Numerical analysis of semilinear elliptic equations with finite spectral interaction

We present an algorithm for solving − Δ u − f ( x , u ) = g with Dirichlet boundary conditions in a bounded domain Ω . The nonlinearities are non-resonant and have finite spectral interaction: no eigenvalue of − Δ D is an endpoint of ∂ 2 f ( Ω , R ) ¯ , which in turn only contains a finite number of eigenvalues. The algorithm is based on ideas used by Berger and Podolak to provide a geometric proof of the Ambrosetti–Prodi theorem and advances work by Smiley and Chun on the same problem.