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The Inclusion of Dirichlet Eigenvalues with Singularity Functions
Author(s) -
Ennenbach R.,
Niemeyer H.
Publication year - 1996
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19960760706
Subject(s) - eigenvalues and eigenvectors , mathematics , bounded function , domain (mathematical analysis) , singularity , logarithm , lambda , boundary (topology) , dirichlet boundary condition , dirichlet eigenvalue , ansatz , dirichlet distribution , mathematical analysis , pure mathematics , combinatorics , mathematical physics , boundary value problem , physics , dirichlet's principle , quantum mechanics
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 = y m 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.