Advertisement

Lobachevskii Journal of Mathematics

, Volume 40, Issue 11, pp 1959–1966 | Cite as

Finite Element Approximation of the Minimal Eigenvalue and the Corresponding Positive Eigenfunction of a Nonlinear Sturm—Liouville Problem

  • D. M. KorostelevaEmail author
  • P. S. Solov’evEmail author
  • S. I. Solov’evEmail author
Article
  • 5 Downloads

Abstract

The problem of finding the minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm—Liouville problem for the ordinary differential equation with coefficients nonlinear depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. A sufficient condition for the existence of a minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm— Liouville problem is established. The original differential eigenvalue problem is approximated by the finite element method with Lagrangian finite elements of arbitrary order on a uniform grid. The error estimates of the approximate eigenvalue and the approximate positive eigenfunction to exact ones are proved. Investigations of this paper generalize well known results for the Sturm—Liouville problem with linear entrance on the spectral parameter.

Keywords and phrases

radio-frequency induction discharge eigenvalue positive eigenfunction nonlinear eigenvalue problem ordinary differential equation finite element method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding

This work was supported by Russian Science Foundation, project no. 16-11-10299.

References

  1. 1.
    I. Sh. Abdullin, V. S. Zheltukhin, and N. F. Kashapov, Radio-Frequency Plasma-Jet Processing of Materials at Reduced Pressures: Theory and Practice of Applications (Izd. Kazan. Univ., Kazan, 2000) [in Russian].Google Scholar
  2. 2.
    V. S. Zheltukhin, S. I. Solov’ev, P. S. Solov’ev, and V. Yu. Chebakova, “Existence of solutions for electron balance problem in the stationary high-frequency induction discharges,” IOP Conf. Sen: Mater. Sci. Eng. 158,012103-1-6 (2016).Google Scholar
  3. 3.
    V. S. Zheltukhin, S. I. Solov’ev, P. S. Solov’ev, V. Yu. Chebakova, and A. M. Sidorov, “Third type boundary conditions for steady state ambipolar diffusion equation,” IOP Conf. Sen: Mater. Sci. Eng. 158, 012102-1—4 (2016).CrossRefGoogle Scholar
  4. 4.
    S. I. Solov’ev, P. S. Solov’ev, and V. Yu. Chebakova, “Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges,” MATEC Web Conf. 129, 06014-1—4 (2017).CrossRefGoogle Scholar
  5. 5.
    S. I. Solov’ev and P. S. Solov’ev, “Finite element approximation of the minimal eigenvalue of a nonlinear eigenvalue problem,” Lobachevskii J. Math. 39 (7), 949–956 (2018).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    S. I. Solov’ev, “ Eigenvibrations of a beam with elastically attached load,” Lobachevskii J. Math. 37 (5), 597–609 (2016).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    S. I. Solov’ev, “ Eigenvibrations of a bar with elastically attached load,” Differ. Equat. 53, 409–423 (2017).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    A. V. Goolin and S. V. Kartyshov, “Numerical study of stability and nonlinear eigenvalue problems,” Surv. Math. Ind. 3, 29–48 (1993).MathSciNetzbMATHGoogle Scholar
  9. 9.
    T. Betcke, N. J. Higham, V. Mehrmann, C. Schroder, and F. Tisseur, “NLEVP: A collection of nonlinear eigenvalue problems,” ACM Trans. Math. Software 39 (2), 7 (2013).Google Scholar
  10. 10.
    V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations (Am. Math. Soc, Providence, 2001).Google Scholar
  11. 11.
