An analytical and numerical study of the two-dimensional Bratu equation. (English) Zbl 0649.65057

Bratu’s problem, which is the nonlinear eigenvalue equation \(\Delta u+\lambda \exp (u)=0\) with \(u=0\) on the walls of the unit square and \(\lambda\) as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance of symmetry: because of invariance under the \(C_ 4\) rotation group and parity in both x and y, one can slash the size of the basis set by a factor of eight and reduce the CPU time by three orders of magnitude. Second, the pseudospectral method is an analytical as well as a numerical tool: the simple approximation \(\lambda\approx 3.2A \exp (- 0.64A)\), where A is the maximum value of u(x,y), is derived via collocation with but a single interpolation point, but is a quantitatively acurate for small and moderate A.
Third, the Newton-Kantorovich/Chebyshev pseudospectral algorithm is so efficient that it is possible to compute good numerical solutions - five decimal places - on a microcomputer in BASIC. Fourth, asymptotic estimates of the Chebyshev coefficients can be very misleading: the coefficients for moderately or strongly nonlinear solutions to Bratu’s equations fall off exponentially rather than algebraically with \(\nu\) until \(\nu\) is so large that one has already obtained several decimal places of accuracy. The corner singularities, which dominate the behavior of the Chebyshev coefficients in the limit \(\nu\to \infty\), are so weak as to be irrelevant, and replacing Bratu’s problem by a more complicated and realistic equation would merely exaggerate the unimportance of the corner branch points even more.


65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs


Full Text: DOI


[1] Abbott, J. P. (1978). An efficient algorithm for the determination of certain bifurcation points,J. Comp. Appl. Math. 4, 19. · Zbl 0384.65022 · doi:10.1016/0771-050X(78)90015-3
[2] Bank, R. E., and Chan, T. F. (1986). PLTMGC: A multi-grid continuation program for parameterized nonlinear elliptic systems,SIAM J. Sci. Stat. Comp. 7, 540-559. · Zbl 0589.65074 · doi:10.1137/0907036
[3] Birkhoff, G., and Lynch, R. E. (1984).Numerical Solutions of Elliptic Problems, Soc. Ind. and Appl. Math., Philadelphia, pp. 43-44, 73-74, 249-250. · Zbl 0556.65078
[4] Boyd, J. P. (1978). Spectral and pseudospectral methods for eigenvalue and nonseparable boundary value problems,Mon. Wea. Rev. 106, 1192-1203. · doi:10.1175/1520-0493(1978)106<1192:SAPMFE>2.0.CO;2
[5] Boyd, J. P. (1986a). Solitons from sine waves: Analytical and numerical methods for nonintegrable solitary and sinoidal waves,Physica 21D, 227-246. · Zbl 0611.35080
[6] Boyd, J. P. (1986b). Polynomial series versus sinc expansions for functions with corner or endpoint singularities,J. Comp. Phys. 64, 266-269. · Zbl 0608.65010 · doi:10.1016/0021-9991(86)90031-8
[7] Boyd, J. P. (1986c). Spectral methods using rational basis functions on an infinite interval,J. Comp. Phys., in press.
[8] Brandt, A., Fulton, S. R., and Taylor, G. D. (1985).J. Comp. Phys. 58, 96-112. · Zbl 0569.65084 · doi:10.1016/0021-9991(85)90159-7
[9] Chan, T. F., and Keller, H. B. (1982). Arc-length continuation and multigrid techniques for nonlinear elliptic eigenvalue problems,SIAM J. Sci. Stat. Comp. 3, 173-194. · Zbl 0497.65028 · doi:10.1137/0903012
[10] Cotton, F. A. (1963).Chemical Applications of Group Theory, Wiley, New York, p. 84.
[11] Finlayson, B. A. (1972).The Method of Weighted Residuals and Variational Principles, Academic Press, New York, pp. 97-106. · Zbl 0319.49020
[12] Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, Soc. Ind. and Appl. Math., Philadelphia, pp. 1-24. · Zbl 0412.65058
[13] Haidvogel, D. B., and Zang, T. (1979). The efficient solution of Poisson’s equation in two dimensions via Chebyshev approximation,J. Comp. Phys. 30, 167-180. · Zbl 0397.65077 · doi:10.1016/0021-9991(79)90097-4
[14] Merzbacher, E. (1970).Quantum Mechanics, Wiley, New York, p. 438. · Zbl 0102.42701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.