Hostname: page-component-74d7c59bfc-6nqbt Total loading time: 0 Render date: 2026-02-02T22:00:42.375Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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

References

Adams, R. A. and Fournier, J. J. F. (2003), Sobolev Spaces, Vol. 140 of Pure and Applied Mathematics (Amsterdam), second edition, Elsevier/Academic Press.Google Scholar
Ainsworth, M. (2010), A framework for obtaining guaranteed error bounds for finite element approximations, J. Comput. Appl. Math. 234, 2618–2632.10.1016/j.cam.2010.01.037CrossRefGoogle Scholar
Ainsworth, M. and Oden, J. T. (2000), A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics (New York), Wiley-Interscience.Google Scholar
Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D. (2002), Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39, 1749–1779.10.1137/S0036142901384162CrossRefGoogle Scholar
Babuška, I. (1971), Error-bounds for finite element method, Numer. Math. 16, 322–333.10.1007/BF02165003CrossRefGoogle Scholar
Babuška, I. (1971), The finite element method for elliptic differential equations, in Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Hubbard, B., ed.), Academic Press, pp. 69–106.10.1016/B978-0-12-358502-8.50007-4CrossRefGoogle Scholar
Babuška, I. and Aziz, A. K. (1972), Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Aziz, A. K., ed.), Academic Press, pp. 1–359.Google Scholar
Babuška, I. and Miller, A. (1987), A feedback finite element method with a posteriori error estimation, I: The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg 61, 1–40.10.1016/0045-7825(87)90114-9CrossRefGoogle Scholar
Babuška, I. and Rheinboldt, W. C. (1978), Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15, 736–754.10.1137/0715049CrossRefGoogle Scholar
Babuška, I., Kellogg, R. B. and Pitkäranta, J. (1979), Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math. 33, 447–471.10.1007/BF01399326CrossRefGoogle Scholar
Bänsch, E., Morin, P. and Nochetto, R. H. (2002), An adaptive Uzawa FEM for the Stokes problem: Convergence without the inf-sup condition, SIAM J. Numer. Anal. 40, 1207–1229.10.1137/S0036142901392134CrossRefGoogle Scholar
da Veiga, L. Beirão, Canuto, C., Nochetto, R. H., Vacca, G. and Verani, M. (2023), Adaptive VEM: Stabilization-free a posteriori error analysis and contraction property, SIAM J. Numer. Anal. 61, 457–494.10.1137/21M1458740CrossRefGoogle Scholar
da Veiga, L. Beirão, Canuto, C., Nochetto, R. H., Vacca, G. and Verani, M. (2024), Adaptive VEM for variable data: Convergence and optimality, IMA J. Numer. Anal. Available at https://doi.org/10.1093/imanum/drad085.CrossRefGoogle Scholar
Bergh, J. and Löfström, J. (1976), Interpolation Spaces: An Introduction, Vol. 223 of Grundlehren der mathematischen Wissenschaften, Springer.10.1007/978-3-642-66451-9CrossRefGoogle Scholar
Bernardi, C. and Girault, V. (1998), A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35, 1893–1916.10.1137/S0036142995293766CrossRefGoogle Scholar
Bhatia, R. (2000), Pinching, trimming, truncating, and averaging of matrices, Amer. Math. Monthly 107, 602–608.10.1080/00029890.2000.12005245CrossRefGoogle Scholar
Binev, P. (2018), Tree approximation for hp-adaptivity, SIAM J. Numer. Anal. 56, 3346–3357.10.1137/18M1175070CrossRefGoogle Scholar
Binev, P. and DeVore, R. (2004), Fast computation in adaptive tree approximation, Numer. Math. 97, 193–217.10.1007/s00211-003-0493-6CrossRefGoogle Scholar
Binev, P., Dahmen, W. and DeVore, R. (2004), Adaptive finite element methods with convergence rates, Numer. Math. 97, 219–268.10.1007/s00211-003-0492-7CrossRefGoogle Scholar
