliation:
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA E-mail: rhn@umd.edu
Andreas Veeser
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy E-mail: andreas.veeser@unimi.it
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Abstract

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This is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second-order elliptic PDEs and dimension d > 1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs, and beyond coercive problems to inf-sup stable AFEMs.

MSC classification

Primary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N50: Mesh generation and refinement
Secondary: 65N35: Spectral, collocation and related methods
Type
Research Article
Information
Acta Numerica , Volume 33 , July 2024 , pp. 163 - 485
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

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