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COMPUTATION OF THE LEADING EIGENVALUE AND THE CORRESPONDING EIGENELEMENT OF EIGENVALUE PROBLEMS WITH NONLINEAR DEPENDENCE ON THE SPECTRAL PARAMETER
- Authors: Solov’ev P.S.1, Solov’ev S.I.1
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Affiliations:
- Kazan (Volga region) Federal University
- Issue: Vol 60, No 7 (2024)
- Pages: 967–989
- Section: NUMERICAL METHODS
- URL: https://ogarev-online.ru/0374-0641/article/view/265852
- DOI: https://doi.org/10.31857/S037406412407009
- EDN: https://elibrary.ru/KNEQAH
- ID: 265852
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Abstract
The paper studies the symmetric eigenvalue problem with nonlinear dependence on the spectral parameter in a Hilbert space which is a vector lattice with a cone of positive elements. The existence of a positive simple minimum eigenvalue corresponding to a single normalised positive eigenelement is established. The approximation of the problem in a finite-dimensional subspace is investigated. Results on the convergence and error of approximations to the minimum eigenvalue and the corresponding positive eigenelement are obtained. Computational methods for solving matrix eigenvalue problems with nonlinear dependence on the spectral parameter are developed and justified. The results of numerical experiments illustrating theoretical conclusions are given.
About the authors
P. S. Solov’ev
Kazan (Volga region) Federal University
Email: pavel.solovev.kpfu@mail.ru
Russia
S. I. Solov’ev
Kazan (Volga region) Federal University
Email: sergey.solovev.kpfu@mail.ru
Russia
References
- Абдуллин, И.Ш. Высокочастотная плазменно-струйная обработка материалов при пониженных давлениях. Теория и практика применения / И.Ш. Абдуллин, В.С. Желтухин, Н.Ф. Кашапов. — Казань : Изд-во Казанск. ун-та, 2000. — 348 с.
- Abdullin, I.Sh., Zheltukhin, V.S., and Kashapov, N.F., Vysokochastotnaya plazmenno-struinaya obrabotka materialov pri ponizhennykh davleniyakh. Teoriya i praktika primeneniya (Radio-Frequency Plasma-Jet Processing of Materials at Reduced Pressures. Theory and Practice of Application), Kazan: Izd. Kazan. Univ., 2000.
- Соловьев, С.И. Аппроксимация вариационных задач на собственные значения / С.И. Соловьев // Дифференц. уравнения. — 2010. — Т. 46, № 7. — С. 1022–1032.
- Solov’ev, S.I., Approximation of variational eigenvalue problems, Differ. Equat., 2010, vol. 46, no. 7, pp. 1030–1041.
- Соловьев, С.И. Аппроксимация нелинейных спектральных задач в гильбертовом пространстве / С.И. Соловьев // Дифференц. уравнения. — 2015. — Т. 51, № 7. — С. 937–950.
- Solov’ev, S.I., Approximation of nonlinear spectral problems in the Hilbert space, Differ. Equat., 2015, vol. 51, no. 7, pp. 934–947.
- Solov’.ev, S.I. Preconditioned iterative methods for a class of nonlinear eigenvalue problems / S.I. Solov’.ev // Linear Algebra Appl. — 2006. — V. 41, № 1. — P. 210–229.
- Solov’¨ev, S.I., Preconditioned iterative methods for a class of nonlinear eigenvalue problems, Linear Algebra Appl., 2006, vol. 415, no. 1, pp. 210–229.
- Богатов, Е.М. Об истории положительных операторов (1900-е—1960-е гг.) и вкладе М.А. Красносельского / Е.М. Богатов // Прикл. математика & Физика. — 2020. — Т. 52, № 2. — С. 105–127.
- Bogatov, E.M., On the history of the positive operators (1900s—1960s) and the contribution of M.A. Krasnosel’skii, Applied Mathematics & Physics, 2020, vol. 52, no. 2, pp. 105–127.
- Вулих, Б.З. Введение в функциональный анализ / Б.З. Вулих. — М. : Наука, 1967. — 416 с.
- Vulikh, B.Z., Vvedenie v funktsional’nyi analiz (Introduction to Functional Analysis), Moscow: Nauka, 1967.
- Gazzola, F. Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains / F. Gazzola, H.-C. Grunau, G. Sweers. — Berlin : Springer, 2010. — 423 p.
- Gazzola, F., Grunau, H.-C., and Sweers, G., Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Berlin: Springer, 2010.
- B.atkai, A. Positive operator semigroups. From finite to infinite dimensions / A. B.atkai, M.K. Fijavˇz, A. Rhandi. — Cham : Springer, 2017. — 364 p.