Binev, P., Dahmen, W., DeVore, R. and Petrushev, P. (2002), Approximation classes for adaptive methods, Serdica Math. J. 28, 391–416.Google Scholar
Binev, P., Fierro, F. and Veeser, A. (2023), Near-best adaptive approximation on conforming meshes, Constr. Approx. 57, 327–349.10.1007/s00365-022-09612-2CrossRefGoogle Scholar
Blechta, J., Málek, J. and Vohralík, M. (2020), Localization of the ${W}_q^{-1}$ -norm for local a posteriori efficiency, IMA J. Numer. Anal. 40, 914–950.10.1093/imanum/drz002CrossRefGoogle Scholar
Boffi, D., Brezzi, F. and Fortin, M. (2013), Mixed Finite Element Methods and Applications, Vol. 44 of Springer Series in Computational Mathematics, Springer.10.1007/978-3-642-36519-5CrossRefGoogle Scholar
Bonito, A. and Devaud, D. (2015), Adaptive finite element methods for the Stokes problem with discontinuous viscosity, Math. Comp. 84, 2137–2162.10.1090/S0025-5718-2015-02935-4CrossRefGoogle Scholar
Bonito, A. and Nochetto, R. H. (2010), Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal. 48, 734–771.10.1137/08072838XCrossRefGoogle Scholar
Bonito, A., Cascón, J. M., Mekchay, K., Morin, P. and Nochetto, R. H. (2016), High-order AFEM for the Laplace–Beltrami operator: Convergence rates, Found. Comput. Math. 16, 1473–1539.10.1007/s10208-016-9335-7CrossRefGoogle Scholar
Bonito, A., Cascón, J. M., Morin, P. and Nochetto, R. H. (2013a), AFEM for geometric PDE: The Laplace–Beltrami operator, in Analysis and Numerics of Partial Differential Equations (Brezzi, F. et al., eds), Springer, pp. 257–306.10.1007/978-88-470-2592-9_15CrossRefGoogle Scholar
Bonito, A., DeVore, R. A. and Nochetto, R. H. (2013b), Adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Numer. Anal. 51, 3106–3134.10.1137/130905757CrossRefGoogle Scholar
Bonito, A., Nochetto, R. H. and Ntogkas, D. (2021), DG approach to large bending plate deformations with isometry constraint, Math. Models Methods Appl. Sci. 31, 133–175.10.1142/S0218202521500044CrossRefGoogle Scholar
Bornemann, F., Erdmann, B. and Kornhuber, R. (1996), A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal. 33, 1188–1204.10.1137/0733059CrossRefGoogle Scholar
Braess, D. (2007), Finite Elements: Theory, Fast solvers, and Applications in Elasticity Theory, third edition, Cambridge University Press.10.1017/CBO9780511618635CrossRefGoogle Scholar
Braess, D., Pillwein, V. and Schöberl, J. (2009), Equilibrated residual error estimates are p-robust, Comput. Methods Appl. Mech. Engrg 198, 1189–1197.10.1016/j.cma.2008.12.010CrossRefGoogle Scholar
Brenner, S. C. (2003), Poincaré–Friedrichs inequalities for piecewise H 1 functions, SIAM J. Numer. Anal. 41, 306–324.10.1137/S0036142902401311CrossRefGoogle Scholar
Brenner, S. C. and Scott, L. R. (2008), The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics, third edition, Springer.10.1007/978-0-387-75934-0CrossRefGoogle Scholar
Brezzi, F. (1974), On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Fr. Autom. Inform. Rech. Opér. Anal. Numér. 8, 129–151.Google Scholar
Brezzi, F., Douglas, J. Jr and Marini, L. D. (1985), Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47, 217–235.10.1007/BF01389710CrossRefGoogle Scholar
Brezzi, F., Douglas, J. Jr, Fortin, M. and Marini, L. D. (1987), Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21, 581–604.10.1051/m2an/1987210405811CrossRefGoogle Scholar
Brezzi, F., Manzini, G., Marini, D., Pietra, P. and Russo, A. (2000), Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations 16, 365–378.10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y3.0.CO;2-Y>CrossRefGoogle Scholar
