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Nitsche’s Method for Defective Boundary Value Problems in Incompressibile Fluid-dynamics

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Abstract

In this work we present a unified formulation for the prescription of defective boundary conditions in fluid-dynamics, by means of the Nitsche’s method. We study the well-posedness of the discrete problem and the convergence of the numerical solution. Finally, we present several numerical results, focusing on the validation of the proposed method, on a comparison with a pre-existing strategy for the prescription of the flow rate, and on the application to the fluid-structure interaction case.

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References

  1. Badia, S., Nobile, F., Vergara, C.: Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems. Comput. Methods Appl. Mech. Eng. 198(33–36), 2768–2784 (2009)

    Article  MathSciNet  Google Scholar 

  2. Badia, S., Nobile, F., Vergara, C.: Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227, 7027–7051 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Benzi, M., Golub, H.G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blanco, P.J., Feijòo, R.A., Urquiza, S.A.: A unified variational approach for coupling 3D–1D models and its blood flow applications. Comput. Methods Appl. Mech. Eng. 196, 4391–4410 (2007)

    Article  MATH  Google Scholar 

  5. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    MATH  Google Scholar 

  6. Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximation of the Stokes problem. In: Hackbush, W. (ed.) Efficient Solution of Elliptic Systems, pp. 11–19. Vieweg, Braunschweig (1984)

    Google Scholar 

  7. Cahouet, J., Chabard, J.-P.: Some fast 3D finite element solvers for the generalized Stokes problem. Int. J. Numer. Methods Fluids 8, 869–895 (1988)

    Article  MathSciNet  Google Scholar 

  8. Fernández, M.A., Gerbeau, J.F., Grandmont, C.: A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Numer. Methods Eng. 69(4), 794–821 (2007)

    Article  MATH  Google Scholar 

  9. Formaggia, L., Gerbeau, J.-F., Nobile, F., Quarteroni, A.: On the coupling of 3D an 1D Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191(6–7), 561–582 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Formaggia, L., Gerbeau, J.-F., Nobile, F., Quarteroni, A.: Numerical treatment of defective boundary conditions for the Navier-Stokes equation. SIAM J. Numer. Anal. 40(1), 376–401 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Formaggia, L., Quarteroni, A., Veneziani, A. (eds.): Cardiovascular Mathematics—Modeling and Simulation of the Circulatory System. Springer, Berlin (2009)

    MATH  Google Scholar 

  12. Formaggia, L., Veneziani, A., Vergara, C.: Flow rate boundary problems for an incompressible fluid in deformable domains: formulations and solution methods. Comput. Methods Appl. Mech. Eng. 199(9–12), 677–688 (2010)

    Article  Google Scholar 

  13. Formaggia, L., Veneziani, A., Vergara, C.: A new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics. SIAM J. Numer. Anal. 46(6), 2769–2794 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986)

    MATH  Google Scholar 

  15. Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 22, 325–352 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hughes, T.J.R., Liu, W.K., Zimmermann, T.K.: Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29(3), 329–349 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuŝka-Brezzi condition: a stable Petrov-Galerkin formulation for the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59, 85–99 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  18. Juntunen, M., Stenberg, R.: Nitsche’s method for general boundary conditions. Math. Comput. 78, 1353–1374 (2009)

    MATH  MathSciNet  Google Scholar 

  19. Kim, H.J., Figueroaa, C.A., Hughes, T.J.R., Jansen, K.E., Taylor, C.A.: Augmented Lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three-dimensional finite element simulations of blood flow. Comput. Methods Appl. Mech. Eng. 198, 3551–3566 (2009)

    Article  MATH  Google Scholar 

  20. Nitsche, J.A.: Uber ein variationsprinzip zur lozung von dirichlet-problemen bei verwendung von teilraumen, die keinen randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1970/71)

    Article  MathSciNet  Google Scholar 

  21. Nobile, F., Vergara, C.: An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. SIAM J. Sci. Comput. 30(2), 731–763 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Quarteroni, A., Saleri, F., Veneziani, A.: Analysis of the Yosida method for the incompressible Navier-Stokes equations. J. Math. Pures Appl. 78, 473–503 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Quarteroni, A., Saleri, F., Veneziani, A.: Factorization methods for the numerical approximation of Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 188, 505–526 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  25. Thomée, V.: Galerkin Finite Element Method for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin (2001)

    Google Scholar 

  26. Veneziani, A.: Mathematical and numerical modeling of blood flow problems. PhD thesis, University of Milan (1998)

  27. Veneziani, A., Vergara, C.: Flow rate defective boundary conditions in haemodinamics simulations. Int. J. Numer. Methods Fluids 47, 803–816 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Veneziani, A., Vergara, C.: An approximate method for solving incompressible Navier-Stokes problems with flow rate conditions. Comput. Methods Appl. Mech. Eng. 196(9–12), 1685–1700 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zunino, P.: Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty technique. Comput. Methods Appl. Mech. Eng. 198(37–40), 3026–3038 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Information Technology and Mathematical Methods, Università degli Studi di Bergamo, Viale Marconi 5, 24044, Dalmine, (BG), Italy

    Christian Vergara

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  1. Christian Vergara
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Correspondence to Christian Vergara.

Additional information

This work has been (partially) supported by the ERC Advanced Grant N.227058 MATHCARD and by the Italian MURST, through a project COFIN07.

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Vergara, C. Nitsche’s Method for Defective Boundary Value Problems in Incompressibile Fluid-dynamics. J Sci Comput 46, 100–123 (2011). https://doi.org/10.1007/s10915-010-9389-7

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  • DOI: https://doi.org/10.1007/s10915-010-9389-7

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