Abstract
An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal.
Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods.
The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems.
The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one.
As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes.
Authors
Tiberiu Popoviciu Institute of Numerical Analysis
School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Keywords
References
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Paper coordinates
C.I. Gheorghiu, B. Zinsou, Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem, J. Numer. Anal. Approx. Theory, 48 (2019) no. 2, 159-174. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1201
About this paper
Journal
J. Numer. Anal. Approx. Theory
Publisher Name
Editura Academiei Romane
DOI
not available yet.
Print ISSN
2457-6794
Online ISSN
2501-059X
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[1] K. Aydemir and O.S. Mukhtarov, Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem, Bound. Value Probl. 82 (2016).
[2] I. Babuska and J.E. Osborn, Numerical Treatment of Eigenvalue Problems for Differential Equations with Discontinuous Coefficients, Math. Comput. 32 (1978), pp. 991–1023, DOI: 10.2307/2006330
[3] P.B. Bailey, M.K. Gordon and L.F. Shampine, Automatic Solution of the Sturm-Liouville Problem, ACM. T. Math. Software. 4 (1978), pp. 193–208, https://doi.org/10.1016/0377-0427(92)90222-J
[4] P.B. Bailey, W.N. Everitt and A. Zettl, Algorithm 810: The SLEIGN2 Sturm-Liouville Code, ACM T. Math. Software. 27 (2001) pp. 143–192, https://doi.org/10.1145/383738.383739
[5] J.P. Boyd, Chebyshev and Fourier spectral methods, 2nd rev. ed. Mineola, NY: Dover Publications. 2001.
[6] P.A.M. Boomkamp, B.J. Boersma, R.H.M. Miesen and G.V. Beijnon, A Chebyshev Collocation Method for Solving Two-Phase Flow Stability Problems, J. Comput. Phys. 132 (1997), pp. 191–200, https://doi.org/10.1006/jcph.1996.5571
[7] S.C. Brenner, L.R. Scott, The mathematical theory of finite element methods, 3rd ed. Texts in Applied Mathematics 15. New-York, NY: Springer 2008, https://doi.org/10.1007/978-0-387-75934-0
[8] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New-York, Oxford 1978.
[9] W.N. Everitt, A. Zettl, Sturm–Liouville differential operators in direct sum spaces, Rocky Mountain J. Math. 16 (1986), pp. 497–516, https://doi.org/10.1216/rmj-1986-16-3-497
[10] F. Gesztesyt, C. Macdeo and L. Streit, An exactly solvable periodic Schrodinger operator J. Phys. A: Math. Gen. 18 (1985), pp. L503–L507, https://doi.org/10.1088/0305-4470/18/9/003
[11] C. I. Gheorghiu, J. Rommes, Application of the Jacobi-Davidson method to accurate analysis of singular linear hydrodynamic stability problems, Int. J. Numer. Meth. Fl. 71 (2012), pp. 358–369, https://doi.org/10.1002/fld.3669
[12] C.I. Gheorghiu, Spectral Methods for Non-Standard Eigenvalue Problems. Fluid and Structural Mechanics and Beyond, Springer Cham Heidelberg New York Dondrecht London 2014, https://doi.org/10.1007/978-3-319-06230-3
[13] Y. He, D.P. Nicholls and J. Shen, An efficient and stable spectral method for electromagnetic scattering from a layered periodic sructure, J. Comput. Phys. 231 (2012), pp. 3007–3022, https://doi.org/10.1016/j.jcp.2011.10.033
[14] C.O. Horgan, J.P. Spence, A. N. Andry, Lower bounds for eigenvalues of Sturm-Liouville problems with discontinuous coefficients: integral equation methods, Q. Appl. Math. 39 (1982), pp. 455–465, https://doi.org/10.1090/qam/644100
[15] Multi-precision Computing Toolbox for MATLAB. Yokohama: Advanpix LLC.; 2008-2017.
[16] M. Marletta, J.D. Pryce, Automatic solution of Sturm-Liouville problems using the Pruess method, J. Comput. Appl. Math. 39 (1992), pp. 57–78, https://doi.org/10.1016/0377-0427(92)90222-j
[17] O.S. Mukhtarov, M. Kadakal and F.S. Muhtarov, On discontinuous Sturm-Liouville problems with transmission conditions, J. Math. Kyoto Univ. (JMKYAZ) 44 (2004), pp. 779–798, https://doi.org/10.1215/kjm/1250281698
[18] J. Necas, Direct Methods in the Theory of Elliptic Equations, Springer Monographs in Mathematics. Berlin 2012, https://doi.org/10.1007/978-3-642-10455-8
[19] S. Pruess, C.T. Fulton, Mathematical Software for Sturm-Liouville Problems, ACM T. Math. Software. 19 (1993), pp. 360–376, https://doi.org/10.1145/155743.155791
[20] J. D. Pryce, Numerical solutions of Sturm-Liouville problem, Oxford University Press,Oxford, U. K., 1993.
[21] D.G. Shepelsky, The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions, Adv. Soviet Math. 19 (1994), pp.209–231.
[22] I. Titeux, Ya. Yakubov, Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients Math. Models Methods Appl. Sc., 7 (1997), pp. 1035–1050, https://doi.org/10.1142/S0218202597000529
[23] J.A.C. Weideman, S. C. Reddy, A MATLAB Differentiation Matrix Suite, ACM T. Math. Software. 26 (2000), pp. 465–519,
[24] A. Zettl, Adjoint and Self-Adjoint Boundary Value Problems with Interface Conditions,SIAM J. Appl. Math. 16 (1968), pp. 851–859.
[25] A. Zettl, Sturm—Liouville Theory, Math. Surveys Monogr., vol. 121, Amer. Math. Soc., Providence, RI 2005.
[26] H. Yserentant, A Short Theory of the Rayleigh-Ritz Method, C.M.A.M. 13 (2013) pp. 496–502, https://doi.org/10.1515/cmam-2013-0013