    Th. Apel, A.-M. Sändig, and S. I. Solov’ev, “Computation of 3D vertex singularities for linear elasticity: error estimates for a finite element method on graded meshes,” Math. Model. Numer. Anal. 36, 1043–1070 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    S.I. Solov’ev, “Fast methods for solving mesh schemes of the finite element method of second order accuracy forthe Poisson equation in a rectangle,” Izv Vyssh. Uchebn. Zaved. Mat., No. 10, 71–74 (1985).Google Scholar
  13. 13.
    S. I. Solov’ev, “A fast direct method for solving finite element method schemes with Hermitian bicubic elements,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 87–89 (1990).MathSciNetzbMATHGoogle Scholar
  14. 14.
    A. D. Lyashko and S. I. Solov’ev, “Fourier method of solution of FE systems with Hermite elements for Poisson equation,” Russ. J. Numer. Anal. Math. Model. 6, 121–130 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    S. I. Solov’ev, “Fast direct methods of solving finite-element grid schemes with bicubic elements for the Poisson equation,” J. Math. Sci. 71, 2799–2804 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    S. I. Solov’ev, “A fast direct method of solving Hermitian fourth-order finite-element schemes for the Poisson equation,” J. Math. Sci. 74, 1371–1376 (1995).MathSciNetCrossRefGoogle Scholar
  17. 17.
    E. M. Karchevskii and S. I. Solov’ev, “Investigation of a spectral problem for the Helmholtz operator on the plane,” Differ. Equation. 36, 631–634 (2000).MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. A. Samsonov and S. I. Solov’ev, “Eigenvibrations of a beam with load,” Lobachevskii J. Math. 38 (5), 849–855 (2017).MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    I. B. Badriev, G. Z. Garipova, M. V. Makarov, and V. N. Paymushin, “Numerical solution of the issue about geometrically nonlinear behavior of sandwich plate with transversal soft filler,” Res. J. Appl. Sci. 10, 428–435 (2015).Google Scholar
  20. 20.
    A. A. Samsonov, S. I. Solov’ev, and P. S. Solov’ev, “Eigenvibrations of a bar with load,” MATEC Web Conf. 129, 06013-1-4 (2017).Google Scholar
  21. 21.
    A. A. Samsonov, S. I. Solov’ev, and P. S. Solov’ev, “Eigenvibrations of a simply supported beam with elastically attached load,” MATEC Web Conf. 224, 04012-1-6 (2018).CrossRefGoogle Scholar
  22. 22.
    A. A. Samsonov and S. I. Solov’ev, “Investigation of eigenvibrations of a loaded bar,” MATEC Web Conf. 224, 04013-1-5 (2018).CrossRefGoogle Scholar
  23. 23.
    A. A. Samsonov, S. I. Solov’ev, and P. S. Solov’ev, “Finite element modeling of eigenvibrations of a bar with elastically attached load,” AIP Conf. Proc. 2053, 040082-1-4 (2018).Google Scholar
  24. 24.
    A. A. Samsonov and S. I. Solov’ev, “Investigation of eigenvibrations of a simply supported beam with load,” AIP Conf. Proc. 2053, 040083-1-4 (2018).Google Scholar
  25. 25.
    A. A. Samsonov, D. M. Korosteleva, and S. I. Solov’ev, “Approximation of the eigenvalue problem on eigenvibration of a loaded bar,” J. Phys.: Conf. Se. 1158, 042009-1-5 (2019).Google Scholar
  26. 26.
    A. A. Samsonov, D. M. Korosteleva, and S. I. Solov’ev, “Investigation of the eigenvalue problem on eigenvibration of a loaded string,” J. Phys.: Conf. Se. 1158, 042010-1—5 (2019).Google Scholar
  27. 27.
    A. V. Gulin and A. V. Kregzhde, “On the applicability of the bisection method to solve nonlinear difference Eigenvalue problems,” Preprint No. 8 (Inst. Appl. Math., USSR Science Academy, Moscow, 1982).Google Scholar
  28. 28.
    A. V. Gulin and S. A. Yakovleva, “On a numerical solution of a nonlinear eigenvalue problem,” in Computational Processes and Systems (Nauka, Moscow, 1988), Vol. 6, pp. 90–97 [in Russian].MathSciNetzbMATHGoogle Scholar
  29. 29.
