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We consider the numerical solution of nonlinear elliptic boundary valueproblems with Kansa's method. We derive analytic formulas for the Jacobian andHessian of the resulting nonlinear collocation system and exploit them withinthe framework of the trust-region algorithm. This ansatz is tested onsemilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau'sproblem, Hele-Shaw flow and the Monge-Amp\`ere equation) with excellentresults. The new approach distinctly outperforms previous ones based onlinearization or finite-difference Jacobians.

Trust-Region Methods for Nonlinear Elliptic Equations with Radial Basis Functions