Eigenvalue approximation by mixed and hybrid methods

Authors:
B. Mercier, J. Osborn, J. Rappaz and P.-A. Raviart

Journal:
Math. Comp. **36** (1981), 427-453

MSC:
Primary 65N25; Secondary 65N15, 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606505-9

MathSciNet review:
606505

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Rate of convergence estimates are derived fro the approximation of eigenvalues and eigenvectors by mixed and hybrid methods. Several closely related abstract results on spectral approximation are proved. These results are then applied to a variety of finite element methods of mixed and hybrid type: a mixed method for 2nd order problems, mixed methods for 4th order problems, a hybrid method for 2nd order problems, and two mixed methods for the Stokes eigenvalue problem.

**[1]**I. Babuška, "Error-bounds for finite element methods,"*Numer. Math.*, v. 16, 1971, pp. 322-333. MR**0288971 (44:6166)****[2]**I. Babuška & A. Aziz, "Survey lectures on the mathematical foundations of the finite element method" in*The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations*(A. K. Aziz, Ed.), Academic Press, New York, 1973, pp. 5-359.**[3]**I. Babuška & J. Osborn, "Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients,"*Math. Comp.*, v. 32, 1978, pp. 991-1023. MR**0501962 (58:19171)****[4]**J. Bramble & J. Osborn, "Rate of convergence estimates for non-selfadjoint eigenvalue approximations,"*Math. Comp.*, v. 27, 1973, pp. 525-549. MR**0366029 (51:2280)****[5]**I. Babuška, J. Osborn & J. Pitkäranta, "Analysis of mixed methods using mesh dependent norms,"*Math. Comp.*, v. 35, 1980, pp. 1039-1062. MR**583486 (81m:65166)****[6]**F. Brezzi, "On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers,"*R.A.I.R.O.*, v. 8, R-2, 1974, pp. 129-151. MR**0365287 (51:1540)****[7]**F. Brezzi, "Sur la méthode des éléments finis hybrides pour le problème biharmonique,"*Numer. Math.*, v. 24, 1975, pp. 103-131. MR**0391538 (52:12359)****[8]**F. Brezzi & P. A. Raviart, "Mixed finite element methods for 4th order elliptic equations,"*Topics in Numerical Analysis*III (J. Miller, Ed.), Academic Press, New York, 1978. MR**0657975 (58:31905)****[9]**C. Canuto, "Eigenvalue approximations by mixed methods,"*R.A.I.R.O. Anal. Numér.*, v. 12, 1978, pp. 27-50. MR**0488712 (58:8229)****[10]**P. Ciarlet & P. A. Raviart, "A mixed finite element method for the biharmonic equation,"*Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations*(C. de Boor, Ed.), Academic Press, New York, 1974, pp. 125-143. MR**0657977 (58:31907)****[11]**M. Crouzeix & P. A. Raviart, "Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I,"*R.A.I.R.O.*, v. 7, R-3, 1973, pp. 33-76. MR**0343661 (49:8401)****[12]**J. Descloux, N. Nassif & J. Rappaz, "On spectral approximation, Part 2: Error estimates for the Galerkin method,"*R.A.I.R.O. Anal. Numér.*, v. 12, 1978, pp. 113-119. MR**0483401 (58:3404b)****[13]**J. Descloux, N. Nassif & J. Rappaz,*Various Results on Spectral Approximation*, Rapport du Dept. de Math. de l'Ecole Polytechnique Fédérale de Lausanne, Suisse, 1977.**[14]**R. Falk, "Approximation of the biharmonic equation by a mixed finite element method,"*SIAM J. Numer. Anal.*, v. 15, 1978, pp. 556-567. MR**0478665 (57:18142)****[15]**R. Falk & J. Osborn, "Error estimates for mixed methods,"*R.A.I.R.O. Anal. Numér.*, v. 14, 1980, pp. 249, 277. MR**592753 (82j:65076)****[16]**G. Fix, "Eigenvalue approximation by the finite element method,"*Adv. in Math.*, v. 10, 1973, pp. 300-316. MR**0341900 (49:6646)****[17]**M. Fortin, "Analysis of the convergence of mixed finite element methods,"*R.A.I.R.O.*, v. 11, 1977, pp. 341-354. MR**0464543 (57:4473)****[18]**V. Girault & P. A. Raviart, "An analysis of a mixed finite element method for the Navier-Stokes equations,"*Numer. Math.*, v. 33, 1979, pp. 235-271. MR**553589 (81a:65100)****[19]**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 résolutions des problèmes approchés," in*Topics in Numerical Analysis*(J. J. H. Miller, Ed.), Academic Press, New York, 1973, pp. 123-171. MR**0351120 (50:3609)****[20]**L. Herrmann, "Finite element bending analysis for plates,"*J. Eng. Mech.*, Div. ASCE EM5, v. 93, 1967, pp. 49-83.**[21]**L. Herrmann, "A bending analysis for plates,"*Proc. Conf. on Matrix Methods in Structural Mechanics*, AFFDL-TR-66-88, pp. 577-604.