Canuto, C. and Fassino, D. (2023), Higher-order adaptive virtual element methods with contraction properties, Math. Engrg 5, 1–33.10.3934/mine.2023101CrossRefGoogle Scholar
Carroll, R., Duff, G., Friberg, J., Gobert, J., Grisvard, P., Nečas, J. and Seeley, R. (1966), Equations aux Dérivées Partielles, Vol. 19 of Séminaire de mathématiques supérieures, Les Presses de l’Université de Montréal.Google Scholar
Carstensen, C. (1997), A posteriori error estimate for the mixed finite element method, Math. Comp. 66, 465–476.10.1090/S0025-5718-97-00837-5CrossRefGoogle Scholar
Carstensen, C., Feischl, M., Page, M. and Praetorius, D. (2014), Axioms of adaptivity, Comput. Math. Appl. 67, 1195–1253.10.1016/j.camwa.2013.12.003CrossRefGoogle ScholarPubMed
Cascón, J. M. and Nochetto, R. H. (2012), Quasioptimal cardinality of AFEM driven by nonresidual estimators, IMA J. Numer. Anal. 32, 1–29.10.1093/imanum/drr014CrossRefGoogle Scholar
Cascón, J. M., Kreuzer, C., Nochetto, R. H. and Siebert, K. G. (2008), Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46, 2524–2550.10.1137/07069047XCrossRefGoogle Scholar
Ciarlet, P. G. (2002), The Finite Element Method for Elliptic Problems, Vol. 40 of Classics in Applied Mathematics, SIAM. Reprint of the 1978 original.10.1137/1.9780898719208CrossRefGoogle Scholar
Clément, P. (1975), Approximation by finite element functions using local regularization, Rev. Fr. Autom. Inform. Rech. Opér. Anal. Numér. 9, 77–84.Google Scholar
Cohen, A., DeVore, R. and Nochetto, R. H. (2012), Convergence rates of AFEM with H −1 data, Found. Comput. Math. 12, 671–718.10.1007/s10208-012-9120-1CrossRefGoogle Scholar
Daniel, P. and Vohralík, M. (2023), Guaranteed contraction of adaptive inexact hp-refinement strategies with realistic stopping criteria, ESAIM Math. Model. Numer. Anal. 57, 329–366.10.1051/m2an/2022082CrossRefGoogle Scholar
Davies, E. B. (1988), Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. ( 2) 37, 148–157.10.1112/jlms/s2-37.121.148CrossRefGoogle Scholar
Dekel, S. and Leviatan, D. (2004), Whitney estimates for convex domains with applications to multivariate piecewise polynomial approximation, Found. Comput. Math. 4, 345–368.10.1007/s10208-004-0096-3CrossRefGoogle Scholar
DeVore, R. A. (1998), Nonlinear approximation, Acta Numer. 7, 51–150.10.1017/S0962492900002816CrossRefGoogle Scholar
DeVore, R. A. and Lorentz, G. G. (1993), Constructive Approximation, Vol. 303 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer.10.1007/978-3-662-02888-9CrossRefGoogle Scholar
DeVore, R. A. and Popov, V. A. (1988), Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305, 397–414.10.1090/S0002-9947-1988-0920166-3CrossRefGoogle Scholar
Diening, L., Gehring, L. and Storn, J. (2023), Adaptive mesh refinement for arbitrary initial triangulations. Available at arXiv:2304.02674.Google Scholar
Diening, L., Kreuzer, C. and Stevenson, R. (2016), Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16, 33–68.10.1007/s10208-014-9236-6CrossRefGoogle Scholar
Ditzian, Z. (1988), On the Marchaud-type inequality, Proc. Amer. Math. Soc. 103, 198–202.10.1090/S0002-9939-1988-0938668-8CrossRefGoogle Scholar
Dörfler, W. (1996), A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33, 1106–1124.10.1137/0733054CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T. (1988), Linear Operators, part II, Spectral Theory: Selfadjoint Operators in Hilbert Space, Wiley Classics Library, Wiley. Reprint of the 1963 original.Google Scholar
Dupont, T. and Scott, R. (1980), Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34, 441–463.10.1090/S0025-5718-1980-0559195-7CrossRefGoogle Scholar
