Numerische Mathematik

, 114:491

A finite element method for surface PDEs: matrix properties



We consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces. The main idea of this method is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a problem in an outer domain that contains the surface, for example, two-phase flow problems. It has been proved that the method has optimal order of convergence both in the H1 and in the L2-norm. In this paper, we address linear algebra aspects of this new finite element method. In particular the conditioning of the mass and stiffness matrix is investigated. For the two-dimensional case we present an analysis which proves that the (effective) spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h−3| ln h| and h−2| ln h|, respectively, where h is the mesh size of the outer triangulation.

Mathematics Subject Classification (2000)

58J32 65N15 65N30 76D45 76T99 


  1. 1.
    Braess D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, London (2007)MATHGoogle Scholar
  2. 2.
    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., doi:10.1093/imanum/drn049 (2009)
  3. 3.
    Demlow A., Dziuk G.: An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45, 421–442 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
  5. 5.
    Dziuk G.: Finite elements for the beltrami operator on arbitrary surfaces. In: Hildebrandt, S., Leis, R. (eds) Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol. 1357, pp. 142–155. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  6. 6.
    James A., Lowengrub J.: A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comp. Phys. 201, 685–722 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Muradoglu M., Tryggvason G.: A front-tracking method for computation of interfacial flows with soluble surfactant. J. Comput. Phys. 227, 2238–2262 (2008)MATHCrossRefGoogle Scholar
  8. 8.
    Olshanskii, M.A., Reusken, A.: A finite element method for surface PDEs: Matrix properties, Preprint 287, IGPM, RWTH Aachen, earlier version of this paper (2008)Google Scholar
  9. 9.
    Olshanskii, M.A., Reusken, A., Grande, J.: An eulerian finite element method for elliptic equations on moving surfaces. Accepted to SIAM J. Numer. Anal. (2009)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsMoscow State M.V. Lomonosov UniversityMoscowRussia
  2. 2.Institut für Geometrie und Praktische MathematikRWTH-Aachen UniversityAachenGermany

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