nach oben

Journal of Scientific Computing

Erschienen in:

25.10.2017

Analysis of the Finite Element Method for the Laplace–Beltrami Equation on Surfaces with Regions of High Curvature Using Graded Meshes

verfasst von: Johnny Guzman, Alexandre Madureira, Marcus Sarkis, Shawn Walker

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2018

Einloggen

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

search-config

KI-gestützte Suche

Aus

Abstract

We derive error estimates for the piecewise linear finite element approximation of the Laplace–Beltrami operator on a bounded, orientable, \(C^3\), surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.

Vorheriger Artikel Dispersion Analysis of HDG Methods

Nächster Artikel A Finite Element Method with Strong Mass Conservation for Biot’s Linear Consolidation Model

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 "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

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

Nur mit Berechtigung zugänglich

Antonietti, P.F., Dedner, A., Madhavan, P., Stangalino, S., Stinner, B., Verani, M.: High order discontinuous Galerkin methods for elliptic problems on surfaces. SIAM J. Numer. Anal. 53(2), 1145–1171 (2015). https://doi.org/10.1137/140957172 MathSciNetCrossRefMATH

Bertalmío, M., Cheng, L.-T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759–780 (2001). https://doi.org/10.1006/jcph.2001.6937 MathSciNetCrossRefMATH

Bonito, A., Manuel Cascón, J., Morin, P., Mekchay, K., Nochetto, R.H.: High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates (2016). arXiv:1511.05019 MathSciNetCrossRef

Burman, E., Hansbo, P., Larson, M.G.: A stabilized cut finite element method for partial differential equations on surfaces: the Laplace–Beltrami operator. Comput. Methods Appl. Mech. Eng. 285, 188–207 (2015). https://doi.org/10.1016/j.cma.2014.10.044 MathSciNetCrossRefMATH

Camacho, F., Demlow, A.: L2 and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces. IMA J. Numer. Anal. 35(3), 1199–1227 (2015). https://doi.org/10.1093/imanum/dru036 MathSciNetCrossRefMATH

Chernyshenko, A.Y., Olshanskii, M.A.: An adaptive octree finite element method for PDEs posed on surfaces. Comput. Methods Appl. Mech. Eng. 291, 146–172 (2015). https://doi.org/10.1016/j.cma.2015.03.025 MathSciNetCrossRef

Chernyshenko, A.Y., Olshanskii, M.A.: Non-degenerate Eulerian finite element method for solving PDEs on surfaces. Russian J. Numer. Anal. Math. Model. 28(2), 101–124 (2013). https://doi.org/10.1515/rnam-2013-0007 MathSciNetCrossRefMATH

Cockburn, B., Demlow, A.: Hybridizable discontinuous Galerkin and mixed finite element methods for elliptic problems on surfaces. Math. Comput. 85(302), 2609–2638 (2016). https://doi.org/10.1090/mcom/3093 MathSciNetCrossRefMATH

Deckelnick, K., Dziuk, G., Elliott, C.M., Heine, C.-J.: An h-narrow band finite-element method for elliptic equations on implicit surfaces. IMA J. Numer. Anal. 30(2), 351–376 (2010). https://doi.org/10.1093/imanum/drn049 MathSciNetCrossRefMATH

10.

Dedner, A., Madhavan, P.: Adaptive discontinuous Galerkin methods on surfaces. Numer. Math. 132(2), 369–398 (2016). https://doi.org/10.1007/s00211-015-0719-4 MathSciNetCrossRefMATH

11.

Dedner, A., Madhavan, P., Stinner, B.: Analysis of the discontinuous Galerkin method for elliptic problems on surfaces. IMA J. Numer. Anal. 33(3), 952–973 (2013). https://doi.org/10.1093/imanum/drs033 MathSciNetCrossRefMATH

12.

Demlow, A.: Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47(2), 805–827 (2009). https://doi.org/10.1137/070708135 MathSciNetCrossRefMATH

13.

Demlow, A., Dziuk, G.: An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45(1), 421–442 (2007). https://doi.org/10.1137/050642873 MathSciNetCrossRefMATH

14.

