A Legendre spectral method based on a hybrid format and its error estimation for fourth-order eigenvalue problems

Yuanqiang Chen; Jihui Zheng; Jing An; Yuanqiang Chen; Jihui Zheng; Jing An

doi:10.3934/math.2024367

AIMS Mathematics

2024, Volume 9, Issue 3: 7570-7588. doi: 10.3934/math.2024367

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Research article

A Legendre spectral method based on a hybrid format and its error estimation for fourth-order eigenvalue problems

1.
School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China

Received: 13 January 2024 Revised: 08 February 2024 Accepted: 19 February 2024 Published: 22 February 2024
MSC : 65N15, 65N35

In this paper, we developed and studied an efficient Legendre spectral method for fourth order eigenvalue problems with the boundary conditions of a simply supported plate. Initially, a new variational formulation based on a hybrid format and its discrete variational form were established. We then employed the spectral theory of complete continuous operators to establish the prior error estimates of the approximate solutions. By integrating approximation results of some orthogonal projection operators in weighted Sobolev spaces, we further gave the error estimation for the approximating eigenvalues and eigenfunctions. In addition, we developed an effective set of basis functions by utilizing the orthogonal properties of Legendre polynomials, and subsequently derived the matrix eigenvalue system of the discrete variational form for both two-dimensional and three-dimensional cases, based on a tensor product. Finally, numerical examples were provided to demonstrate the exponential convergence and efficiency of the algorithm.
- fourth order eigenvalue problem,
- hybrid format,
- Legendre spectral methods,
- error estimation
Citation: Yuanqiang Chen, Jihui Zheng, Jing An. A Legendre spectral method based on a hybrid format and its error estimation for fourth-order eigenvalue problems[J]. AIMS Mathematics, 2024, 9(3): 7570-7588. doi: 10.3934/math.2024367

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Abstract

In this paper, we developed and studied an efficient Legendre spectral method for fourth order eigenvalue problems with the boundary conditions of a simply supported plate. Initially, a new variational formulation based on a hybrid format and its discrete variational form were established. We then employed the spectral theory of complete continuous operators to establish the prior error estimates of the approximate solutions. By integrating approximation results of some orthogonal projection operators in weighted Sobolev spaces, we further gave the error estimation for the approximating eigenvalues and eigenfunctions. In addition, we developed an effective set of basis functions by utilizing the orthogonal properties of Legendre polynomials, and subsequently derived the matrix eigenvalue system of the discrete variational form for both two-dimensional and three-dimensional cases, based on a tensor product. Finally, numerical examples were provided to demonstrate the exponential convergence and efficiency of the algorithm.

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