## Abstract

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu’s system, Lamé’s system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently.

The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.

## Authors

Călin-Ioan **Gheorghiu**

(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Bor **Plestenjak**

(IMFM and Department of Mathematics, University of Ljubljana)

Michiel E. **Hochstenbach**

(Department of Mathematics and Computer Science, TU Eindhoven)

## Keywords

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## Cite this paper as

B. Plestenjak, C.I. Gheorghiu, M.E. Hochstenbach, Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems. J. Comput. Phys. 298 (2015) 595-601.

doi: 10.1016/j.jcp.2015.06.015

## About this paper

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0021-9991

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[1] A.A. Abramov, A.L. Dyshko, N.B. Konyukhova, T.V. Levitina, *Computation of angular wave functions of Lame by means of solution of auxiliary differential equations*, Comput. Math. Math. Phys. 29 (1989) 119–131.

[2] A.A. Abramov, V.I. Ul’yanova, *A method for solving self-adjoint multiparameter spectral problems for weakly coupled sets of ordinary differential equations*, Comput. Math. Math. Phys. 37 (1997) 552–557.

[3] P. Amodio, T. Levitina, G. Settani, E.B. Weinmuller, *Numerical simulation of the whispering gallery modes in prolate spheriods*, Comput. Phys. Commun. 185 (2014) 1200–1206.

[4] F.M. Arscott, P.J. Taylor, R.V.M. Zahar, *On the numerical construction of ellipsoidal wave functions*, Math. Comp. 40 (1983) 367–380.

[5] F.V. Atkinson, *Multiparameter Eigenvalue Problems*, Academic Press, New York, 1972.

[6] F.V. Atkinson, A.B. Mingarelli, *Multiparameter Eigenvalue Problems. Sturm–Liouville Theory*, CRC Press, Boca Raton, 2010.

[7] P.B. Bailey, *The automatic solution of two-parameter Sturm-Liouville eigenvalue problems in ordinary differential equations*, Appl. Math. Comput. 8 (1981) 251–259.

[8] R.H. Bartels, G. W. Stewart, *Algorithm 432: Solution of matrix equation AX + XB = C*, Comm. ACM 15 (1972) 820–826.

[9] J. Boersma, J.K.M. Jansen, *Electromagnetic field singularities at the tip of an elliptic cone*, EUT Report 90-WSK-Ol, TU Eindhoven, 1991.

[10] J.P. Boyd, *Chebyshev and Fourier Spectral Methods*, 2nd ed., Dover Publications, Mineola, 2001.

[11] J.P. Boyd, *Chebyshev spectral methods and the Lane-Emden problem*, Numer. Math. Theor. Meth. Appl. 4 (2011) 142–157.

[12] M. Faierman, *Two-parameter Eigenvalue Problems in Ordinary Differential Equations*, volume 205 of Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, Harlow, 1991.

[13] B. Fornberg, *A Practical Guide to Pseudospectral Methods*, Cambridge University Press, Cambridge, 1998.

[14] C.I. Gheorghiu, *Spectral Methods for Differential Problem*s, Casa Cartii de Stiinta, Cluj-Napoca, 2007.

[15] C.I. Gheorghiu, *Spectral Methods for Non-Standard Eigenvalue Problems. Fluid and Structural Mechanics and Beyond*, Springer, Cham, Heidelberg, New York, Dordrecht, London, 2014.

[16] C.I. Gheorghiu, M.E. Hochstenbach, B. Plestenjak, J. Rommes, *Spectral collocation solutions to multiparameter Mathieu’s system*, Appl. Math. Comp. 218 (2012) 11990–12000.

[17] G.H. Golub, C.F. Van Loan, *Matrix Computations*, 3rd ed., The Johns Hopkins University Press, Baltimore, 1996.

[18] D. Gottlieb, S. A. Orszag, *Numerical Analysis of Spectral Methods: Theory and Applications*, SIAM, Philadelphia, 1977.

[19] M.E. Hochstenbach, T. Kosir, B. Plestenjak. *A Jacobi–Davidson type method for the nonsingular two-parameter eigenvalue problem*, SIAM J. Matrix Anal. Appl. 26 (2005) 477–497.

[20] M. E. Hochstenbach, B. Plestenjak, *Harmonic Rayleigh-Ritz for the multiparameter eigenvalue problem*, Elec. Trans. Numer. Anal. 29 (2008) 81–96.

