Some results in lumped mass finite-element approximation of eigenvalue problems using numerical quadrature formulas. (English) Zbl 0762.65056

Andreev, A. B.; Kascieva, V. A.; Vanmaele, M.

J. Comput. Appl. Math. 43, No. 3, 291-311 (1992).

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.

Reviewer: W.Velte (Würzburg)

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

Keywords:

lumped mass discrete eigenvalue problem; second order elliptic eigenvalue problems; discrete spectrum; Lobatto quadrature formula; Lagrange finite elements; eigenpairs; order of convergence; numerical results; optimal error estimate

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