A note on the effect of numerical quadrature in finite element eigenvalue approximation
- 60 Downloads
In a recent work by the author and J.E. Osborn, it was shown that the finite element approximation of the eigenpairs of differential operators, when the elements of the underlying matrices are approximated by numerical quadrature, yield optimal order of convergence when the numerical quadrature satisfies a certain precision requirement. In this note we show that this requirement is indeed sharp for eigenvalue approximation. We also show that the optimal order of convergence for approximate eigenvectors can be obtained, using numerical quadrature with less precision.
Mathematics Subject Classification (1991)65N25
Unable to display preview. Download preview PDF.
- 1.Babuška, I., Osborn, J.E. (1991): Eigenvalue problems. In: P.G. Ciarlet, J.L. Lions eds., Handbook of numerical analysis. Finite Element Methods, vol. II. Amsterdam, North HollandGoogle Scholar
- 2.Banerjee, U., Osborn, J.E. (1990): Estimation of the effect of numerical integration in finite element eigenvalue approximation. Numer. Math.56, 735–762Google Scholar
- 3.Birkhoff, G., de Boor, C., Swartz, B., Wendroff, B. (1966): Rayleigh-Ritz approximation by piecewise cubic polynomials. SIAM J. Numer. Anal.3, 188–203Google Scholar
- 4.Chatelin, F. (1983): Spectral approximation of linear operators. Academic Press, New YorkGoogle Scholar
- 5.Ciarlet, P.G. (1978): The finite element method for elliptic problems. Amsterdam, North HollandGoogle Scholar
- 6.Ciarlet, P.G., Raviart, P.-A. (1972): The mathematical foundation of the finite element method with application to partial differential equations. In: A.K. Aziz, ed., The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. Academic Press, New York, pp. 404–474Google Scholar
- 7.Fix, F.J. (1977): The mathematical foundation of the finite element method with application to partial differential equations. In: A.K. Aziz, ed., Effect of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems. Academic Press, New York, pp. 525–556Google Scholar
- 8.Descloux, J., Nassif, N., Rappaz, J. (1973): On spectral approximation, Part I. The problem of convergence. RAIRO Anal. Numer.12, 97–112Google Scholar