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Quadrature errors in finite element eigenvalue computations

Published online by Cambridge University Press:  17 February 2009

Alan L. Andrew
Alan L. Andrew
Affiliation:
Mathematics Department, La Trobe University, Bundoora, VIC 3083, Australia.
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Abstract

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Some recent results on the optimal choice of quadrature rules for the finite element solution of eigenvalue problems are discussed in the light of some results of the author and J. Paine.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 1 , July 2000 , pp. 136 - 140
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Andrew, A. L., “Asymptotic correction of finite difference eigenvalues”, in Computational Techniques and Applications: CTAC-85 (eds Noye, J. and May, R.), (North-Holland, Amsterdam, 1986) 333–341.Google Scholar
[2]Andrew, A. L., “Correction of finite element eigenvalues for problems with natural or periodic boundary conditions”, BIT 25 (1988), 254–269.CrossRefGoogle Scholar
[3]Andrew, A. L., “Some recent developments in finite element eigenvalue computation”, in Computational Techniques and Applications: CTAC-87 (eds Noye, J. and Fletcher, C.), (North-Holland, Amsterdam, 1988) 83–91.Google Scholar
[4]Andrew, A. L., “Asymptotic correction of computed eigenvalues of differential equations”, Ann. Numer. Math. 1 (1994) 41–51.Google Scholar
[5]Andrew, A. L. and Paine, J. W., “Correction of finite element estimates for Sturm-Liouville eigenvalues”, Numer. Math. 50 (1986) 205–215.CrossRefGoogle Scholar
[6]Banerjee, U., “A note on the effect of numerical quadrature in finite element eigenvalue approximation”, Numer. Math. 61 (1992) 145–152.CrossRefGoogle Scholar
[7]Banerjee, U. and Osborn, J. E., “Estimation of the effect of numerical integration in finite element eigenvalue approximation”, Numer. Math. 56 (1990) 735–762.CrossRefGoogle Scholar
[8]Banerjee, U. and Suri, M., “Analysis of numerical integration in p-version finite element eigenvalue approximation”, Numer. Methods Partial Differential Equations 8 (1992) 381–394.CrossRefGoogle Scholar
[9]Dun, C. R., “Algebraic correction methods for two-dimensional eigenvalue problems”, Ph. D. Thesis, Australian National University, 1995.Google Scholar
[10]Elliott, D. and Paget, D. F., “Product-integration rules and their convergence”, BIT 16(1976) 32–40.CrossRefGoogle Scholar
[11]Elliott, D. and Paget, D. F., “The convergence of product integration rules”, BIT 18 (1978) 137–141.CrossRefGoogle Scholar
[12]Fix, G. J., “Effects of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems”, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (ed. Aziz, A. K.), (Academic Press, New York, 1972) 525–556.CrossRefGoogle Scholar
[13]Lewinska, E., “Approximation of the eigenvalue problem for elliptic operator by finite element method with numerical integration”, Mat. Stos. 37 (1994) 3–14.Google Scholar
[14]Mikhlin, S. G., The Numerical Performance of Variational Methods, (English transl. Anderssen, R. S.), (Wolters-Noordhoff, Gröningen, 1972).Google Scholar
[15]Paine, J. W., de Hoog, F. R. and Anderssen, R. S., “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems”, Computing 26 (1981) 123–139.CrossRefGoogle Scholar
[16]Pryce, J. D., Numerical Solution of Sturm-Liouville Problems (Oxford Univ. Press, London, 1993).Google Scholar
[17]Sloan, I. H. and Smith, W. E., “Properties of interpolatory product integration”, SIAM J. Numer. Anal. 19(1982)427–442.CrossRefGoogle Scholar
[18]Vanmaele, M. and Van Keer, R., “An operator method for a numerical quadrature finite element approximation for a class of second-order elliptic eigenvalue problems in composite structures”, RAIRO Modél. Math. Anal. Numér. 29 (1995) 339–365.CrossRefGoogle Scholar
[19]Vanmaele, M. and Van Keer, R., “On a variational approximation method for a class of elliptic eigenvalue problems in composite structures”, Math. Comput. 65 (1996) 999–1017.CrossRefGoogle Scholar
[20]Vanmaele, M. and Ženíšek, A., “The combined effect of numerical integration and approximation of the boundary condition in the finite element method for eigenvalue problems”, Numer. Math. 71 (1995) 253–273.CrossRefGoogle Scholar