An Analytical and Numerical Study of the Two-Dimensional Bratu Equation

Bratu's problem, which is the nonlinear eigenvalue equation Au + 2 exp(u)= 0 with u = 0 on the walls of the unit square and 2 as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance of symmett:v: 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 2 ~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 quantitatively accurate for small and moderate A. Third, the NewtonKantorovich/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 ~ co, 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.