Legendre-Fourier spectral approximation and error analysis for nonlinear eigenvalue problems in complex domains

In this paper, we develop and analyze an efficient Legendre-Fourier spectral approximation for solving nonlinear eigenvalue problems in complex domains. The main idea is to employ the domain mapping method to convert the nonlinear eigenvalue problem on a complex domain into an equivalent form on a s...

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Bibliographic Details
Published in:Journal of applied mathematics & computing Vol. 71; no. 4; pp. 5477 - 5504
Main Authors: Zheng, Jihui, An, Jing
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2025
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Coordinate transformations
Eigenvalues
Eigenvectors
Error analysis
Estimates
Fourier series
Mapping
Mathematical and Computational Engineering
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical analysis
Original Research
Polynomials
Spectral methods
Theory of Computation
65N25
Complex domains
65N15
81Q05
Error analysis
Nonlinear eigenvalue problems
Legendre-fourier spectral method
35Q55
65N30
ISSN:1598-5865, 1865-2085
Online Access:Get full text
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Summary:In this paper, we develop and analyze an efficient Legendre-Fourier spectral approximation for solving nonlinear eigenvalue problems in complex domains. The main idea is to employ the domain mapping method to convert the nonlinear eigenvalue problem on a complex domain into an equivalent form on a standard circular domain. Based on this, an effective Legendre-Fourier spectral method is implemented by utilizing Legendre polynomials and Fourier series approximations in the radial and tangential directions, respectively. As the initial step, we establish a priori error estimates for standard circular regions. Then, we define a new class of projection operators, demonstrate their approximation properties, and further prove the error estimates for approximating eigenvalues and their corresponding eigenfunctions. Subsequently, by employing region mapping techniques, we extend the algorithm to address nonlinear eigenvalue problems in two-dimensional complex domains, and validate its convergence and spectral accuracy through numerical examples.
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ISSN:1598-5865
1865-2085
DOI:10.1007/s12190-025-02444-w