Stabilization of a non standard FETI-DP mortar method for the Stokes problem

E. Chacón Vera; T. Chacón Rebollo

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 1, page 285-304
  • ISSN: 0764-583X

Abstract

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In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.

How to cite

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Chacón Vera, E., and Chacón Rebollo, T.. "Stabilization of a non standard FETI-DP mortar method for the Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.1 (2014): 285-304. <http://eudml.org/doc/273306>.

@article{ChacónVera2014,
abstract = {In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.},
author = {Chacón Vera, E., Chacón Rebollo, T.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {incompressible Stokes problem; non-standard FETI-DP; non-standard FETI-DP mortar method},
language = {eng},
number = {1},
pages = {285-304},
publisher = {EDP-Sciences},
title = {Stabilization of a non standard FETI-DP mortar method for the Stokes problem},
url = {http://eudml.org/doc/273306},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Chacón Vera, E.
AU - Chacón Rebollo, T.
TI - Stabilization of a non standard FETI-DP mortar method for the Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 1
SP - 285
EP - 304
AB - In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.
LA - eng
KW - incompressible Stokes problem; non-standard FETI-DP; non-standard FETI-DP mortar method
UR - http://eudml.org/doc/273306
ER -

References

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  1. [1] R.A. Adams, Sobolev Spaces. In vol. 65 of Pure and Applied Mathematics. Academic Press, New York, London (1975). Zbl0314.46030MR450957
  2. [2] D. Braess, W. Dahmen and C. Wieners, A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal.37 (1999) 48–69. Zbl0942.65139MR1721260
  3. [3] F. Ben Belgacem, The Mortar finite element method with Lagrange multipliers. Numerische Mathematik84 (1999) 173–197. Zbl0944.65114MR1730018
  4. [4] C. Bernardi, T. Chacón Rebollo and E. Chacón Vera, A FETI method with a mesh independent condition number for the iteration matrix. Comput. Methods Appl. Mech. Engrg.197 (2008) 1410–1429. Zbl1186.65147MR2387009
  5. [5] C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, edited by H. Brezis and J.-L. Lions. Collège de France Seminar XI, Pitman (1994) 13–51. Zbl0797.65094MR1268898
  6. [6] E. Chacón Vera, A continuous framework for FETI-DP with a mesh independent condition number for the dual problem. Comput. Methods Appl. Mech. Engrg.198 (2009) 2470–2483. Zbl1229.65210MR2530841
  7. [7] E. Chacón Vera, and D. Franco Coronil, A non standard FETI-DP mortar method for Stokes Problem. Proceedings of the 3rd FreeFem++ days, Paris, 2011. J. Numer. Math. 20 (2012) 161–182. Zbl1299.76132MR3043635
  8. [8] http://www.freefem.org/ff++ 
  9. [9] L.P. Franca, T.J.R. Hughes and R. Stenberg, Stabilized Finite Element Methods, in Incompressible Computational Fluid Dynamics, chap. 4, edited by M. Gunzburger and R.A. Nicolaides. Cambridge Univ. Press, Cambridge (1993) 87–107. Zbl1189.76339MR2504357
  10. [10] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, vol. 5 of Springer Series in Comput. Math. Springer-Verlag, Berlin (1986). Zbl0585.65077MR851383
  11. [11] P. Grisvard, Singularities in Boundary value problems, vol. 22 of Recherches en Mathématiques Appliquées, Masson (1992). Zbl0766.35001MR1173209
  12. [12] C.O. Lee and E.H. Park, A dual iterative substructuring method with a penalty term, Numerische Mathematik V.112 (2009) 89–113. Zbl1165.65078MR2481531
  13. [13] P.A. Raviart and J.-M. Thomas, Primal Hybrid Finite Element Methods for second order elliptic equations. Math. Comput.31 (1977) 391–413. Zbl0364.65082MR431752

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