    R. Z. Dautov, A. D. Lyashko, and S. I. Solov’ev, “The bisection method for symmetric eigenvalue problems with a parameter entering nonlinearly,” Russ. J. Numer. Anal. Math. Model. 9, 417–427 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. 10, 674–689 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    F. Tisseur and K. Meerbergen, “ The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    V. Mehrmann and H. Voss, “ Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods,” GAMM-Mit. 27, 1029–1051 (2004).MathSciNetzbMATHGoogle Scholar
  33. 33.
    S. I. Solov’ev, “Preconditioned iterative methods fora class of nonlinear eigenvalue problems,” Linear Algebra Appl. 415, 210–229 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    D. Kressner, “A block Newton method for nonlinear eigenvalue problems,” Numer. Math. 114, 355–372 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    X. Huang, Z. Bai, and Y. Su, “ Nonlinear rank-one modification of the symetric eigenvalue problem,” J. Comput. Math. 28, 218–234 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    H. Schwetlick and K. Schreiber, “Nonlinear Rayleigh functionals,” Linear Algebra Appl. 436, 3991–4016 (2012).MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    W.-J. Beyn, “An integral method for solving nonlinear eigenvalue problems,” Linear Algebra Appl. 436, 3839–3863 (2012).MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    A. Leblanc and A. Lavie, “ Solving acoustic nonlinear eigenvalue problems with a contour integral method,” Eng. Anal. Bound. Elem. 37, 162–166 (2013).MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    X. Qian, L. Wang, and Y. Song, “ A successive quadratic approximations method for nonlinear eigenvalue problems,” J. Comput. Appl. Math. 290, 268–277 (2015).MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    A. A. Samsonov, P. S. Solov’ev, and S. I. Solov’ev, “The bisection method for solving the nonlinear bar eigenvalue problem,” J. Phys.: Conf. Se. 1158, 042011-1-5 (2019).Google Scholar
  41. 41.
    A. A. Samsonov, P. S. Solov’ev, and S. I. Solov’ev, “Spectrum division for eigenvalue problems with nonlinear dependence on the parameter,” J. Phys.: Conf. Se. 1158, 042012-1—5 (2019).Google Scholar
  42. 42.
    A. V. Gulin and A. V. Kregzhde, “Difference schemes for some nonlinear spectral problems,” KIAM Preprint No. 153 (Keldysh Inst. Appl. Math., USSR Science Academy, Moscow, 1981).Google Scholar
  43. 43.
    A. V. Kregzhde, “On difference schemes for the nonlinear Sturm—Liouville problem,” Differ. Uravn. 17, 1280–1284 (1981).MathSciNetGoogle Scholar
  44. 44.
    S. I. Solov’ev and P. S. Solov’ev, “Error estimates of the finite difference method for eigenvalue problems with nonlinear entrance of the spectral parameter,” J. Phys.: Conf. Sen 158, 042020-1—5 (2019).Google Scholar
  45. 45.
    A. A. Samsonov, P. S. Solov’ev, and S. I. Solov’ev, “Error investigation of a finite element approximation for a nonlinear Sturm—Liouville problem,” Lobachevskii J. Math. 39 (7), 1460–1465 (2018).MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    R. Z. Dautov, A. D. Lyashko, and S. I. Solov’ev, “Convergence of the Bubnov—Galerkin method with perturbations for symmetric spectral problems with parameter entering nonlinearly,” Differ. Equat. 27, 799–806 (1991).zbMATHGoogle Scholar
  47. 47.
    S. I. Solov’ev, “The error of the Bubnov—Galerkin method with perturbations for symmetric spectral problems with a non-linearly occurring parameter,” Comput. Math. Math. Phys. 32, 579–593 (1992).MathSciNetzbMATHGoogle Scholar
  48. 48.
    S. I. Solov’ev, “Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter,” Differ. Equation. 50, 947–954 (2014).MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    S. I. Solov’ev, “ Superconvergence of finite element approximations of eigenfunctions,” Differ. Equat. 30, 1138–1146 (1994).MathSciNetzbMATHGoogle Scholar
