Numerical analysis of semilinear elliptic equations with finite spectral interaction
For researchers in numerical PDEs, this provides a method for a class of problems with finite spectral interaction, but the contribution is incremental as it advances existing approaches without reporting concrete numerical results or SOTA comparisons.
The paper presents an algorithm for solving semilinear elliptic equations with non-resonant nonlinearities that interact with only finitely many eigenvalues of the Laplacian. The algorithm builds on geometric ideas from the Ambrosetti-Prodi theorem and extends prior work by Smiley and Chun.
We present an algorithm to solve $- \lap u - f(x,u) = g$ with Dirichlet boundary conditions in a bounded domain $Ω$. The nonlinearities are non-resonant and have finite spectral interaction: no eigenvalue of $-\lap_D$ is an endpoint of $\bar{\partial_2f(Ω,\RR)}$, which in turn only contains a finite number of eigenvalues. The algorithm is based in ideas used by Berger and Podolak to provide a geometric proof of the Ambrosetti-Prodi theorem and advances work by Smiley and Chun for the same problem.