Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem
A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter has been studied. The differential eigenvalue problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms nonlinearly depending on the spectral p...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2017-09-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/error-investigation-of-finite-element_332935.html |
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| author | A.A. Samsonov P.S. Solov'ev S.I. Solov'ev |
| author_facet | A.A. Samsonov P.S. Solov'ev S.I. Solov'ev |
| author_sort | A.A. Samsonov |
| collection | DOAJ |
| description | A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter has been studied. The differential eigenvalue problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms nonlinearly depending on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues, which correspond to a normalized system of eigenfunctions. The variational problem has been approximated by a mesh scheme of the finite element method on the uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions in dependence on mesh size and eigenvalue size have been established. The obtained results are generalizations of the well-known results for differential eigenvalue problems with linear dependence on the spectral parameter. |
| format | Article |
| id | doaj-art-e8a8f86bfe9c43d69a26cf417e99e145 |
| institution | OA Journals |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2017-09-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-e8a8f86bfe9c43d69a26cf417e99e1452025-08-20T02:17:57ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982017-09-011593354363Error investigation of finite element approximation for a nonlinear Sturm–Liouville problemA.A. Samsonov0P.S. Solov'ev1S.I. Solov'ev2Kazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaA positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter has been studied. The differential eigenvalue problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms nonlinearly depending on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues, which correspond to a normalized system of eigenfunctions. The variational problem has been approximated by a mesh scheme of the finite element method on the uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions in dependence on mesh size and eigenvalue size have been established. The obtained results are generalizations of the well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.https://kpfu.ru/error-investigation-of-finite-element_332935.htmleigenvalueeigenfunctioneigenvalue problemmesh approximationfinite element method |
| spellingShingle | A.A. Samsonov P.S. Solov'ev S.I. Solov'ev Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem Учёные записки Казанского университета: Серия Физико-математические науки eigenvalue eigenfunction eigenvalue problem mesh approximation finite element method |
| title | Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem |
| title_full | Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem |
| title_fullStr | Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem |
| title_full_unstemmed | Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem |
| title_short | Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem |
| title_sort | error investigation of finite element approximation for a nonlinear sturm liouville problem |
| topic | eigenvalue eigenfunction eigenvalue problem mesh approximation finite element method |
| url | https://kpfu.ru/error-investigation-of-finite-element_332935.html |
| work_keys_str_mv | AT aasamsonov errorinvestigationoffiniteelementapproximationforanonlinearsturmliouvilleproblem AT pssolovev errorinvestigationoffiniteelementapproximationforanonlinearsturmliouvilleproblem AT sisolovev errorinvestigationoffiniteelementapproximationforanonlinearsturmliouvilleproblem |