zbMATH — the first resource for mathematics

Some results in lumped mass finite-element approximation of eigenvalue problems using numerical quadrature formulas. (English) Zbl 0762.65056
The paper deals with second order elliptic eigenvalue problems defined on rectangular domains and exhibiting a discrete spectrum \(0<\lambda_ 1\leq \lambda_ 2\leq\dots\), \(\lambda_ n\to\infty\) as \(n\to\infty\). Starting from the variational formulation \(\lambda\in\mathbb{R}\), \(u\in V\): \(a(u,v)=\lambda(u,v)\) for all \(v\in V\), a “lumped mass” discrete eigenvalue problem is introduced. The latter arises from a Lobatto quadrature formula for the inner product \((\;,\;)\) together with rectangular Lagrange finite elements of order \(k\geq 2\).
For simple eigenvalues \(\lambda\) and approximate eigenpairs \((\lambda^ h,u^ h)\to(\lambda,u)\) as \(h\to 0\), various results on the order of convergence are obtained. In particular it is shown that the lumped mass approximation of an eigenvalue is as good as the consistent mass approximation, at least if the bilinear form \(a(\;,\;)\) is “regular” so that the Aubin-Nitsche argument can be applied. An example is presented where the numerical results indicate the optimal error estimate to hold.

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
Full Text: DOI
[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Banerjee, U.; Osborn, J.E., Estimation of the effect of numerical integration in finite element eigenvalue approximation, Numer. math., 56, 735-762, (1990) · Zbl 0693.65071
[3] Bramble, J.H.; Hilbert, S., Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. math., 16, 362-369, (1971) · Zbl 0214.41405
[4] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[5] Davis, P.J.; Rabinowitz, P., Methods of numerical integration, (), 73-89
[6] Fix, G.J., Effects of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems, (), 525-556
[7] Ishihara, K., Convergence of the finite element method applied to the eigenvalue problem δu + λu = 0, RIMS Kyoto univ., 13, 47-60, (1977) · Zbl 0364.65080
[8] Pierce, J.G.; Varga, R.S., Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems, SIAM J. numer. anal., 9, 137-151, (1972) · Zbl 0301.65063
[9] Raviart, P.A.; Thomas, J.-M., Introduction à l’analyse numérique des équations aux Dérivées partielles, (), 11-153
[10] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[11] Tong, P.; Pian, T.H.H.; Bucciarelli, L.L., Mode shapes and frequencies by the finite element method using consistent and lumped masses, J. comput. struct., 1, 623-638, (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.