Advertisement

Journal of Scientific Computing

, Volume 77, Issue 3, pp 1736–1761 | Cite as

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

  • Johnny Guzman
  • Alexandre Madureira
  • Marcus Sarkis
  • Shawn Walker
Article
  • 95 Downloads

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.

Keywords

Finite elements Laplace–Beltrami Curvature 

References

  1. 1.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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 MathSciNetCrossRefGoogle Scholar
  4. 4.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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 MathSciNetCrossRefGoogle Scholar
  7. 7.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976). Translated from the PortugueseGoogle Scholar
  16. 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 Google Scholar
  17. 17.
    Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)MathSciNetCrossRefGoogle Scholar
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 21.
    Hansbo, P., Larson, M.G.: A Stabilized Finite Element Method for the Darcy Problem on Surfaces (2015). arXiv:1511.03747
  22. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lang, S.: Undergraduate Analysis, Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (1997)Google Scholar
  25. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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)MathSciNetzbMATHGoogle Scholar
  32. 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)MathSciNetzbMATHGoogle Scholar
  33. 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)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Johnny Guzman
    • 1
  • Alexandre Madureira
    • 2
    • 3
  • Marcus Sarkis
    • 4
  • Shawn Walker
    • 5
  1. 1.Brown UniversityProvidenceUSA
  2. 2.Laboratório Nacional de Computação CientíficaPetropolisBrazil
  3. 3.Fundacao Getulio VargasRio de JaneiroBrazil
  4. 4.Mathematical Sciences DepartmentWorcester Polytechnic InstituteWorcesterUSA
  5. 5.Department of Mathematics and Center for Computation and Technology (CCT)Louisiana State UniversityBaton RougeUSA

Personalised recommendations