Max Tran


We investigate the convergence properties of a mixed finite element method approximation to the Biharmonic eigenvalue problem under the presence of numerical integration. We give a brief overview of the results obtained when exact integration is used in a finite element method, then develop related theories and obtained the convergence rates when numerical quadrature is taken into account. The standard approach to obtaining error estimate of variational eigenvalue problems is based on the error estimate of the solution operators of the source problems. The important issues are the rate of convergence of the solution operators and the conditions required for convergence. Paralleling the work of Babuska, Osborn and Pikaranta to overcome some technical difficulties, we will use mesh dependent norms to obtain error estimates between the solutions operators. We then use these estimates to get errors estimates between the approximate and the actual eigenvalues and eigenvectors.

Full Text:



I. Babuska, Error Bounds for the Finite Element Method, Numer. Math. 16 (1971), 322-333.

I. Babuska and A. Aziz, Survey Lectures on the Mathematical Foundations of the Finite Element Method, in: A. K. Aziz, ed., The Mathematical Foundations of the Finite Element Method With Application to Partial Differential Equations,

Academic Press, New York, 1973, 5-359.

I. Babuska and J. E. Osborn, Estimates for the Errors in Eigenvalue And Eigenvector Approximation by Galerkin Methods, With Particular Attention to the Case of Multiple Eigenvalues, SIAM J. Numer. Anal. 24 (1987), 1249-1276.

I. Babuska and J. E. Osborn, Eigenvalue Problems in: P.G. Ciarlet and J.L. Lyons, eds. Handbook of Numerical Analysis,vol. 2, North-Holland, Amsterdam, 1991, 645-787.

I. Babuska, J. Osborn and J. Pitkaranta, Analysis of Mixed Methods Using Mesh Dependent Norms, Math. Comp. 35 (1990), 1039-1062.

U. Banerjee and J. E. Osborn, Estimation in the Effect of Numerical Integration in Finite Element Eigenvalue Approximation, Numer. Math. 56 (1990), 735-762.

P.K. Bhattacharyya and N. Nataraj, Isoparametric Mixed Finite Element Approximation of Eigenvalues and Eigenvectors of 4th Order Eigenvalue Problems With Variable Coefficients, Mathematical Modelling and Numerical Analysis (M2AN),

-32 (2002).

J.H. Bramble and J.E. Osborn, Rate of Convergence Estimates for Nonselfadjoint Eigenvalue Approximations, Math. Comp., 27 (1973), 525-549.

F. Brezzi, On the Existence, Uniqueness And Approximation of Saddle-point Problems Arising From Lagrangian Multipliers, R.A.I.R.O., R2, v. 8 (1974), 129-151.

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.

C. Canuto, Eigenvalue Approximation by Mixed Methods, R.A.I.R.O. Anal Numer., v. 12 (1978), 27-50.

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

P. Cl ement, Approximation by Finite Element Functions Using Local Regularization, R.A.I.R.O. R2 (1975), 77-84.

J. Descloux, N. Nassif, and J. Rappaz, On Spectral Approximation Part 1. The Problem of Convergence, R.A.I.R.O. Anal. Numer 12 (1978), 97-112.

J. Descloux, N. Nassif, and J. Rappaz, On Spectral Approximation Part 2. Error Estimates for the Galerkin Method, R.A.I.R.O. Anal. Numer. 12 (1978), 113-119.

N. Dunford and J.T. Schwartz, Linear Operators, II, Wiley-Interscience, New York, 1963.

R. Falk and J. Osborn, Error Estimates of Mixed Methods, R.A.I.R.O. 14 (1980), 249-277.

G. Fix, Effects of Quadrature Errors in Finite Element Approximation of Steady State, Eigenvalue, and Parabolic Problems in: A. K. Aziz, ed., The Mathematical Foundations of the Finite Element Method With Application to Partial

Differential Equations, Academic Press, New York, 1973, 525-556.

P. Grisvard, Elliptic Problems in Nonsmooth Domain, Pitman, Boston, MA., 1985.

V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Computational Mathematics 749, Springer Berlin, 1979.

K. Ishihara, A Mixed Finite Element Method for the Biharmonic Eigenvalue Problem of Plate Bending, publ. Res. Inst. Math. Sci. Kyoto University, v. 14 (1978), 399-414.

T. Kato, Peturbation Theory for Nullity, Deficiency and Other Quantities Of Linear Operators, J. Anal. Math. 6 (1958), 261-322.

T. Kato, Peturbation Theory for Linear Operators, Lectures Notes in Mathematics 132, Berlin New York, Springer 1966.

S. Kesavan and M. Vanninathan, Sur Une Methodes d’ elements Finis Mixte Pour l’ equation Biharmonique, R.A.I.R.O. 11 (1977), 255-270.

M. P. Lebaud, Error Estimates In An Isoparametric Finite Element Eigenvalue Problem, Math. Comp. 63 (1994), 19-40.

B. Mercier, J. Osborn, J. Rappaz and P. A. Raviart, Eigenvalue Approximation by Mixed and Hybrid Methods, Math. Comp. 36, (1981), 427-452.

J. Necas, Les Methods Directes en Theorie des Equations Elliptiques, Masson, Paris; Academia, Prague, 1967. MR 37 #3118.

M. Vanmaele and A. Zensek, External Finite Element Approximations of Eigenvalue Problems, M.M.A.N. 27 (1993), 565-589.

J. E. Osborn, Spectral Approximation for Compact Operators, Math. Comp. 29 (1975), 712-725.


  • There are currently no refbacks.


Free Web Counter