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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| Published in: | Journal of applied mathematics & computing Vol. 71; no. 4; pp. 5477 - 5504 |
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| Language: | English |
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Berlin/Heidelberg
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01.08.2025
Springer Nature B.V |
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| ISSN: | 1598-5865, 1865-2085 |
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| Abstract | 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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| AbstractList | 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. |
| Author | An, Jing Zheng, Jihui |
| Author_xml | – sequence: 1 givenname: Jihui surname: Zheng fullname: Zheng, Jihui organization: School of Mathematical Sciences, Guizhou Normal University – sequence: 2 givenname: Jing orcidid: 0000-0003-1855-9761 surname: An fullname: An, Jing email: anjing@gznu.edu.cn organization: School of Mathematical Sciences, Guizhou Normal University |
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| Cites_doi | 10.1017/CBO9780511805769 10.1051/m2an/2011038 10.1007/s10444-011-9235-y 10.1016/j.cam.2008.08.034 10.1007/978-3-540-71041-7 10.3934/krm.2013.6.1 10.1002/mma.793 10.1016/j.jcp.2023.112352 10.1007/BF02105068 10.1137/S1064827503422956 10.1142/S0129183123500523 10.1063/1.4869598 10.1016/j.cma.2011.02.008 10.1137/20M1345153 10.1016/j.rinp.2022.105891 10.1007/s10915-010-9358-1 10.1016/j.apnum.2018.04.012 10.1088/0951-7715/16/1/307 10.1016/j.heliyon.2024.e25903 10.1103/RevModPhys.53.603 10.1088/0951-7715/17/2/010 10.1007/s10915-020-01189-x 10.1002/mma.1292 10.1016/j.rinam.2024.100504 |
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| SubjectTerms | 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 |
| Title | Legendre-Fourier spectral approximation and error analysis for nonlinear eigenvalue problems in complex domains |
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