Ern, A., Smears, I. and Vohralík, M. (2017), Discrete p-robust H(div)-liftings and a posteriori estimates for elliptic problems with H −1 source terms, Calcolo 54, 1009–1025.10.1007/s10092-017-0217-4CrossRefGoogle Scholar
Evans, L. C. (2010), Partial Differential Equations, Vol. 19 of Graduate Studies in Mathematics, second edition, American Mathematical Society.Google Scholar
Faermann, B. (2000), Localization of the Aronszajn–Slobodeckij norm and application to adaptive boundary element methods I: The two-dimensional case, IMA J. Numer. Anal. 20, 203–234.10.1093/imanum/20.2.203CrossRefGoogle Scholar
Faermann, B. (2002), Localization of the Aronszajn–Slobodeckij norm and application to adaptive boundary element methods II: The three-dimensional case, Numer. Math. 92, 467–499.10.1007/s002110100319CrossRefGoogle Scholar
Feischl, M. (2019), Optimality of a standard adaptive finite element method for the Stokes problem, SIAM J. Numer. Anal. 57, 1124–1157.10.1137/17M1153170CrossRefGoogle Scholar
Feischl, M. (2022), Inf-sup stability implies quasi-orthogonality, Math. Comp. 91, 2059–2094.10.1090/mcom/3748CrossRefGoogle Scholar
Fierro, F. and Veeser, A. (2003), A posteriori error estimators for regularized total variation of characteristic functions, SIAM J. Numer. Anal. 41, 2032–2055.10.1137/S0036142902408283CrossRefGoogle Scholar
Funken, S., Praetorius, D. and Wissgott, P. (2011), Efficient implementation of adaptive P1-FEM in Matlab, Comput. Methods Appl. Math. 11, 460–490.10.2478/cmam-2011-0026CrossRefGoogle Scholar
Galdi, G. P. (1994), An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Vol. I, Linearized Steady Problems, Vol. 38 of Springer Tracts in Natural Philosophy, Springer.Google Scholar
Gaspoz, F. D. and Morin, P. (2014), Approximation classes for adaptive higher order finite element approximation, Math. Comp. 83, 2127–2160.10.1090/S0025-5718-2013-02777-9CrossRefGoogle Scholar
Gaspoz, F. D. and Morin, P. (2017), Errata to ‘Approximation classes for adaptive higher order finite element approximation’, Math. Comp. 86, 1525–1526.10.1090/mcom/3243CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S. (2001), Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer. Reprint of the 1998 edition.10.1007/978-3-642-61798-0CrossRefGoogle Scholar
Grisvard, P. (1985), Elliptic Problems in Nonsmooth Domains, Vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program).Google Scholar
Grisvard, P. (2011), Elliptic Problems in Nonsmooth Domains, Vol. 69 of Classics in Applied Mathematics, SIAM. Reprint of the 1985 original.10.1137/1.9781611972030CrossRefGoogle Scholar
Hackbusch, W. (1992), Elliptic Differential Equations: Theory and Numerical Treatment, Vol. 18 of Springer Series in Computational Mathematics, Springer.10.1007/978-3-642-11490-8CrossRefGoogle Scholar
Houston, P., Schötzau, D. and Wihler, T. P. (2004), Mixed hp-discontinuous Galerkin finite element methods for the Stokes problem in polygons, in Numerical Mathematics and Advanced Applications: Proceedings of the 5th European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2003), Springer, pp. 493–501.10.1007/978-3-642-18775-9_46CrossRefGoogle Scholar
Houston, P., Schötzau, D. and Wihler, T. P. (2007), Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci. 17, 33–62.10.1142/S0218202507001826CrossRefGoogle Scholar
Jerison, D. and Kenig, C. E. (1995), The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130, 161–219.10.1006/jfan.1995.1067CrossRefGoogle Scholar
