An analytical and numerical study of the two-dimensional Bratu equation

Abstract

Bratu's problem, which is the nonlinear eigenvalue equation Δu+λ exp( u )=0 with u =0 on the walls of the unit square and λ 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 λ ≈3.2A exp(−0.64 A ), where A is the maximum value of u(x, y) , is derived via collocation with but a single interpolation point, but is quantitatively accurate 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 v until v 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 v →∞, 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.