Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem

A.A. Samsonov^*, P.S. Solov'ev^**, S.I. Solov'ev^***

Kazan Federal University, Kazan, 420008 Russia

E-mail: ^* anton.samsonov.kpfu@mail.ru, ^** pavel.solovev.kpfu@mail.ru, ^*** sergei.solovyev@kpfu.ru

Received June 5, 2017

Full text PDF

Abstract

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.

Keywords: eigenvalue, eigenfunction, eigenvalue problem, mesh approximation, finite element method

Acknowledgments. The study was supported by the Russian Science Foundation (project no. 16-11-10299).

References

1. Solov'ev S.I. Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter. Differ. Equations, 2014, vol. 50, no. 7, pp. 947–954. doi: 10.1134/S0012266114070106.

2. Lyashko A.D., Solov'ev S.I. Fourier method of solution of FE systems with Hermite elements for Poisson equation. Sov. J. Numer. Anal. Math. Modell., 1991, vol. 6, no. 2, pp. 121–129.

3. Solov'ev S.I. Fast direct methods of solving finite-element grid schemes with bicubic elements for the Poisson equation. J. Math. Sci., 1994, vol. 71, no. 6, pp. 2799–2804.

4. Solov'ev S.I. A fast direct method of solving Hermitian fourth-order finite-element schemes for the Poisson equation. J. Math. Sci., 1995, vol. 74, no. 6, pp. 1371–1376.

5. Dautov R.Z., Lyashko A.D., Solov'ev S.I. The bisection method for symmetric eigenvalue problems with a parameter entering nonlinearly. Russ. J. Numer. Anal. Math. Modell., 1994, vol. 9, no. 5, pp. 417–427.

6.  Solov'ev S.I. The finite element method for symmetric nonlinear eigenvalue problems. Comput. Math. Math. Phys., 1997, vol. 37, no. 11, pp. 1269–1276.

7. Dautov R.Z., Lyashko A.D., Solov'ev S.I. Convergence of the Bubnov–Galerkin method with perturbations for symmetric spectral problems with parameter entering nonlinearly. Differ. Equations, 1991, vol. 27, no. 7, pp. 799–806.

8. Solov'ev S.I. The error of the Bubnov–Galerkin method with perturbations for symmetric spectral problems with a non-linearly occurring parameter. Comput. Math. Math. Phys., 1992, vol. 32, no. 5, pp. 579–593.

9. Solov'ev S.I. Superconvergence of finite element approximations of eigenfunctions. Differ. Equations, 1994, vol. 30, no. 7, pp. 1138–1146.

10. Solov'ev S.I. Superconvergence of finite element approximations to eigenspaces. Differ. Equations, 2002, vol. 38, no. 5, pp. 752–753.

11. Solov'ev S.I. Approximation of differential eigenvalue problems. Differ. Equations, 2013, vol. 49, no. 7, pp. 908–916.

12. Solov'ev S.I. Finite element approximation with numerical integration for differential eigenvalue problems. Appl. Numer. Math., 2015, vol. 93, pp. 206–214.

13. Solov'ev S.I. Approximation of nonlinear spectral problems in a Hilbert space. Differ. Equations, 2015, vol. 51, no. 7, pp. 934–947.

14. Solov'ev S.I. Approximation of variational eigenvalue problems. Differ. Equations, 2010, vol. 46, no. 7, pp. 1030–1041.

15. Solov'ev S.I. Approximation of positive semidefinite spectral problems. Differ. Equations, 2011, vol. 47, no. 8, pp. 1188–1196. doi: 10.1134/S001226611108012X.

16. Solov'ev S.I. Approximation of sign-indefinite spectral problems. Differ. Equations, 2012, vol. 48, no. 7, pp. 1028–1041.

17. Badriev I.B., Banderov V.V., Gnedenkova V.L., Kalacheva N.V., Korablev A.I., Tagirov R.R. On the finite dimensional approximations of some mixed variational inequalities. Appl. Math. Sci., 2015, vol. 9, no. 113–116, pp. 5697–5705.

18. Badriev I.B., Garipova G.Z., Makarov M.V., Paymushin V.N. Numerical solution of the issue about geometrically nonlinear behavior of sandwich plate with transversal soft filler. Res. J. Appl. Sci., 2015, vol. 10, no. 8, pp. 428–435.

19. Badriev I.B., Garipova G.Z., Makarov M.V., Paimushin V.N., Khabibullin R.F. On solving physically nonlinear equilibrium problems for sandwich plates with a transversely soft filler. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, vol. 157, no. 1, pp. 15–24.

20. Badriev I.B., Makarov M.V., Paimushin V.N. Solvability of physically and geometrically nonlinear problem of the theory of sandwich plates with transversally-soft core. Russ. Math., 2015, vol. 59, no. 10, pp. 57–60. doi: 10.3103/S1066369X15100072.

21. Badriev I.B., Makarov M.V., Paimushin V.N. Numerical investigation of physically nonlinear problem of sandwich plate bending. Proc. Eng., 2016, vol. 150, pp. 1050–1055. doi: 10.1016/j.proeng.2016.07.213.

22. Badriev I.B., Makarov M.V., Paimushin V.N. Mathematical simulation of nonlinear problem of three-point composite sample bending test. Proc. Eng., 2016, vol. 150, pp. 1056–1062. doi: 10.1016/j.proeng.2016.07.214.

23. Badriev I.B., Nechaeva L.A. Mathematical simulation of steady filtration with multivalued law. Vestn. Permsk. Nats. Issled. Politekh. Univ., Mekh., 2013, no. 3, pp. 35–62. (In Russian)

24. Mikhailov V.P. Differential Equations in Partial Derivatives. Moscow, Nauka, 1983. 424 p. (In Russian)

25. Ciarlet P.G. The Finite Element Method for Elliptic Problems. Philadelphia, Am. Math. Soc., 2002. xxiii, 529 p.

For citation:  Samsonov A.A., Solov'ev P.S., Solov'ev S.I. Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 3, pp. 354–363. (In Russian)

The content is available under the license Creative Commons Attribution 4.0 License.