Kahane, J.-P. (1961), Teoria Constructiva de Funciones, Universidad de Buenos Aires.Google Scholar
Karakashian, O. A. and Pascal, F. (2007), Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems, SIAM J. Numer. Anal. 45, 641–665.10.1137/05063979XCrossRefGoogle Scholar
Kellogg, R. B. (1974/75), On the Poisson equation with intersecting interfaces, Applicable Anal. 4, 101–129.10.1080/00036817408839086CrossRefGoogle Scholar
Kreuzer, C. and Siebert, K. G. (2011), Decay rates of adaptive finite elements with Dörfler marking, Numer. Math. 117, 679–716.10.1007/s00211-010-0324-5CrossRefGoogle Scholar
Kreuzer, C. and Veeser, A. (2019), Convergence of adaptive finite element methods with error-dominated oscillation, in Numerical Mathematics and Advanced Applications (ENUMATH 2017), Vol. 126 of Lecture Notes in Computational Science and Engineering, Springer, pp. 471–479.10.1007/978-3-319-96415-7_42CrossRefGoogle Scholar
Kreuzer, C. and Veeser, A. (2021), Oscillation in a posteriori error estimation, Numer. Math. 148, 43–78.10.1007/s00211-021-01194-8CrossRefGoogle Scholar
Kreuzer, C., Veeser, A. and Zanotti, P. (2024), Accurate error bounds for finite element methods. In preparation.Google Scholar
Lax, P. D. and Milgram, A. N. (1954), Parabolic equations, in Contributions to the Theory of Partial Differential Equations, Vol. 33 of Annals of Mathematics Studies, Princeton University Press, pp. 167–190.Google Scholar
Leoni, G. (2009), A First Course in Sobolev Spaces, Vol. 105 of Graduate Studies in Mathematics, American Mathematical Society.Google Scholar
Luce, R. and Wohlmuth, B. (2004), A local a posteriori error estimator based on equilibrated fluxes, SIAM J. Numer. Anal. 42, 1394–1414.10.1137/S0036142903433790CrossRefGoogle Scholar
Maubach, J. M. (1995), Local bisection refinement for n-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16, 210–227.10.1137/0916014CrossRefGoogle Scholar
Meyers, N. G. (1963), An L p-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. ( 3) 17, 189–206.Google Scholar
Mitchell, W. F. (1989), A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software 15, 326–347.10.1145/76909.76912CrossRefGoogle Scholar
Morin, P., Nochetto, R. and Siebert, K. (2003), Local problems on stars: A posteriori error estimators, convergence, and performance, Math. Comp. 72, 1067–1097.10.1090/S0025-5718-02-01463-1CrossRefGoogle Scholar
Morin, P., Nochetto, R. H. and Siebert, K. G. (2000), Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38, 466–488.10.1137/S0036142999360044CrossRefGoogle Scholar
Morin, P., Nochetto, R. H. and Siebert, K. G. (2002), Convergence of adaptive finite element methods, SIAM Rev. 44, 631–658. Revised reprint of ‘Data oscillation and convergence of adaptive FEM’.10.1137/S0036144502409093CrossRefGoogle Scholar
Morin, P., Siebert, K. G. and Veeser, A. (2008), A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci. 18, 707–737.10.1142/S0218202508002838CrossRefGoogle Scholar
Nédélec, J.-C. (1980), Mixed finite elements in ℝ3, Numer. Math. 35, 315–341.10.1007/BF01396415CrossRefGoogle Scholar
Nečas, J. (1962), Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa Cl. Sci. ( 3) 16, 305–326.Google Scholar
Nochetto, R. H. and Veeser, A. (2012), Primer of adaptive finite element methods, in Multiscale and Adaptivity: Modeling, Numerics and Applications, Vol. 2040 of Lecture Notes in Mathematics, Springer, pp. 125–225.Google Scholar
Nochetto, R. H., Siebert, K. G. and Veeser, A. (2009), Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation (DeVore, R. and Kunoth, A., eds), Springer, pp. 409–542.10.1007/978-3-642-03413-8_12CrossRefGoogle Scholar
