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A fourth-order finite-difference approximation for the fixed membrane eigenproblem

Author: J. R. Kuttler
Journal: Math. Comp. 25 (1971), 237-256
MSC: Primary 65N25
MathSciNet review: 0301955
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Abstract: The fixed membrane problem $\Delta u + \lambda u = 0$ in $\Omega ,u = 0$ on $\partial \Omega$, for a bounded region $\Omega$ of the plane, is approximated by a finite-difference scheme whose matrix is monotone. By an extension of previous methods for schemes with matrices of positive type, $O({h^4})$ convergence is shown for the approximating eigenvalues and eigenfunctions, where h is the mesh width. An application to an approximation of the forced vibration problem $\Delta u + qu = f$ in $\Omega ,u = 0$ in $\partial \Omega$, is also given.

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Keywords: Finite-differences, membrane, fixed membrane, eigenvalues, elliptic partial differential equations, monotone matrices, forced vibration problem, discrete Green’s function
Article copyright: © Copyright 1971 American Mathematical Society