Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Foundations of Computational Mathematics
Erschienen in: Foundations of Computational Mathematics 6/2016

01.12.2016

High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

verfasst von: Andrea Bonito, J. Manuel Cascón, Khamron Mekchay, Pedro Morin, Ricardo H. Nochetto

Erschienen in: Foundations of Computational Mathematics | Ausgabe 6/2016

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We present a new AFEM for the Laplace–Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally \(W^1_\infty \) and piecewise in a suitable Besov class embedded in \(C^{1,\alpha }\) with \(\alpha \in (0,1]\). The idea is to have the surface sufficiently well resolved in \(W^1_\infty \) relative to the current resolution of the PDE in \(H^1\). This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in \(W^1_\infty \) and PDE error in \(H^1\).
Vorheriger Artikel Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations
Nächster Artikel The L-Functions and Modular Forms Database Project

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Literatur
1.
R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, 2003.
2.
M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics. Wiley-Interscience [Wiley], New York, 2000.
3.
P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97(2) (2004), 219–268.MathSciNetCrossRefMATH
4.
P. Binev, W. Dahmen, R.A. DeVore and P. Petrushev, Approximation classes for adaptive methods, Serdica Math. J. 28(4) (2002), 391–416.MathSciNetMATH
5.
M.S. Birman and M. Solomyak, Piecewise-polynomial approximations of functions of the classes \(W_p^\alpha \), Mat. Sb. (N.S.) 73(115)(3) (1967), 331–355.
6.
A. Bonito, J.M. Cascón, P. Morin and R.H. Nochetto, AFEM for geometric PDE: the Laplace–Beltrami operator, in Analysis and Numerics of Partial Differential Equations, (F. Brezzi, P. Coli, U. Gianazza and G. Gilardi, eds.), Springer INdAM Ser., vol. 4, Springer, Milan, 2013, pp. 257–306.
7.
A. Bonito, R.A. DeVore and R.H. Nochetto, Adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Numer. Anal. 51(6) (2013), 3106–3134.MathSciNetCrossRefMATH
8.
A. Bonito and R.H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal. 48(2) (2010), 734–771.MathSciNetCrossRefMATH
9.
A. Bonito and J. Pasciak, Convergence analysis of variational and non-variational multigrid algorithm for the Laplace-Beltrami operator, Math. Comp. 81 (2012), 1263–1288.MathSciNetCrossRefMATH
10.
G. Bourdaud and W. Sickel, Composition operators on function spaces with fractional order of smoothness, in Harmonic Analysis and Nonlinear Partial Differential Equations, (T. Ozawa and M. Sugimoto, eds.), RIMS Kôkyûroku Bessatsu, B26, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 93–132.
11.
12.
J.M. Cascón, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46(5) (2008), 2524–2550.MathSciNetCrossRefMATH
13.
J.M. Cascón and R.H. Nochetto, Quasioptimal cardinality of AFEM driven by nonresidual estimators, IMA J. Numer. Anal. 3 2(1) (2012), 1–29.
14.
P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, vol. 4, North-Holland Publishing Co., Amsterdam, 1978.
15.
A. Cohen, W. Dahmen, I. Daubechies, and R.A. DeVore, Tree approximation and optimal encoding, Appl. Comput. Harmon. Anal. 11(2) (2001), 192–226.MathSciNetCrossRefMATH
16.
A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal. 47(2) (2009), 805–827.MathSciNetCrossRefMATH
17.
A. Demlow and G. Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal. 45(1) (2007), 421–442.MathSciNetCrossRefMATH
18.
R.A. DeVore and G.G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993.
19.
20.
G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, in Partial Differential Equations and Calculus of Variations (S. Hildebrandt and R. Leis, eds.), Lecture Notes in Math., vol. 1357, Springer, Berlin, 1988 , pp. 142–155.
21.
22.
E.M. Garau and P. Morin, Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA J. Numer. Anal. 31(3) (2011), 914–946.MathSciNetCrossRefMATH
23.
F.D. Gaspoz and P. Morin, Approximation classes for adaptive higher order finite element approximation, Math. Comp. 83(289) (2014), 2127–2160.MathSciNetCrossRefMATH
24.
R. Kornhuber and H. Yserentant, Multigrid methods for discrete elliptic problems on triangular surfaces, Comput. Vis. Sci. 11(4-6) (2008), 251–257.MathSciNetCrossRef
25.
K. Mekchay, P. Morin and R. H. Nochetto, AFEM for the Laplace-Beltrami operator on graphs: design and conditional contraction property, Math. Comp. 80(274) (2011), 625–648.MathSciNetCrossRefMATH
26.
P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38(2) (2000), 466–488.MathSciNetCrossRefMATH
27.
P. Morin, R.H. Nochetto and K.G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44(4) (2002), 631–658. Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38(2) (2000), 466–488].
28.
R.H. Nochetto, K.G. Siebert and A. Veeser, Theory of adaptive finite element methods: an introduction, in: Multiscale, Nonlinear and Adaptive Approximation (R. DeVore and A. Kunoth, eds.), Springer, Berlin, 2009, pp. 409–542.
29.
L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54(190) (1990), 483–493.MathSciNetCrossRefMATH
30.
31.
R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77(261) (2008), 227–241.MathSciNetCrossRefMATH
32.
H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers, Rev. Mat. Complut. 15(2) (2002), 475–524.MathSciNetCrossRefMATH
33.
A. Veeser, Approximating gradients with continuous piecewise polynomial functions, Found. Comput. Math. 16(3) (2016), 723–750.MathSciNetCrossRefMATH
34.
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Technique, Wiley-Teubner, Chichester, 1996.MATH
Metadaten
Titel
High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates
verfasst von
Andrea Bonito
J. Manuel Cascón
Khamron Mekchay
Pedro Morin
Ricardo H. Nochetto
Publikationsdatum
01.12.2016
Verlag
Springer US
Erschienen in
Foundations of Computational Mathematics / Ausgabe 6/2016
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-016-9335-7

Weitere Artikel der Ausgabe 6/2016

Foundations of Computational Mathematics 6/2016 Zur Ausgabe

OriginalPaper

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

OriginalPaper

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

OriginalPaper

Condition Length and Complexity for the Solution of Polynomial Systems

OriginalPaper

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

OriginalPaper

The L-Functions and Modular Forms Database Project

OriginalPaper

Structural Approach to Subset Sum Problems