Payne, L. E. and Weinberger, H. F. (1960), An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5, 286–292.10.1007/BF00252910CrossRefGoogle Scholar
Perugia, I. and Schötzau, D. (2003), The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations, Math. Comp. 72, 1179–1214.10.1090/S0025-5718-02-01471-0CrossRefGoogle Scholar
Raviart, P.-A. and Thomas, J. M. (1977), A mixed finite element method for 2-nd order elliptic problems, in Mathematical Aspects of Finite Element Methods, Vol. 606 of Lecture Notes in Mathematics, Springer, pp. 292–315.10.1007/BFb0064470CrossRefGoogle Scholar
Sacchi, R. and Veeser, A. (2006), Locally efficient and reliable a posteriori error estimators for Dirichlet problems, Math. Models Methods Appl. Sci. 16, 319–346.10.1142/S0218202506001170CrossRefGoogle Scholar
Scott, L. R. and Zhang, S. (1990), Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54, 483–493.10.1090/S0025-5718-1990-1011446-7CrossRefGoogle Scholar
Siebert, K. and Veeser, A. (2007), A unilaterally constrained quadratic minimization with adaptive finite elements, SIAM J. Optim. 18, 260–289.10.1137/05064597XCrossRefGoogle Scholar
Siebert, K. G. (2012), Mathematically founded design of adaptive finite element software, in Multiscale and Adaptivity: Modeling, Numerics and Applications, Vol. 2040 of Lecture Notes in Mathematics, Springer, pp. 227–309.Google Scholar
Stevenson, R. (2007), Optimality of a standard adaptive finite element method, Found. Comput. Math. 7, 245–269.10.1007/s10208-005-0183-0CrossRefGoogle Scholar
Stevenson, R. (2008), The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77, 227–241.10.1090/S0025-5718-07-01959-XCrossRefGoogle Scholar
Szyld, D. B. (2006), The many proofs of an identity on the norm of oblique projections, Numer. Algorithms 42, 309–323.10.1007/s11075-006-9046-2CrossRefGoogle Scholar
Tantardini, F., Veeser, A. and Verfürth, R. (2024), Best error localization in the approximation of functionals with piecewise polynomials. In preparation.Google Scholar
Taylor, C. and Hood, P. (1973), A numerical solution of the Navier–Stokes equations using the finite element technique, Int. J. Comput. Fluids 1, 73–100.10.1016/0045-7930(73)90027-3CrossRefGoogle Scholar
Traxler, C. T. (1997), An algorithm for adaptive mesh refinement in n dimensions, Computing 59, 115–137.10.1007/BF02684475CrossRefGoogle Scholar
Triebel, H. (2010), Theory of Function Spaces, Modern Birkhäuser Classics, Birkhäuser/ Springer. Reprint of 1983 edition.Google Scholar
Veeser, A. (2002), Convergent adaptive finite elements for the nonlinear Laplacian, Numer. Math. 92, 743–770.10.1007/s002110100377CrossRefGoogle Scholar
Veeser, A. (2016), Approximating gradients with continuous piecewise polynomial functions, Found. Comput. Math. 16, 723–750.10.1007/s10208-015-9262-zCrossRefGoogle Scholar
Veeser, A. and Verfürth, R. (2009), Explicit upper bounds for dual norms of residuals, SIAM J. Numer. Anal. 47, 2387–2405.10.1137/080738283CrossRefGoogle Scholar
Verfürth, R. (2013), A Posteriori Error Estimation Techniques for Finite Element Methods, Numerical Mathematics and Scientific Computation, Oxford University Press.10.1093/acprof:oso/9780199679423.001.0001CrossRefGoogle Scholar
Xu, J. and Zikatanov, L. (2003), Some observations on Babuška and Brezzi theories, Numer. Math. 94, 195–202.10.1007/s002110100308CrossRefGoogle Scholar
Xu, J., Chen, L. and Nochetto, R. H. (2009), Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation (DeVore, R. and Kunoth, A., eds), Springer, pp. 599–659.10.1007/978-3-642-03413-8_14CrossRefGoogle Scholar