{"title":"各向同性非均质介质传输特征值问题的频谱近似和误差分析","authors":"","doi":"10.1016/j.cam.2024.116163","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we propose an effective Legendre-Fourier spectral method for the transmission eigenvalue problem in polar geometry with an isotropic inhomogeneous medium. The basic idea of this methodology is to rewrite the initial problem into its equivalent form by using polar coordinates and some specially designed polar conditions. A variational method and its discrete version (i.e., Legendre-Fourier spectral method) are then presented within a class of weighted Sobolev spaces. With the help of the spectral theory of compact operators and the approximation properties of some specially designed projections in non-uniformly weighted Sobolev spaces, error estimates with spectral accuracy of the Legendre-Fourier spectral method for both the eigenvalue and eigenfunction approximations are established. Numerical experiments are presented to confirm the theoretical findings and the efficiency of our algorithm.</p></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Spectral approximation and error analysis for the transmission eigenvalue problem with an isotropic inhomogeneous medium\",\"authors\":\"\",\"doi\":\"10.1016/j.cam.2024.116163\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we propose an effective Legendre-Fourier spectral method for the transmission eigenvalue problem in polar geometry with an isotropic inhomogeneous medium. The basic idea of this methodology is to rewrite the initial problem into its equivalent form by using polar coordinates and some specially designed polar conditions. A variational method and its discrete version (i.e., Legendre-Fourier spectral method) are then presented within a class of weighted Sobolev spaces. With the help of the spectral theory of compact operators and the approximation properties of some specially designed projections in non-uniformly weighted Sobolev spaces, error estimates with spectral accuracy of the Legendre-Fourier spectral method for both the eigenvalue and eigenfunction approximations are established. Numerical experiments are presented to confirm the theoretical findings and the efficiency of our algorithm.</p></div>\",\"PeriodicalId\":50226,\"journal\":{\"name\":\"Journal of Computational and Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-07-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0377042724004126\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724004126","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Spectral approximation and error analysis for the transmission eigenvalue problem with an isotropic inhomogeneous medium
In this paper, we propose an effective Legendre-Fourier spectral method for the transmission eigenvalue problem in polar geometry with an isotropic inhomogeneous medium. The basic idea of this methodology is to rewrite the initial problem into its equivalent form by using polar coordinates and some specially designed polar conditions. A variational method and its discrete version (i.e., Legendre-Fourier spectral method) are then presented within a class of weighted Sobolev spaces. With the help of the spectral theory of compact operators and the approximation properties of some specially designed projections in non-uniformly weighted Sobolev spaces, error estimates with spectral accuracy of the Legendre-Fourier spectral method for both the eigenvalue and eigenfunction approximations are established. Numerical experiments are presented to confirm the theoretical findings and the efficiency of our algorithm.
期刊介绍:
The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest.
The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.