Numerical analysis of semilinear elliptic equations with finite spectral interaction

We present an algorithm for solving −Δu−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 −ΔD is an endpoint of ∂2f(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.