Abstract
We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no á priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form A + iB where A is self-adjoint, B is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrödinger and Dirac operators.
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References
Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: a collection of nonlinear eigenvalue problems (2011). (MIMS EPrint 2011.116)
Boffi D., Brezzi F., Gastaldi L.: On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69(229), 121–140 (2000)
Boffi D., Duran R.G., Gastaldi L.: A remark on spurious eigenvalues in a square. Appl. Math. Lett. 12(3), 107–114 (1999)
Boulton L.: Limiting set of second order spectrum. Math. Comput. 75, 1367–1382 (2006)
Boulton L., Boussaid N.: Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials. LMS J. Comput. Math. 13, 10–32 (2010)
Boulton L., Levitin M.: On approximation of the eigenvalues of perturbed periodic Schrödinger operators. J. Phys. A Math. Theor. 40, 9319–9329 (2007)
Boulton L., Strauss M.: On the convergence of second-order spectra and multiplicity. Proc. R. Soc. A 467, 264–275 (2011)
Boulton L., Strauss M.: Eigenvalues enclosures and convergence for the linearized MHD operator. BIT Numer. Math. 52, 801–825 (2012)
Chatelin F.: Spectral Approximation of Linear Operators. Academic Press, New York (1983)
Dauge M., Suri M.: Numerical approximation of the spectra of non-compact operators arising in buckling problems. J. Numer. Math. 10, 193–219 (2002)
Davies E.B.: Spectral enclosures and complex resonances for general self-adjoint operators. LMS J. Comput. Math. 1, 42–74 (1998)
Davies E.B., Plum M.: Spectral pollution. IMA J. Numer. Anal. 24, 417–438 (2004)
Hansen A.C.: On the approximation of spectra of linear operators on Hilbert spaces. J. Funct. Anal. 254(8), 2092–2126 (2008)
Hansen A.C.: Infinite dimensional numerical linear algebra; theory and applications. Proc. R. Soc. Lond. Ser. A. 466(2124), 3539–3559 (2010)
Hansen A.C.: On the solvability complexity index, the n-pseudospectrum and approximations of spectra of operators. J. Am. Math. Soc. 24(1), 81–124 (2011)
Kato T.: On the upper and lower bounds of eigenvalues. J. Phys. Soc. Jpn. 4, 334–339 (1949)
Kato T.: Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Anallyse Math. 6, 261–322 (1958)
Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1995)
Levitin M., Shargorodsky E.: Spectral pollution and second order relative spectra for self-adjoint operators. IMA J. Numer. Anal. 24, 393–416 (2004)
Marletta M.: Neumann–Dirichlet maps and analysis of spectral pollution for non-self-adjoint elliptic PDEs with real essential spectrum. IMA J. Numer. Anal. 30, 917–939 (2010)
Marletta M., Naboko S.: The finite section method for dissipative operators. Mathematika 60(2), 415–443 (2014)
Marletta M., Scheichl R.: Eigenvalues in spectral gaps of differential operators. J. Spectr. Theory 2(3), 293–320 (2012)
Rappaz J., Sanchez Hubert J., Sanchez Palencia E., Vassiliev D.: On spectral pollution in the finite element approximation of thin elastic membrane shells. Numer. Math. 75, 473–500 (1997)
Schmidt K.M.: Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm–Liouville operators. Commun. Math. Phys. 211, 465–485 (2000)
Shargorodsky E.: Geometry of higher order relative spectra and projection methods. J. Oper. Theory 44, 43–62 (2000)
Shargorodsky E.: On the limit behaviour of second order relative spectra of self-adjoint operators. J. Spectr. Theory 3(4), 535–552 (2013)
Strauss M.: Quadratic projection methods for approximating the spectrum of self-adjoint operators. IMA J. Numer. Anal. 31, 40–60 (2011)
Strauss M.: The second order spectrum and optimal convergence. Math. Comput. 82, 2305–2325 (2013)
Strauss M.: The Galerkin method for perturbed self-adjoint operators and applications. J. Spectr. Theory 4(1), 113–151 (2014)
Strauss M.: A new approach to spectral approximation. J. Funct. Anal. 267(8), 3084–3103 (2014)
Tretter C.: Spectral Theory Of Block Operator Matrices And Applications. Imperial College Press, London (2008)
Zimmermann S., Mertins U.: Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwend. 14, 327–345 (1995)
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Hinchcliffe, J., Strauss, M. Spectral Enclosure and Superconvergence for Eigenvalues in Gaps. Integr. Equ. Oper. Theory 84, 1–32 (2016). https://doi.org/10.1007/s00020-015-2247-0
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DOI: https://doi.org/10.1007/s00020-015-2247-0