A Multigrid Algorithm for an Elliptic Problem with a Perturbed Boundary Condition

  • Andrea Bonito
  • Joseph E. PasciakEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 45)


We discuss the preconditioning of systems coupling elliptic operators in \(\Omega \subset {\mathbb{R}}^{d}\), d=2,3, with elliptic operators defined on hypersurfaces. These systems arise naturally when physical phenomena are affected by geometric boundary forces, such as the evolution of liquid drops subject to surface tension. The resulting operators are sums of interior and boundary terms weighted by parameters. We investigate the behavior of multigrid algorithms suited to this context and demonstrate numerical results which suggest uniform preconditioning bounds that are level and parameter independent.


Multigrid Laplace-Beltrami Surface Laplacian Parameter dependent problems 



This work was supported in part by award number KUS-C1-016-04 made by King Abdulla University of Science and Technology (KAUST). It was also supported in part by the National Science Foundation through Grant DMS-0914977 and DMS-1216551.


  1. 1.
    Bänsch, E.: Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math. 88(2), 203–235 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bonito, A., Pasciak, J.E.: Analysis of a multigrid algorithm for an elliptic problem with a perturbed boundary condition. Technical report (in preparation)Google Scholar
  3. 3.
    Bonito, A., Pasciak, J.E.: Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator. Math. Comp. 81(279), 1263–1288 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bonito, A., Nochetto, R.H., Pauletti, M.S.: Dynamics of biomembranes: effect of the bulk fluid. Math. Model. Nat. Phenom. 6(5), 25–43 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp. 50(181), 1–17 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bramble, J.H., Pasciak, J.E.: The analysis of smoothers for multigrid algorithms. Math. Comp. 58(198), 467–488 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bramble, J., Zhang, X.: The analysis of multigrid methods. In: Ciarlet, P.C., Lions, J.L. (eds.) Handbook of Numerical Analysis, Techniques of Scientific Computing (Part 3). Elsevier, Amsterdam (2000)Google Scholar
  8. 8.
    Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34(3), 1072–1092 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bramble, J.H., Pasciak, J.E., Vassilevski, P.S.: Computational scales of Sobolev norms with application to preconditioning. Math. Comp. 69(230), 463–480 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Du, Q., Liu, Ch., Ryham, R., Wang, X.: Energetic variational approaches in modeling vesicle and fluid interactions. Phys. D 238(9–10), 923–930 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58(6), 603–611 (1991)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionGoogle Scholar
  14. 14.
    Haase, G., Langer, U., Meyer, A., Nepomnyaschikh, S.V.: Hierarchical extension operators and local multigrid methods in domain decomposition preconditioners. East-West J. Numer. Math. 2(3), 173–193 (1994)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hysing, S.: A new implicit surface tension implementation for interfacial flows. Int. J. Numer. Methods Fluids 51(6), 659–672 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lee, Y.-Ju., Wu, J., Xu, J., Zikatanov, L.: Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci. 17(11), 1937–1963 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lee, Y.-Ju., Wu, J., Xu, J., Zikatanov, L.: A sharp convergence estimate for the method of subspace corrections for singular systems of equations. Math. Comp. 77(262), 831–850 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Pauletti, M.S.: Parametric AFEM for geometric evolution equation and coupled fluid-membrane interaction. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), University of Maryland, College Park (2008)Google Scholar
  19. 19.
    Rusten, T., Winther, R.: A preconditioned iterative method for saddlepoint problems. SIAM J. Matrix Anal. Appl. 13(3), 887–904 (1992). Iterative Methods in Numerical Linear Algebra, Copper Mountain, CO (1990)Google Scholar
  20. 20.
    Sohn, J.S., Tseng, Y.-H., Li, S., Voigt, A., Lowengrub, J.S.: Dynamics of multicomponent vesicles in a viscous fluid. J. Comput. Phys. 229(1), 119–144, (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Walker, S.W., Bonito, A., Nochetto, R.H.: Mixed finite element method for electrowetting on dielectric with contact line pinning. Interfaces Free Bound. 12(1), 85–119 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations