Accurate Chebyshev collocation solutions for the biharmonic eigenproblem on a rectangle


We are concerned with accurate Chebyshev collocation (ChC) solutions to fourth order eigenvalue problems. We consider the 1D case as well as the 2D case. In order to improve the accuracy of computation we use the precondtitioning strategy for second order differential operator introduced by Labrosse in 2009. The fourth order differential operator is factorized as a product of second order operators. In order to asses the accuracy of our method we calculate the so called drift of the first five eigenvalues. In both cases ChC method with the considered preconditioners provides accurate eigenpairs of interest.


Imre Boros
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


spectral methods; Chebyshev collocation; preconditioning; fourth order eigenvalue problems;

Paper coordinates

I. Boros, Accurate Chebyshev collocation solutions for the biharmonic eigenproblem on a rectangle, J. Numer. Anal. Approx. Theory, 46 (2017) no. 1, pp. 38-46.


About this paper


Journal of Numerical Analysis and Approximation Theory

Publisher Name

Editura Romanian Academy

Print ISSN


Online ISSN


google scholar link

[1] L. Bauer, E.L. Reiss, Block five diagonal matrices and the fast numerical solution of the biharmonic equation, Math. Comp., 26 (1972) no. 118, 311–326.

[2] P.E. Bjorstad, P. Tjostheim, A note on high precision solutions of two fourth order eigenvalue problems, 1998.

[3] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Courier Corporation, New York, 2000.

[4] C.I. Gheorghiu, Spectral Methods for Non-Standard Eigenvalue Problems: Fluid and Structural Mechanics and Beyond, Springer Science & Business, 2014.

[5] J. Hoepffner, Implementation of boundary conditions, 2007, (accessed on August 2, 2017 hoepffner/boundarycondition.pdf).

[6] G. Labrosse, The piecewise-linear Finite Volume scheme: The best known lowest-order preconditioner for the d2 dx2 Chebyshev spectral operator, J. Comp. Phys., 228 (2009) no. 12, 4491–4509.
[7] W. Kahan, B. Parlett, E. Jiang, Residual bounds on approximate eigensystems of nonnormal matrices, SIAM J. Numer. Anal., 19 (1982) no. 3, 470–484.

[8] M.P. Owen, Asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle, J. Differential Equations, 190 (1997) no. 1, 166–190.

[9] S.A. Orszag, Spectral methods for problems in complex geometries, J. Comp. Phys, 37 (1980) no. 1, 70–92.

[10] L.N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000.

[11] J.A.C. Weideman, S.C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software, 26 (2000) no. 4, 465-519.

Related Posts