An accurate algorithm for computing the eigenvalues of a polygonal membrane

Descloux, J.; Tolley, Michael

doi:10.1016/0045-7825(83)90072-5

Cited by 41 publications

(38 citation statements)

References 9 publications

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“…Such an approach was used by Driscoll with great success in a related method [12]. But this approach makes it necessary to accurately compute derivatives of F (λ) and G(λ), which might not always be possible.…”

Section: Comparison To Generalized Eigenvalue Formulations Based On mentioning

confidence: 99%

The Generalized Singular Value Decomposition and the Method of Particular Solutions

Betcke¹

2008

SIAM J. Sci. Comput.

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Abstract. A powerful method for solving planar eigenvalue problems is the Method of Particular Solutions (MPS), which is also well known under the name "point matching method". The implementation of this method usually depends on the solution of one of three types of linear algebra problems: singular value decomposition, generalized eigenvalue decomposition, or generalized singular value decomposition. We compare and give geometric interpretations of these different variants of the MPS. It turns out that the most stable and accurate of them is based on the Generalized Singular Value Decomposition. We present results to this effect and demonstrate the behavior of the generalized singular value decomposition in the presence of a highly ill-conditioned basis of particular solutions.

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Section: Comparison To Generalized Eigenvalue Formulations Based On mentioning

confidence: 99%

The Generalized Singular Value Decomposition and the Method of Particular Solutions

Betcke¹

2008

SIAM J. Sci. Comput.

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“…Our algorithm will only find one of them, but for the computed eigenvalue r 2 1 , the second smallest singular value of A(r 1 , M , m 0 ) is quite small, indicating that there exists a linear combination of functions of the form J m (r 1 ( ))e im that, while orthogonal to the computed eigenfunction, nearly satisfies the boundary conditions. It is therefore reasonable to conclude that there is a second eigenvalue r 2 2 nearby, which can be found by scanning the graph of 1 (A(r , M , m 0 )) near r 1 .…”

Section: Detecting Multiple and Clustered Eigenvaluesmentioning

confidence: 99%

“…Second, the size of the discretization matrix grows with the number of eigenvalues to be computed. For this type of approach we refer to [1,2] for algorithms using finite elements and finite differences, respectively. One advantage of this method is of course that it can be applied to a wide class of operators, not necessarily with constant coefficients.…”

Section: P Guidotti and J V Lambersmentioning

confidence: 99%

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Eigenvalue Characterization and Computation for the Laplacian on General 2-D Domains

Guidotti

Lambers

2008

Numerical Functional Analysis and Optimization

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“…Hence, h-refinement is not necessary. This is similar to the domain decomposition method of Descloux and Tolley for the Laplace eigenvalue problem on polygonal domains [12].…”

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confidence: 99%

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

Barnett¹,

Betcke²

2010

SIAM J. Sci. Comput.

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Abstract. In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners, and representing the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

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An accurate algorithm for computing the eigenvalues of a polygonal membrane

Cited by 41 publications

References 9 publications

The Generalized Singular Value Decomposition and the Method of Particular Solutions

The Generalized Singular Value Decomposition and the Method of Particular Solutions

Eigenvalue Characterization and Computation for the Laplacian on General 2-D Domains

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

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