**[22]**K. Ishihara, "Convergence of the finite element method applied to the eigenvalue problem ,"*Publ. Res. Inst. Math. Sci.*, v. 13, 1977, pp. 48-60. MR**0455475 (56:13713)****[23]**K. Ishihara, "The buckling of plates by the mixed finite element method,"*Mem. Numer. Math.*, v. 5, 1978, pp. 73-82. MR**0483544 (58:3538)****[24]**K. Ishihara, "A mixed finite element method for the biharmonic eigenvalue problem of plate bending,"*Publ. Res. Inst. Math. Sci.*, v. 14, 1978, pp. 399-414. MR**509196 (80c:73047)****[25]**C. Johnson, "On the convergence of a mixed finite element method for plate bending problems,"*Numer. Math.*, v. 21, 1973, pp. 43-62. MR**0388807 (52:9641)****[26]**W. Kolata, "Approximation of variationally posed eigenvalue problems,"*Numer. Math.*, v. 29, 1978, pp. 159-171. MR**482047 (80a:49077)****[27]**B. Mercier, "Numerical solution of the biharmonic problems by mixed finite elements of class ,"*Boll. Un. Mat. Ital.*, v. 10, 1974, pp. 133-149. MR**0378442 (51:14610)****[28]**B. Mercier & J. Rappaz,*Eigenvalue approximation via non-conforming and hybrid finite element methods*, Rapport du Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France, 1978.**[29]**T. Miyoshi, "A finite element method for the solution of fourth order partial differential equations,"*Kumamoto J. Sci.*(*Math.*), v. 9, 1973, pp. 87-116. MR**0386298 (52:7156)****[30]**S. Nemat-Nasser, "General variational methods for elastic waves in composites,"*J. Elasticity*, v. 2, 1972, pp. 73-90.**[31]**S. Nemat-Nasser, "Harmonic waves in layered composites,"*J. Appl. Mech.*, v. 39, 1972, pp. 850-852.**[32]**S. Nemat-Nasser,*General Variational Principles in Nonlinear and Linear Elasticity with Applications. Mechanics Today*1, Pergamon Press, New York, 1974, pp. 214-261.**[33]**T. Oden, "Some contributions to the mathematical theory of mixed finite element approximations," in*Theory and Practice in Finite Element Structural Analysis*, Univ. of Tokyo Press, Tokyo, 1973, pp. 3-23.**[34]**T. Oden, "Some contributions to the mathematical theory of mixed finite element approximations," in*Theory and Practice in Finite Element Structural Analysis*, Univ. of Tokyo Press, Tokyo, 1973, pp. 3-23.**[35]**T. Oden & J. Reddy, "On mixed finite element approximations,"*SIAM J. Numer. Anal.*, v. 13, 1976, pp. 393-404. MR**0413551 (54:1665)****[36]**J. Osborn, "Spectral approximation for compact operators,"*Math. Comp.*, v. 29, 1975, pp. 712-725. MR**0383117 (52:3998)****[37]**J. Osborn, "Approximation of the eigenvalues of a non-selfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations,"*SIAM J. Numer. Anal.*, v. 13, 1976, pp. 185-197. MR**0447842 (56:6152)****[38]**J. Osborn,*Advances in Computer Methods for Partial Differential Equations*III, Proc. Third IMACS Internat. Sympos. on Computer Methods for Partial Differential Equations held at Lehigh University, Bethlehem, Pennsylvania, June 1979, (R. Vichnevetsky, R. Stepleman, Eds.). MR**603451 (82a:65005)****[39]**R. Rannacher, "On nonconforming and mixed finite element methods for plate bending problems--the linear case." (Preprint.) MR**555385 (80i:65125)****[40]**P. A. Raviart & J. M. Thomas,*A Mixed Finite Element Method for 2nd Order Elliptic Problems*, Lecture Notes in Math., Vol. 606, Springer-Verlag, Berlin and New York, 1977, pp. 292-315. MR**0483555 (58:3547)****[41]**P. A. Raviart & J. M. Thomas, "Primal hybrid finite element method for 2nd order elliptic problems,"*Math. Comp.*, v. 31, 1977, pp. 391-413. MR**0431752 (55:4747)****[42]**R. Scholz,*Approximation von Sattelpunkten mit Finiten Elementen*, Tagungsband, Bonn. Math. Schr., Vol. 89, 1976, pp. 53-66. MR**0471377 (57:11111)****[43]**R. Scholz, "A mixed method for 4th order problems using linear finite elements,"*R.A.I.R.O. Numér. Anal.*, v. 12 1978, pp. 85-90. MR**0483557 (58:3549)****[44]**J. M. Thomas,*Sur l'Analyse Numérique des Méthodes d'Eléments Finis Hybrides et Mixtes*, Thesis, Univ. P & M Curie, Paris, 1977.**[45]**J. Wilkinson,*The Algebraic Eigenvalue Problem*, Oxford Univ. Press, 1965. MR**0184422 (32:1894)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N25,
65N15,
65N30

Retrieve articles in all journals with MSC: 65N25, 65N15, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606505-9

Article copyright:
© Copyright 1981
American Mathematical Society