Demlow, A., Olshanskii, M.A.: An adaptive surface finite element method based on volume meshes. SIAM J. Numer. Anal. 50(3), 1624–1647 (2012). https://doi.org/10.1137/110842235 MathSciNetCrossRefMATH

15.

do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976). Translated from the Portuguese

16.

Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol. 1357, pp. 142–155. Springer, Berlin (1988). https://doi.org/10.1007/BFb0082865

17.

Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)MathSciNetCrossRef

18.

Efendiev, Y., Galvis, J., Sebastian Pauletti, M.: Multiscale finite element methods for flows on rough surfaces. Commun. Comput. Phys. 14(4), 979–1000 (2013)MathSciNetCrossRef

19.

Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRef

20.

Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer, Berlin (1983)CrossRef

21.

Hansbo, P., Larson, M.G.: A Stabilized Finite Element Method for the Darcy Problem on Surfaces (2015). arXiv:1511.03747

22.

Hansbo, P., Larson, M.G., Zahedi, S.: Stabilized finite element approximation of the mean curvature vector on closed surfaces. SIAM J. Numer. Anal. 53(4), 1806–1832 (2015). https://doi.org/10.1137/140982696 MathSciNetCrossRefMATH

23.

Holst, M., Stern, A.: Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Found. Comput. Math. 12(3), 263–293 (2012). https://doi.org/10.1007/s10208-012-9119-7 MathSciNetCrossRefMATH

24.

Lang, S.: Undergraduate Analysis, Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (1997)

25.

Mantegazza, C., Mennucci, A.C.: Hamilton–Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47(1), 1–25 (2003). https://doi.org/10.1007/s00245-002-0736-4 MathSciNetCrossRefMATH

26.

Olshanskii, M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47(5), 3339–3358 (2009). https://doi.org/10.1137/080717602 MathSciNetCrossRefMATH

27.

Olshanskii, M.A., Reusken, A.: A finite element method for surface PDEs: matrix properties. Numer. Math. 114(3), 491–520 (2010). https://doi.org/10.1007/s00211-009-0260-4 MathSciNetCrossRefMATH

28.

Olshanskii, M.A., Reusken, A., Xu, X.: An Eulerian space–time finite element method for diffusion problems on evolving surfaces. SIAM J. Numer. Anal. 52(3), 1354–1377 (2014). https://doi.org/10.1137/130918149 MathSciNetCrossRefMATH

29.

Olshanskii, M.A., Reusken, A., Xu, X.: A stabilized finite element method for advection–diffusion equations on surfaces. IMA J. Numer. Anal. 34(2), 732–758 (2014). https://doi.org/10.1093/imanum/drt016 MathSciNetCrossRefMATH

30.

Olshanskii, M.A., Safin, D.: A narrow-band unfitted finite element method for elliptic PDEs posed on surfaces. Math. Comput. 85(300), 1549–1570 (2016). https://doi.org/10.1090/mcom/3030 MathSciNetCrossRefMATH

31.

Schatz, A.H., Wahlbin, L.B.: Maximum norm estimates in the finite element method on plane polygonal domains. I. Math. Comput. 32(141), 73–109 (1978)MathSciNetMATH

32.

Schatz, A.H., Wahlbin, L.B.: Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements. Math. Comput. 33(146), 465–492 (1979)MathSciNetMATH

33.

Walker, S.W.: The Shapes of Things, Advances in Design and Control. A Practical Guide to Differential Geometry and the Shape Derivative, vol. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2015)CrossRef

Titel: Analysis of the Finite Element Method for the Laplace–Beltrami Equation on Surfaces with Regions of High Curvature Using Graded Meshes
verfasst von: Johnny Guzman
Alexandre Madureira
Marcus Sarkis
Shawn Walker
Publikationsdatum: 25.10.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0580-y

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

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

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3/2018

Conservative High Order Positivity-Preserving Discontinuous Galerkin Methods for Linear Hyperbolic and Radiative Transfer Equations

A Finite Element Method with Strong Mass Conservation for Biot’s Linear Consolidation Model

Hybrid Discretization Methods with Adaptive Yield Surface Detection for Bingham Pipe Flows

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

Discontinuous Galerkin Methods for Acoustic Wave Propagation in Polygons

Globally Divergence-Free Discontinuous Galerkin Methods for Ideal Magnetohydrodynamic Equations