  50. 50.
    S. I. Solov’ev, “ Superconvergence of finite element approximations to eigenspaces,” Differ. Equat. 38, 752–753 (2002).zbMATHCrossRefGoogle Scholar
  51. 51.
    S. I. Solov’ev, “Approximation of differential eigenvalue problems,” Differ. Equat. 49, 908–916 (2013).MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    S. I. Solov’ev, “Finite element approximation with numerical integration for differential eigenvalue problems,” Appl. Numer. Math. 93, 206–214 (2015).MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    S. I. Solov’ev and P. S. Solov’ev, “Error estimates of the quadrature finite element method with biquadratic finite elements for elliptic eigenvalue problems in the square domain,” J. Phys.: Conf. Se. 1158, 042021 -1—5 (2019).Google Scholar
  54. 54.
    S. I. Solov’ev, “Approximation of nonlinear spectral problems in a Hilbert space,” Differ. Equat. 51, 934–947 (2015).MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    S. I. Solov’ev, “Approximation of variational eigenvalue problems,” Differ. Equat. 46, 1030–1041 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    S. I. Solov’ev, “Approximation of positive semidefinite spectral problems,” Differ. Equat. 47, 1188–1196 (2011).zbMATHCrossRefGoogle Scholar
  57. 57.
    S. I. Solov’ev, “Approximation of sign-indefinite spectral problems,” Differ. Equat. 48, 1028–1041 (2012).MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    S. I. Solov’ev, “Approximation of operator eigenvalue problems in a Hilbert space,” IOP Conf. Sen: Mater. Sci. Eng. 158, 012087-1-6 (2016).Google Scholar
  59. 59.
    S. I. Solov’ev, “Quadrature finite element method for elliptic eigenvalue problems,” Lobachevskii J. Math. 38 (5), 856–863 (2017).MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    I. B. Badriev, V. V. Banderov, and O. A. Zadvornov, “On the equilibrium problem of a soft network shell in the presence of several point loads,” Appl. Mech. Mater. 392, 188–190 (2013).CrossRefGoogle Scholar
  61. 61.
    I. B. Badriev, M. V. Makarov, and V. N. Paimushin, “Geometrically nonlinear problem of longitudinal and transverse bending of a sandwich plate with transversally soft core,” Lobachevskii J. Math. 392(5), 448–457 (2018).MathSciNetCrossRefGoogle Scholar
  62. 62.
    I. B. Badriev, V. V. Banderov, and M. V. Makarov, “Mathematical simulation of the problem of the pre-critical sandwich plate bending in geometrically nonlinear one dimensional formulation,” IOP Conf. Sen: Mater. Sci. Eng. 208, 012002 (2017).CrossRefGoogle Scholar
  63. 63.
    I. B. Badriev, M. V. Makarov, and V. N. Paimushin, “Numerical investigation of a physically nonlinear problem of the longitudinal bending of the sandwich plate with a transversal-soft core,” PNRPU Mech. Bull., No. 1, 39–51 (2017).Google Scholar
  64. 64.
    I. B. Badriev, V. V. Banderov, E. E. Lavrentyeva, and O. V. Pankratova, “On the finite element approximations of mixed variational inequalities of filtration theory,” IOP Conf. Sen: Mater. Sci. Eng. 158, 012012 (2016).CrossRefGoogle Scholar
  65. 65.
    I. B. Badriev, “On the solving of variational inequalities of stationary problems of two-phase flow in porous media,” Appl. Mech. Mater. 392, 183–187 (2013).CrossRefGoogle Scholar
  66. 66.
    I. B. Badriev, O. A. Zadvornov, and A. D. Lyashko, “A study of variable step iterative methods for variational inequalities of the second kind,” Differ. Equat. 40, 971–983 (2004).zbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Kazan State Power Engineering UniversityKazanRussia
  2. 2.Kazan (Volga Region) Federal UniversityKazanRussia

Personalised recommendations