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Finite element approximation of the minimal eigenvalue and the corresponding positive eigenfunction of a nonlinear Sturm-Liouville problem. (English) Zbl 1507.65131

Summary: 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.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
34B24 Sturm-Liouville theory
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations

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[1] Abdullin, I. Sh; Zheltukhin, V. S.; Kashapov, N. F., Radio-Frequency Plasma-Jet Processing of Materials at Reduced Pressures: Theory and Practice of Applications (2000)
[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).
[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).
[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).
[5] Solov’ev, S. I.; Solov’ev, P. S., “Finite element approximation of the minimal eigenvalue of a nonlinear eigenvalue problem,”, Lobachevskii J. Math., 39, 949-956 (2018) · Zbl 1448.65080
[6] Solov’ev, S. I., “ Eigenvibrations of a beam with elastically attached load,”, Lobachevskii J. Math., 37, 597-609 (2016) · Zbl 1388.74063
[7] Solov’ev, S. I., “ Eigenvibrations of a bar with elastically attached load,”, Differ. Equat., 53, 409-423 (2017) · Zbl 06732024
[8] Goolin, A. V.; Kartyshov, S. V., “Numerical study of stability and nonlinear eigenvalue problems,”, Surv. Math. Ind., 3, 29-48 (1993) · Zbl 0790.65092
[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). · Zbl 1295.65140
[10] Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J., Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations (2001) · Zbl 0965.35003
[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). · Zbl 1137.65426
[12] Solov’ev, SI, “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., 10, 71-74 (1985)
[13] Solov’ev, S. I., “A fast direct method for solving finite element method schemes with Hermitian bicubic elements,”, Izv. Vyssh. Uchebn. Zaved. Mat., 8, 87-89 (1990) · Zbl 0711.65022
[14] Lyashko, A. D.; Solov’ev, S. I., “Fourier method of solution of FE systems with Hermite elements for Poisson equation,”, Russ. J. Numer. Anal. Math. Model., 6, 121-130 (1991) · Zbl 0816.65079
[15] Solov’ev, S. I., “Fast direct methods of solving finite-element grid schemes with bicubic elements for the Poisson equation,”, J. Math. Sci., 71, 2799-2804 (1994) · Zbl 1264.65185
[16] Solov’ev, S. I., “A fast direct method of solving Hermitian fourth-order finite-element schemes for the Poisson equation,”, J. Math. Sci., 74, 1371-1376 (1995)
[17] Karchevskii, E. M.; Solov’ev, S. I., “Investigation of a spectral problem for the Helmholtz operator on the plane,”, Differ. Equation., 36, 631-634 (2000) · Zbl 0967.35103
[18] Samsonov, A. A.; Solov’ev, S. I., “Eigenvibrations of a beam with load,”, Lobachevskii J. Math, 38, 849-855 (2017) · Zbl 1379.34026
[19] 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., 10, 428-435 (2015)
[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). · Zbl 1379.34026
[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).
[22] A. A. Samsonov and S. I. Solov’ev, “Investigation of eigenvibrations of a loaded bar,” MATEC Web Conf. 224, 04013-1-5 (2018).
[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).
[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).
[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).
[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).
[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).
[28] Gulin, A. V.; Yakovleva, S. A., “On a numerical solution of a nonlinear eigenvalue problem,”, Computational Processes and Systems, 6, 90-97 (1988) · Zbl 0669.65070
[29] 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. Model., 9, 417-427 (1994) · Zbl 0818.65026
[30] Ruhe, A., “Algorithms for the nonlinear eigenvalue problem,”, SIAM J. Numer. Anal., 10, 674-689 (1973) · Zbl 0261.65032
[31] Tisseur, F.; Meerbergen, K., “ The quadratic eigenvalue problem,”, SIAM Rev., 43, 235-286 (2001) · Zbl 0985.65028
[32] Mehrmann, V.; Voss, H., “ Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods,”, GAMM-Mit., 27, 1029-1051 (2004) · Zbl 1071.65074
[33] Solov’ev, S. I., “Preconditioned iterative methods fora class of nonlinear eigenvalue problems,”, Linear Algebra Appl., 415, 210-229 (2006) · Zbl 1095.65033
[34] Kressner, D., “A block Newton method for nonlinear eigenvalue problems,”, Numer. Math., 114, 355-372 (2009) · Zbl 1191.65054
[35] Huang, X.; Bai, Z.; Su, Y., “ Nonlinear rank-one modification of the symetric eigenvalue problem,”, J. Comput. Math., 28, 218-234 (2010) · Zbl 1224.65093
[36] Schwetlick, H.; Schreiber, K., “Nonlinear Rayleigh functionals,”, Linear Algebra Appl., 436, 3991-4016 (2012) · Zbl 1317.65100
[37] Beyn, W-J, “An integral method for solving nonlinear eigenvalue problems,”, Linear Algebra Appl., 436, 3839-3863 (2012) · Zbl 1237.65035
[38] Leblanc, A.; Lavie, A., “ Solving acoustic nonlinear eigenvalue problems with a contour integral method,”, Eng. Anal. Bound. Elem., 37, 162-166 (2013) · Zbl 1351.76218
[39] Qian, X.; Wang, L.; Song, Y., “ A successive quadratic approximations method for nonlinear eigenvalue problems,”, J. Comput. Appl. Math., 290, 268-277 (2015) · Zbl 1321.65083
[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).
