Isoparametric mixed finite element approximation of eigenvalues and eigenvectors of 4th order eigenvalue problems with variable coefficients

Bhattacharyya, Pulin Kumar; Nataraj, Neela

ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 1, pp. 1-32.

Estimates for the combined effect of boundary approximation and numerical integration on the approximation of (simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficients in convex domains with curved boundary by an isoparametric mixed finite element method, which, in the particular case of bending problems of aniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending moment tensor field $\Psi ={\left({\psi }_{ij}\right)}_{1\le i,j\le 2}$ and displacement field ‘u’, have been developed.

Classification : 35J40, 65N30, 35P99, 74H45
Mots clés : mixed FEM, eigenvalue problem, isoparametric boundary approximation, 4th-order equations, anisotropic plates, convergence analysis, numerical results

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] I. Babuška and J.E. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis. Vol. II, P.G. Ciarlet and J.-L. Lions, Eds., in Finite Element Methods (Part 1), North-Holland Publishing Company, Amsterdam, New York, Oxford (1991) 641-787. | Zbl

[3] S. Balasundaram and P.K. Bhattacharyya, On existence of solution of the Dirichlet problem of fourth-order partial differential equations with variable coefficients. Quart. Appl. Math. 39 (1983) 311-317. | Zbl

[4] S. Balasundaram and P.K. Bhattacharyya, A mixed finite element method for fourth-order elliptic equations with variable coefficients. Comput. Math. Appl. 10 (1984) 245-256. | Zbl

[5] U. Banerjee, A note on the effect of numerical quadrature in finite element eigenvalue approximation. Numer. Math. 61 (1992) 145-152. | Zbl

[6] U. Banerjee and J.E. Osborn, Estimates of the effect of numerical integration in finite element eigenvalue approximation. Numer. Math. 56 (1990) 735-762. | Zbl

[7] M. Bernadou, Méthodes numériques pour les problèmes elliptiques, in Méthodes de Mathématiques Appliquées, Vol. 1, Chap. XI, C.E.A., France (1981).

[8] P.K. Bhattacharyya and N. Nataraj, Isoparametric Mixed Finite Element Approximation of Eigenvalues and Eigenvectors of 4th-Order Eigenvalue Problems with Variable Coefficients. Research Report No. R 98021, Laboratoire d' Analyse Numérique, Université de Paris VI (1998).

[9] P.K. Bhattacharyya and N. Nataraj, On Combined Effect of Boundary Approximation and Numerical Integration on Mixed Finite Element Solution of 4th order Elliptic Problems with Variable Coefficients. ESAIM: M2AN 133 (1999) 807-836. | Numdam | Zbl

[10] P.K. Bhattacharyya and N. Nataraj, Error Estimates for Isoparametric Mixed Finite Element Solution of 4th-Order Elliptic Problems with Variable Coefficients. In preparation. | Zbl

[11] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8 (1974) 129-151. | EuDML | Numdam | Zbl

[12] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin, Heidelberg, New York (1991). | MR | Zbl

[13] F. Brezzi and P.A. Raviart, Mixed finite element methods for 4th-order elliptic equations. Topics Numer. Anal. III, J. Miller, Ed., Academic Press, New York (1978) 33-56. | Zbl

[14] C. Canuto, Eigenvalue approximation by mixed methods. RAIRO Anal. Numér. 12 (1978) 27-50. | EuDML | Numdam | Zbl

[15] F. Chatelin, Spectral Approximation of Linear Operators. Academic Press, New York (1983). | MR | Zbl

[16] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR | Zbl

[17] P.G. Ciarlet and P.A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, in Math. Found. Finite El. Method Appl. Part. Differ. Equations, A.K. Aziz and I. Babuška, Eds., Sympos. Univ. Maryland, Baltimore (1972) 409-474. | Zbl

[18] M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Springer-Verlag, Berlin, Heidelberg, New York (1988). | MR | Zbl

[19] N.J. Decapua and B.C. Sun, Transverse vibration of a class of orthotropic plates. J. Appl. Mech. (1972) 613-615.

[20] S. Gopalsamy and P.K. Bhattacharyya, On existence and uniqueness of solution of boundary value problems of fourth-order elliptic partial differential equations with variable coefficients. J. Math. Anal. Appl. 136 (1988) 589-608. | Zbl

[21] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

[22] K. Ishihara, A mixed finite element method for the biharmonic eigenvalue problem of plate bending. Publ. Res. Inst. Math. Sci. Kyoto Univ. 14 (1978) 399-414. | Zbl

[23] K. Ishihara, The buckling of plates by the mixed finite element method. Mem. Numer. Math. 5 (1978) 73-82. | Zbl

[24] V.A. Kondratev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch. 16 (1967) 209-292. | Zbl

[25] M.P. Lebaud, Error estimate in an isoparametric finite element Eigenvalue Problem. Math. Comp. 63 (1994) 19-40. | Zbl

[26] A.W. Leissa, Vibration of Plates. NASA SP-160 (1969).

[27] S.G. Leknitskii, Anisotropic Plates. Gordon and Breach Science Publishers, New York (1968).

[28] J.-L. Lions, Problèmes aux limites dans les équations aux dérivées partielles. Presses Univ. Montréal, Montreal (1965). | MR | Zbl

[29] B. Mercier, J. Osborn, J. Rappaz and P.A. Raviart, Eigenvalue approximation by mixed and hybrid methods. Math. Comp. 36 (1981) 427-453. | Zbl

[30] T. Miyoshi, A finite element method for the solution of fourth-order partial differential equations. Kumamoto J. Sci. (Math.) 9 (1973) 87-116. | Zbl

[31] N. Nataraj, On Mixed Finite Element Analysis Of Fourth Order Elliptic Source/Eigenvalue Problems in Convex Domains. Ph.D. Thesis, Dept. of Mathematics, Indian Institute of Technology, Delhi (1998).

[32] L.A. Oganesian and L.A. Rukhovec, Variational difference methods for the solution of elliptic problems. Izv. Akad. Nauk Armyan. SSR, Yerevan (1979). In Russian.

[33] A.B. Ouaritini, Méthodes d'éléments finis mixtes pour des problèmes de coques minces1984).

[34] O. Pironneau, Méthodes des éléments finis pour les fluides. Masson, Paris (1988). | Zbl

[35] P.A. Raviart and J.M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, Paris (1983). | Zbl

[36] J.E. Roberts and J.M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis. Vol. II, in Finite Element Methods (Part 1), P.G. Ciarlet and J.-L. Lions, Eds., North-Holland Publishing Company, Amsterdam, New York, Oxford, (1991) 523-633. | Zbl

[37] G. Strang and G.J. Fix, An Analysis of the Finite Element Method. Prentice-Hall, New York (1973). | MR | Zbl

[38] S. Timoshenko and S. Woinowsky-Kreiger, Theory of Plates and Shells. McGraw-Hill Book Company, New York (1959).

[39] M. Vanmaele and A. Ženíšek, External finite-element approximations of eigenvalue problems. RAIRO Modél. Math. Anal. Numér. 27 (1993) 565-589. | EuDML | Numdam | Zbl

[40] M. Vanmaele and A. Ženíšek, The combined effect of numerical integration and approximation of boundary in the finite element method for eigenvalue problems. Numer. Math. (1995). | MR | Zbl

[41] A. Zeníšek, Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. Academic Press, New York (1990). | MR | Zbl