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A note on the effect of numerical quadrature in finite element eigenvalue approximation. (English) Zbl 0748.65078
This (concise!) paper gives a deeper insight into the results of the author’s and J. E. Osborn’s earlier paper (with almost the same title) [Numer. Math. 56, No. 8, 735-762 (1990; Zbl 0693.65071)]. It was shown there that the finite element approximation of the eigenpairs of elliptic differential operators (with dimensions 1, 2, or 3), when the elements of the underlying matrices are approximated by numerical quadrature, yields optimal order of convergence when the precision of the integration is one higher than that used for the finite element approximation of the solution of the corresponding source problem.
Here are the refined results: the original assumption is indeed sharp, to obtain the optimal order of convergence for the approximate eigenvalues. But one does not require to increase the precision of the quadrature to obtain the optimal order of convergence for the eigenvectors.

MSC:
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
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References:
[1] Babu?ka, 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 Holland
[2] Banerjee, U., Osborn, J.E. (1990): Estimation of the effect of numerical integration in finite element eigenvalue approximation. Numer. Math.56, 735-762 · Zbl 0693.65071 · doi:10.1007/BF01405286
[3] Birkhoff, G., de Boor, C., Swartz, B., Wendroff, B. (1966): Rayleigh-Ritz approximation by piecewise cubic polynomials. SIAM J. Numer. Anal.3, 188-203 · Zbl 0143.38002 · doi:10.1137/0703015
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[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. 525-556
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