- B/atkai, A., Fijavˇz, M.K., and Rhandi, A., Positive Operator Semigroups. From Finite to Infinite Dimensions, Cham: Springer, 2017.
- Воеводин, В.В. Линейная алгебра / В.В. Воеводин. — М. : Наука, 1980. — 400 с.
- Voevodin, V.V., Lineinaya algebra (Linear Algebra), Moscow: Nauka, 1980.Dautov, R.Z. The bisection method for symmetric eigenvalue problems with a parameter entering nonlinearly / R.Z. Dautov, A.D. Lyashko, S.I. Solov’ev // Russ. J. Numer. Anal. Math. Model. — 1994. — V. 9, № 5. — P. 417–427.
- Dautov, R.Z., Lyashko, A.D., and Solov’ev, S.I., The bisection method for symmetric eigenvalue problems with a parameter entering nonlinearly, Russ. J. Numer. Anal. Math. Modelling, 1994, vol. 9, no. 5, pp. 417–427.
- Samsonov, A.A. The bisection method for solving the nonlinear bar eigenvalue problem / A.A. Samsonov, P.S. Solov’ev, S.I. Solov’ev // J. Phys.: Conf. Ser. — 2019. — V. 1158. — Art. 042011.
- Samsonov, A.A., Solov’ev, P.S., and Solov’ev, S.I., The bisection method for solving the nonlinear bar eigenvalue problem, J. Phys.: Conf. Ser., 2019, vol. 1158, art. 042011.
- Samsonov, A.A. Spectrum division for eigenvalue problems with nonlinear dependence on the parameter / A.A. Samsonov, P.S. Solov’ev, S.I. Solov’ev // J. Phys.: Conf. Ser. — 2019. — V. 1158. — Art. 042012.
- Samsonov, A.A., Solov’ev, P.S., and Solov’ev, S.I., Spectrum division for eigenvalue problems with nonlinear dependence on the parameter, J. Phys.: Conf. Ser., 2019, vol. 1158, art. 042012.
- Гантмахер, Ф.Р. Теория матриц / Ф.Р. Гантмахер. — М. : Наука, 1988. — 552 с.
- Gantmakher, F.R., Teoriya matrits (Theory of Matrices), Moscow: Nauka, 1988.
- Парлетт, Б. Симметричная проблема собственных значений. Численные методы / Б. Парлетт. — М. : Мир, 1983. — 384 с.
- Parlett, B., The Symmetric Eigenvalue Problem, Englewood Cliffs: Prentice-Hall, 1980.
- Knyazev, A.V. A geometric theory for preconditioned inverse iteration III: a short and sharp convergence estimate for generalized eigenvalue problems / A.V. Knyazev, K. Neymeyr // Linear Algebra Appl. — 2003. — V. 358, № 1–3. — P. 95–114.
- Knyazev, A.V. and Neymeyr, K., A geometric theory for preconditioned inverse iteration III: a short and sharp convergence estimate for generalized eigenvalue problems, Linear Algebra Appl., 2003, vol. 358, no. 1–3, pp. 95–114.
- Adams, R.A. Sobolev spaces / R.A. Adams. — New York : Academic Press, 1975. — 268 p.
- Adams, R.A., Sobolev Spaces, New York: Academic Press, 1975.
- Sauvigny, F. Partial Differential Equations 2. Functional Analytic Methods / F. Sauvigny. — London : Springer-Verlag, 2012. — 453 p.
- Sauvigny, F., Partial Differential Equations 2. Functional Analytic Methods, London: Springer-Verlag, 2012.
- Гилбарг, Д. Эллиптические дифференциальные уравнения с частными производными второго порядка / Д. Гилбарг, Н. Трудингер ; пер. с англ. Л.Н. Купцова ; под ред. А.К. Гущина. — М. : Наука, 1989. — 464 с.
- Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order, Heidelberg: Springer, 1977.
- Сьярле, Ф. Метод конечных элементов для эллиптических задач / Ф. Сьярле ; пер. с англ. Б.И. Квасова. — М. : Мир, 1980. — 512 с.
- Ciarlet P.G., The Finite Element Method for Elliptic Problems, Amsterdam: North Holland, 1978.
- Banerjee, U. Estimation of the effect of numerical integration in finite element eigenvalue approximation / U. Banerjee, J.E. Osborn // Numer. Math. — 1990. — V. 56. — P. 735–762.
- Banerjee, U. and Osborn, J.E., Estimation of the effect of numerical integration in finite element eigenvalue approximation, Numer. Math., 1990, vol. 56, pp. 735–762.
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