On the convergence of eigenvalues for mixed formulations

Daniele Boffi; Franco Brezzi; Lucia Gastaldi

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1997)

  • Volume: 25, Issue: 1-2, page 131-154
  • ISSN: 0391-173X

How to cite

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Boffi, Daniele, Brezzi, Franco, and Gastaldi, Lucia. "On the convergence of eigenvalues for mixed formulations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 25.1-2 (1997): 131-154. <http://eudml.org/doc/84281>.

@article{Boffi1997,
author = {Boffi, Daniele, Brezzi, Franco, Gastaldi, Lucia},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {variational problem; Hilbert spaces; bilinear forms; eigenvalue; convergence},
language = {eng},
number = {1-2},
pages = {131-154},
publisher = {Scuola normale superiore},
title = {On the convergence of eigenvalues for mixed formulations},
url = {http://eudml.org/doc/84281},
volume = {25},
year = {1997},
}

TY - JOUR
AU - Boffi, Daniele
AU - Brezzi, Franco
AU - Gastaldi, Lucia
TI - On the convergence of eigenvalues for mixed formulations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1997
PB - Scuola normale superiore
VL - 25
IS - 1-2
SP - 131
EP - 154
LA - eng
KW - variational problem; Hilbert spaces; bilinear forms; eigenvalue; convergence
UR - http://eudml.org/doc/84281
ER -

References

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  1. [1] I BabuŠka, Error-Bounds for Finite Element Method, Numer. Math.16 (1971), 322-333. Zbl0214.42001MR288971
  2. [2] I Babuška, On the finite element method with Lagrangian multipliers, Numer. Math.20 (1973), 179-192. Zbl0258.65108MR359352
  3. [3] I Babuška - J.E. Osborn, "Handbook of Numerical Analysis", vol. II, ch. Eigenvalue Problems, North-Holland, 1991, pp. 641-788. Zbl0875.65087MR1115240
  4. [4] K.J. Bathe - C. Nitikitpaiboon - X. Wang, A mixed displacement-based finite element formulation for acoustic fluid-structure interaction, Computers &Structures56 (1995), 225-237. Zbl1002.76536MR1336298
  5. [5] D. Boffi - F. Brezzi - L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, submitted to Math. Comp., 1997. Zbl0938.65126MR1642801
  6. [6] J.M. Boland - R. Nicolaides, On the stability of bilinear-constant velocity-pressure finite elements, Numer. Math.44 (1984), 219-222. Zbl0544.76030MR753954
  7. [7] J.H. Bramble - J.E. Osborn, Rate of convergence for nonselfadjoint eigenvalue approximations, Math. Comp.27 (1973), 525-549. Zbl0305.65064MR366029
  8. [8] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, R.A.I.R.O. Anal. Numer.8 (1974), 129-151. Zbl0338.90047MR365287
  9. [9] F. Brezzi - M. Fortin, "Mixed and Hybrid Finite Element Methods", Springer-Verlag, New York, 1991. Zbl0788.73002MR1115205
  10. [10] F. Brezzi - J. DouglasJr. - M. Fortin - L.D. Marini, Efficient rectangular mixed finite elements in two and three space variables, R.A.I.R.O. Model. Math. Anal. Numer.21 (1987), 237-250. Zbl0689.65065MR921828
  11. [11] F. Brezzi - J. DouglasJr. - L.D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math.47 (1985), 217-235. Zbl0599.65072MR799685
  12. [12] P.G. Ciarlet, "The Finite Element Method for Elliptic Problems ", North-Holland, Amsterdam, 1978. Zbl0383.65058MR520174
  13. [13] P.G. Ciarlet - P.-A. Raviart, A mixed finite element methodfor the biharmonic equation, in "Mathematical Aspects of Finite Element in Partial Differential Equations", C. de Boor (ed.), Academic Press, New York, 1974, 125-143. Zbl0337.65058MR657977
  14. [14] J. Falk - J.E. Osborn, Error estimates for mixed methods, R.A.I.R.O. Anal. Numer.4 (1980), 249-277. Zbl0467.65062MR592753
  15. [15] M. Fortin, An analysis of the convergence of mixed finite element methods, R.A.I.R.O. Anal. Numer.11 (1977), 341-354. Zbl0373.65055MR464543
  16. [16] R. Glowinski, Approximations externes par éléments finis de Lagrange d' ordre un et deux, du problème de Dirichlet pour l'opérateur biharmonique. Méthodes itératives de résolution des problèmes approchés, "Topics in Numerical Analysis", J. Miller (ed.), Academic Press, New York, 1973, 123-171. Zbl0277.35003MR351120
  17. [17] P. Grisvard, "Elliptic Problems in Non-Smooth Domains", Pitman, Marshfields, Mass., 1985. Zbl0695.35060
  18. [18] C. Johnson - J. Pitkäranta, Analysis of some mixed finite element methods related to reduced integration, Math. Comp.38 (1982), 375-400. Zbl0482.65058MR645657
  19. [19] R.B. Kellogg - J.E. Osborn, A regularity result for the Stokes problem, J. Funct. Anal.21 (1976), 397-431. Zbl0317.35037MR404849
  20. [20] B. Mercier, Numerical solution of the biharmonic problem by mixed finite elements of class C°, Boll. U.M.I.10 (1974), 133-149. Zbl0332.65058MR378442
  21. [21] B. Mercier - J.E. Osborn - J. Rappaz - P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp.36 (1981), 427-453. Zbl0472.65080MR606505
  22. [22] J.T. Oden - O. Jacquotte, Stability of some mixed finite element methods for Stokesian flows, Comp. Methods Appl. Mech. Eng.43 (1984), 231-247. Zbl0598.76033MR745509
  23. [23] J.E. Osborn, Eigenvalue approximations by mixed methods, in "Advances in Computer Methods for Partial Differential Equations III", R. Vichnevetsky and R. Stepleman (eds.), New Brunswick, 1979, 158-161. MR603467
  24. [24] P.-A. Raviart - J.M. Thomas, A mixed finite element method for second order elliptic problems, in "Mathematical Aspects of the Finite Element Method", I. Galligani and E. Magenes (eds.), Lecture Notes in Math., Springer-Verlag, New York, 1977, 292-315. Zbl0362.65089MR483555
  25. [25] R. Scholz, A mixed method for fourth order problems using linear finite elements, R.A.I.R.O. Anal. Numer.12 (1978), 85-90. Zbl0382.65059MR483557
  26. [26] X. Wang - K.J. Bathe, On mixed elements for acoustic fluid-structure interaction, M3AS, 7 (1997), 329-344. Zbl0881.76053MR1443789

Citations in EuDML Documents

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  1. Qun Lin, Hehu Xie, A Superconvergence result for mixed finite element approximations of the eigenvalue problem
  2. Evgenii V. Chizhonkov, Maxim A. Olshanskii, On the domain geometry dependence of the LBB condition
  3. Qun Lin, Hehu Xie, A Superconvergence result for mixed finite element approximations of the eigenvalue problem
  4. Francesca Gardini, Mixed approximation of eigenvalue problems: A superconvergence result
  5. Evgenii V. Chizhonkov, Maxim A. Olshanskii, On the domain geometry dependence of the LBB condition
  6. Patrick Ciarlet Jr., François Lefèvre, Stéphanie Lohrengel, Serge Nicaise, Weighted regularization for composite materials in electromagnetism
  7. Wei Chen, Qun Lin, Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method

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