Eigenvalue approximation by mixed and hybrid methods
Tóm tắt
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.
Từ khóa
Tài liệu tham khảo
Babuška, Ivo, 1970, Error-bounds for finite element method, Numer. Math., 16, 322, 10.1007/BF02165003
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.
Babuška, I., 1978, Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients, Math. Comp., 32, 991, 10.2307/2006330
Bramble, J. H., 1973, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp., 27, 525, 10.2307/2005658
Babuška, I., 1980, Analysis of mixed methods using mesh dependent norms, Math. Comp., 35, 1039, 10.2307/2006374
Brezzi, F., 1974, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Fran\c{c}aise Automat. Informat. Recherche Op\'{e}rationnelle S\'{e}r. Rouge, 8, 129
Brezzi, F., 1975, Sur la méthode des éléments finis hybrides pour le problème biharmonique, Numer. Math., 24, 103, 10.1007/BF01400961
Brezzi, F., 1977, Mixed finite element methods for 4th order elliptic equations, 33
Canuto, C., 1978, Eigenvalue approximations by mixed methods, RAIRO Anal. Num\'{e}r., 12, 27, 10.1051/m2an/1978120100271
Ciarlet, P. G., 1974, A mixed finite element method for the biharmonic equation, 125
Crouzeix, M., 1973, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Fran\c{c}aise Automat. Informat. Recherche Op\'{e}rationnelle S\'{e}r. Rouge, 7, 33
Descloux, Jean, 1978, On spectral approximation. I. The problem of convergence, RAIRO Anal. Num\'{e}r., 12, 97, 10.1051/m2an/1978120200971
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.
Falk, Richard S., 1978, Approximation of the biharmonic equation by a mixed finite element method, SIAM J. Numer. Anal., 15, 556, 10.1137/0715036
Falk, R. S., 1980, Error estimates for mixed methods, RAIRO Anal. Num\'{e}r., 14, 249, 10.1051/m2an/1980140302491
Fix, George J., 1973, Eigenvalue approximation by the finite element method, Advances in Math., 10, 300, 10.1016/0001-8708(73)90113-8
Fortin, Michel, 1977, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Num\'{e}r., 11, 341, 10.1051/m2an/1977110403411
Girault, V., 1979, An analysis of a mixed finite element method for the Navier-Stokes equations, Numer. Math., 33, 235, 10.1007/BF01398643
Glowinski, R., 1973, Approximations externes, par éléments finis de Lagrange d’ordre un et deux, du problème de Dirichlet pour l’opérateur biharmonique. Méthode itérative de résolution des problèmes approchés, 123
L. Herrmann, "Finite element bending analysis for plates," J. Eng. Mech., Div. ASCE EM5, v. 93, 1967, pp. 49-83.
L. Herrmann, "A bending analysis for plates," Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDL-TR-66-88, pp. 577-604.
Ishihara, Kazuo, 1977, Convergence of the finite element method applied to the eigenvalue problem Δ𝑢+𝜆𝑢=0, Publ. Res. Inst. Math. Sci., 13, 47, 10.2977/prims/1195190100
Ishihara, Kazuo, 1978, The buckling of plates by the mixed finite element method, Mem. Numer. Math., 73
Ishihara, Kazuo, 1978, A mixed finite element method for the biharmonic eigenvalue problems of plate bending, Publ. Res. Inst. Math. Sci., 14, 399, 10.2977/prims/1195189071
Johnson, Claes, 1973, On the convergence of a mixed finite-element method for plate bending problems, Numer. Math., 21, 43, 10.1007/BF01436186
Kolata, William G., 1977, Approximation in variationally posed eigenvalue problems, Numer. Math., 29, 159, 10.1007/BF01390335
Mercier, B., 1974, Numerical solution of the biharmonic problem by mixed finite elements of class 𝐶⁰, Boll. Un. Mat. Ital. (4), 10, 133
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.
Miyoshi, Tetsuhiko, 1972, A finite element method for the solutions of fourth order partial differential equations, Kumamoto J. Sci. (Math.), 9, 87
S. Nemat-Nasser, "General variational methods for elastic waves in composites," J. Elasticity, v. 2, 1972, pp. 73-90.
S. Nemat-Nasser, General Variational Principles in Nonlinear and Linear Elasticity with Applications. Mechanics Today 1, Pergamon Press, New York, 1974, pp. 214-261.
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.
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.
Oden, J. T., 1976, On mixed finite element approximations, SIAM J. Numer. Anal., 13, 393, 10.1137/0713035
Osborn, John E., 1975, Spectral approximation for compact operators, Math. Comput., 29, 712, 10.1090/S0025-5718-1975-0383117-3
Osborn, John E., 1976, Approximation of the eigenvalues of a nonselfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations, SIAM J. Numer. Anal., 13, 185, 10.1137/0713019
1979, Advances in computer methods for partial differential equations. III
Rannacher, Rolf, 1979, On nonconforming and mixed finite element method for plate bending problems. The linear case, RAIRO Anal. Num\'{e}r., 13, 369, 10.1051/m2an/1979130403691
Raviart, P.-A., 1977, A mixed finite element method for 2nd order elliptic problems, 292
Raviart, P.-A., 1977, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp., 31, 391, 10.2307/2006423
Scholz, Reinhard, 1976, Approximation von Sattelpunkten mit finiten Elementen, 53
Scholz, Reinhard, 1978, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Num\'{e}r., 12, 85, 10.1051/m2an/1978120100851
J. M. Thomas, Sur l’Analyse Numérique des Méthodes d’Eléments Finis Hybrides et Mixtes, Thesis, Univ. P & M Curie, Paris, 1977.
Wilkinson, J. H., 1965, The algebraic eigenvalue problem