Analytical solutions to some generalized and polynomial eigenvalue problems

Deng, Quanling

doi:10.48550/arxiv.2007.08130

Cited by 1 publication

(2 citation statements)

References 30 publications

(31 reference statements)

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“…In such a case, the analytical approximate eigenvalues are given in (5.14) and it is a smooth function in terms of the index j, therefore, no outliers. For multiple dimensional problems, the analytical eigenpairs can be obtained by applying Theorem 7 in [18]. For higher-order elements, one requires a reconstruction of the basis functions near the boundaries to recover this consistency.…”

Section: Cubic Elements For the Dirichlet Eigenvalue Problemmentioning

confidence: 99%

“…In the limiting case, the penalty on (−∆) α u = λ α u at the boundaries becomes an extra condition for the matrix system. For C 2 cubic elements for the Dirichlet eigenvalue problem and C 1 quadratic elements for the Neumann eigenvalue problem, based on the work [18], we perform the dispersion analysis (which is unified with the spectrum analysis in [4]) near the boundary and give exact eigenpairs for the resulting matrix problems. For higher-oder elements, a reconstruction of the basis functions near the boundaries could lead to exact eigenpairs of the matrix problems.…”

Section: Introductionmentioning

confidence: 99%

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A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes

Deng,

Calo

2020

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We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The technique removes the outliers appearing in the highfrequency region of the approximate spectrum when using the C p−1 , p-th (p ≥ 3) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multiple dimensional problems. To remove the outliers, we penalize the product of the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis to the Neumann eigenvalue problem (for p ≥ 2). The boundary penalization does not change the test and solution spaces. In the limiting case when the penalty goes to infinity, we perform the dispersion analysis of C 2 cubic elements for Dirichlet eigenvalue problem and C 1 quadratic elements for Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator.

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Section: Cubic Elements For the Dirichlet Eigenvalue Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%