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Dirichlet spectral-Galerkin approximation method for the simply supported vibrating plate eigenvalues

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Abstract

In this paper, we analyze and implement the Dirichlet spectral-Galerkin method for approximating simply supported vibrating plate eigenvalues with variable coefficients. This is a Galerkin approximation that uses the approximation space that is the span of finitely many Dirichlet eigenfunctions for the Laplacian. Convergence and error analysis for this method is presented for two and three dimensions. Here we will assume that the domain has either a smooth or Lipschitz boundary with no reentrant corners. An important component of the error analysis is Weyl’s law for the Dirichlet eigenvalues. Numerical examples for computing the simply supported vibrating plate eigenvalues for the unit disk and square are presented. In order to test the accuracy of the approximation, we compare the spectral-Galerkin method to the separation of variables for the unit disk. Whereas for the unit square we will numerically test the convergence rate for a variable coefficient problem.

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Acknowledgements

The author would like to thank Andreas Kleefeld for helpful feedback and suggestions which lead to a much improved manuscript.

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Isaac Harris

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  1. Isaac Harris
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Correspondence to Isaac Harris.

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This article is part of the section “Computational Approaches” edited by Siddhartha Mishra.

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Harris, I. Dirichlet spectral-Galerkin approximation method for the simply supported vibrating plate eigenvalues. Partial Differ. Equ. Appl. 3, 34 (2022). https://doi.org/10.1007/s42985-022-00171-6

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  • DOI: https://doi.org/10.1007/s42985-022-00171-6

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