[21] J. Hoepffner, *Implementation of boundary conditions*, www.fukagata.mech.keio.ac.jp/~jerome/ (2007).

[22] X. Ji, *On 2D bisection method for double eigenvalue problems*, J. Comp. Phys. 126 (1996) 91–98.

[23] E.D. Kalinin, *Modification of a method for solving the multiparameter eigenvalue problem for systems of loosely coupled ordinary differential equations*, Comput. Math. Math. Phys. 53 (2013) 874–881.

[24] L. Kraus, L. Levine, *Diffraction by an elliptic cone*, Commun. Pure Appl. Math. 9 (1961), 49–68.

[25] J. R. Kuttler, V. G. Sigillito, *Eigenvalues of the Laplacian in two dimensions*, SIAM Rev. 26 (1984) 163–193.

[26] T.V. Levitina, *On numerical solution of multiparameter Sturm-Liouville spectral problems*. Numerical analysis and mathematical modelling, Banach Center Publ., 29, Polish Acad. Sci., Warsaw, 1994, 275–281.

[27] T.V. Levitina, *A numerical solution to some three-parameter spectral problems*, Comput. Math. Math. Phys. 39 (1999) 1715–1729.

[28] L.C. Lew Yan Voon, M. Willatzen, *Helmholtz equation in parabolic rotational coordinates: application to wave problems in quantum mechanics and acoustics*, Math. Comp. Simul. 65 (2004) 337–349.

[29] K. Meerbergen, B. Plestenjak, *A Sylvester–Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants*, Report TW 653, Department of Computer Science, KU Leuven, 2014, to appear in Numer. Linear Algebra Appl.

[30] P. Moon, D.E. Spencer, *Field Theory Handbook*, Springer-Verlag, New York, 1971.

[31] J.A. Morrison, J.A. Lewis, *Charge singularity at the corner of a flat plate*, SIAM J. Appl. Math. 31 (1976) 233–250.

[32] F.W.J. Olver (Ed.), *NIST Handbook of Mathematical Functions*, Cambridge University Press, Cambridge, 2010.

[33] S.H. Patil, *Hydrogen molecular ion and molecule in two dimensions*, J. Chem. Phys. 118 (2003) 2197–2205.

[34] B. Plestenjak, *A continuation method for a right definite two-parameter eigenvalue problem*, SIAM J. Matrix Anal. Appl. 21 (2000) 1163–1184.

[35] B. Plestenjak, *MultiParEig*, www.mathworks.com/matlabcentral/fileexchange/47844-multipareig, MATLAB Central File Exchange. Retrieved September 14, 2014.

[36] B.D. Sleeman, *Multiparameter spectral theory and separation of variable*s, J. Phys. A: Math. Theor. 41 (2008) 1–20.

[37] D.C. Sorensen, *Implicit application of polynomial filters in a k-step Arnoldi method*, SIAM J. Matrix Anal. Appl. 13 (1992) 357–385.

[38] G.W. Stewart, *A Krylov–Schur algorithm for large eigenproblems*, SIAM J. Matrix Anal. Appl., 23 (2001), 601–614.

[39] T. Toolan, *lapack*, www.mathworks.com/matlabcentral/fileexchange/16777-lapack, MATLAB Central File Exchange. Retrieved August 20, 2014.

[40] L.N. Trefethen, *Spectral Methods in MATLAB*, SIAM, Philadelphia, 2000.

[41] H. Volkmer, *Multiparameter Problems and Expansion Theorems*, Lecture Notes in Math. 1356, Springer-Verlag, New York, 1988.

[42] J.A.C. Weideman, *DMSUITE*, www.mathworks.com/matlabcentral/fileexchange/29-dmsuite, MATLAB Central File Exchange. Retrieved August 20, 2014.

[43] J.A.C. Weideman, S.C. Reddy, *A MATLAB differentiation matrix suite*, ACM Trans. Math. Softw. 26 (2000) 465–519.

[44] M. Willatzen, L.C. Lew Yan Voon, *Theory of acoustic eigenmodes in parabolic cylindrical enclosures*, J. Sound Vib. 286 (2005) 251–264.

[45] M. Willatzen, L.C. Lew Yan Voon, *Numerical implementation of the ellipsoidal wave equation and application to ellipsoidal quantum dots*, Comput. Phys. Commun. 171 (2005) 1–18.

[46] M. Willatzen, L. C. Lew Yan Voon, *Separable Boundary-Value Problems in Physics*, WileyVCH, Weinheim, 2011.