Journal of Scientific Computing

, Volume 74, Issue 2, pp 805–825 | Cite as

Distributed Lagrange Multiplier Functions for Fictitious Domain Method with Spectral/hp Element Discretization

  • Riccardo Zamolo
  • Lucia Parussini
  • Valentino Pediroda


A fictitious domain approach for the solution of second-order linear differential problems is proposed; spectral/hp elements have been used for the discretization of the domain. The peculiarity of our approach is that the Lagrange multipliers are particular distributed functions, instead of classical \(\delta \) Dirac (impulsive) multipliers. In this paper we present the formulation and the application of this approach to 1D and 2D Poisson problems and 2D Stokes flow (biharmonic equation).


Fictitious domain Lagrange multipliers Spectral/hp element method Poisson problem Biharmonic problem 

Mathematics Subject Classification

65L10 65N35 


  1. 1.
    Peskin, C.S.: Flow patterns around heart valves: a digital computer method for solving the equations of motion. PhD thesis, Sue Golding Graduate Division of Medical Sciences, Albert Einstein College of Medicine, Yeshiva University (1972)Google Scholar
  2. 2.
    Glowinski, R., Pan, T.W., Periaux, J.: A fictitious domain method for external incompressible viscous flow modeled by Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 112(1–4), 133–148 (1994a)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Glowinski, R., Pan, T.W., Periaux, J.: A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 111(3–4), 283–303 (1994b)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Glowinski, R., Pan, T.W., Periaux, J.: A Lagrange multiplier/fictitious domain method for the Dirichlet problem—generalization to some flow problems. Japan J. Indust. Appl. Math. 12, 87–108 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Babuška, I., Guo, B.: The \(h-p\) version of the finite element method. Comput. Mech. 1(1), 21–41 (1986)CrossRefMATHGoogle Scholar
  6. 6.
    Babuška, I., Suri, M.: The \(h-p\) version of the finite element method with quasiuniform meshes. ESAIM Math. Model. Numer. Anal. 21(2), 199–238 (1987)CrossRefMATHGoogle Scholar
  7. 7.
    Karniadakis, G., Sherwin, S.: Spectral/\(hp\) Element Methods for Computational Fluid Dynamics. Oxford University Press, New York (1999)MATHGoogle Scholar
  8. 8.
    Pontaza, J., Reddy, J.: Space-time coupled spectral/\(hp\) least-squares finite element formulation for the incompressible Navier-Stokes equations. J. Comput. Phys. 197(2), 418–459 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Vos, P., Van Loon, R., Sherwin, S.J.: A comparison of fictitious domain methods appropriate for spectral/\(hp\) element discretization. Comput. Methods Appl. Mech. Eng. 197(25), 2275–2289 (2008)CrossRefMATHGoogle Scholar
  10. 10.
    Parussini, L.: Fictitious domain approach for spectral/\(hp\) element method. Comput. Model. Eng. Sci. 17(2), 95–114 (2007)MathSciNetMATHGoogle Scholar
  11. 11.
    Parussini, L.: Fictitious domain approach via Lagrange multipliers with least squares spectral element method. J. Sci. Comp. 37(3), 316–335 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Parussini, L., Pediroda, V.: Fictitious domain with least-squares spectral element method to explore geometric uncertainties by non-intrusive polynomial chaos method. Comput. Model. Eng. Sci. 22(1), 41–64 (2007)Google Scholar
  13. 13.
    Parussini, L., Pediroda, V.: Investigation of multi geometric uncertainties by different polynomial chaos methodologies using a fictitious domain solver. Comput. Model. Eng. Sci. 23(1), 29–52 (2008)MathSciNetMATHGoogle Scholar
  14. 14.
    Glowinski, R., Pan, T.W., Periaux, J.: Distributed Lagrange multiplier method for incompressible viscous flow around moving rigid bodies. Comput. Methods Appl. Mech. Eng. 151(1–2), 181–194 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Auricchio, F., Boffi, D., Gastaldi, L., Lefieux, A.: On a fictitious domain method with distributed Lagrange multiplier for interface problems. Appl. Numer. Math. 95, 36–50 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Boffi, D., Cavallini, N., Gastaldi, L.: The finite element immersed boundary method with distributed Lagrange multiplier. SIAM J. Numer. Anal. 53(6), 2584–2604 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Glowinski, R.: Finite element methods for incompressible viscous flow. In: Handbook of Numerical Analysis, chap 8, vol. 9, pp. 3–1176. North-Holland, Amsterdam (2003)Google Scholar
  18. 18.
    Dong, S., Liu, D., Maxey, M.R., Karniadakis, G.E.: Spectral distributed Lagrange multiplier method: algorithm and benchmark tests. J. Comput. Phys. 195(2), 695–717 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Buffat, M., Le Penven, L.: A spectral fictitious domain method with internal forcing for solving elliptic pdes. J. Comput. Phys. 230(7), 2433–2450 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Moreau, F., Hellou, M., El Yazidi, M.Z.: Ecoulements cellulaires de Stokes dans un canal plan obstrué par une file de cylindres. Angew. Math. Phys. 49(1), 31–45 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Engineering and Architecture DepartmentUniversity of TriesteTriesteItaly

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