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We present an algorithm to solve $- \lap u - f(x,u) = g$ with Dirichletboundary conditions in a bounded domain $\Omega$. The nonlinearities arenon-resonant and have finite spectral interaction: no eigenvalue of $-\lap_D$is an endpoint of $\bar{\partial_2f(\Omega,\RR)}$, which in turn only containsa finite number of eigenvalues. The algorithm is based in ideas used by Bergerand Podolak to provide a geometric proof of the Ambrosetti-Prodi theorem andadvances work by Smiley and Chun for the same problem.

Numerical analysis of semilinear elliptic equations with finite spectral interaction