We are concerned with the Dirichlet eigenvalue problem Δu + λu = 0 G, u = 0 on Γ, where G is a bounded, two dimensional domain with sufficiently smooth boundary Γ. We deal with the “Ansatz” \documentclass{article}\pagestyle{empty}\begin{document}$ u\left({x,\lambda } \right) = \sum\limits_{m = 1}^M {a_m Y_0 } \left({\sqrt \lambda |x - y_m |} \right) $\end{document} to compute approximate eigenpairs (u*, λ*) by the collocation method. Lower and upper eigenvalue bounds are estimated by an inclusion theorem due to Kuttler‐Sigillito. In contrast to usual choices of trial functions, it is possible to control the numerical stability by placing the source points in dependence of M. The effect arises from the logarithmic singularity for x = ym and allows us to sharpen the eigenvalue bounds by increasing M. We give a strategy for placing the sources, present various application and give a comparison of results which indicates a high efficiency of the method.
We present new inclusion theorems for eigenvalues of the free membrane. These permit a computation of both sided bounds from a given approximate eigenpair (u * , µ * ) in a simple manner. For a properly normalized test function u * , the bounds depend on the quadratic forms G (∆u * + µ * u * ) 2 df and Γ ( ∂ ∂n u * ) 2 ds only. We give very close eigenvalue bounds for the example of the "L"-shaped membran consisting of three unit squares. (1991): 65 N25; 35 P15
Mathematics Subject Classification
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