[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).
[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).
[43] Kregzhde, A. V., “On difference schemes for the nonlinear Sturm—Liouville problem,”, Differ. Uravn., 17, 1280-1284 (1981)
[44] Solov’ev, S. I.; Solov’ev, P. S., “Error estimates of the finite difference method for eigenvalue problems with nonlinear entrance of the spectral parameter,” (2019)
[45] Samsonov, A. A.; Solov’ev, P. S.; Solov’ev, S. I., “Error investigation of a finite element approximation for a nonlinear Sturm—Liouville problem,”, Lobachevskii J. Math., 39, 1460-1465 (2018) · Zbl 1462.65197
[46] 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. Equat., 27, 799-806 (1991) · Zbl 0745.65039
[47] 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., 32, 579-593 (1992) · Zbl 0769.65030
[48] Solov’ev, S. I., “Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter,”, Differ. Equation., 50, 947-954 (2014) · Zbl 1308.65130
[49] Solov’ev, S. I., “ Superconvergence of finite element approximations of eigenfunctions,”, Differ. Equat., 30, 1138-1146 (1994) · Zbl 0852.65093
[50] Solov’ev, S. I., “ Superconvergence of finite element approximations to eigenspaces,”, Differ. Equat., 38, 752-753 (2002) · Zbl 1031.65120
[51] Solov’ev, S. I., “Approximation of differential eigenvalue problems,”, Differ. Equat., 49, 908-916 (2013) · Zbl 1296.65104
[52] Solov’ev, S. I., “Finite element approximation with numerical integration for differential eigenvalue problems,”, Appl. Numer. Math., 93, 206-214 (2015) · Zbl 1326.65099
[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).
[54] Solov’ev, S. I., “Approximation of nonlinear spectral problems in a Hilbert space,”, Differ. Equat., 51, 934-947 (2015) · Zbl 1328.65131
[55] Solov’ev, S. I., “Approximation of variational eigenvalue problems,”, Differ. Equat., 46, 1030-1041 (2010) · Zbl 1205.65183
[56] Solov’ev, S. I., “Approximation of positive semidefinite spectral problems,”, Differ. Equat., 47, 1188-1196 (2011) · Zbl 1228.47071
[57] Solov’ev, S. I., “Approximation of sign-indefinite spectral problems,”, Differ. Equat., 48, 1028-1041 (2012) · Zbl 1277.65049
[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).
[59] Solov’ev, S. I., “Quadrature finite element method for elliptic eigenvalue problems,”, Lobachevskii J. Math, 38, 856-863 (2017) · Zbl 1379.65086
[60] Badriev, I. B.; Banderov, V. V.; Zadvornov, O. A., “On the equilibrium problem of a soft network shell in the presence of several point loads,”, Appl. Mech. Mater., 392, 188-190 (2013)
[61] Badriev, I. B.; Makarov, M. V.; Paimushin, V. N., “Geometrically nonlinear problem of longitudinal and transverse bending of a sandwich plate with transversally soft core,”, Lobachevskii J. Math., 392, 448-457 (2018)
[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).
[63] Badriev, I. B.; Makarov, M. V.; Paimushin, V. N., “Numerical investigation of a physically nonlinear problem of the longitudinal bending of the sandwich plate with a transversal-soft core,”, 39-51 (2017)
[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).
[65] Badriev, I. B., “On the solving of variational inequalities of stationary problems of two-phase flow in porous media,”, Appl. Mech. Mater., 392, 183-187 (2013)
[66] Badriev, I. B.; Zadvornov, O. A.; Lyashko, A. D., “A study of variable step iterative methods for variational inequalities of the second kind,”, Differ. Equat., 40, 971-983 (2004) · Zbl 1081.65061
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