A nonlinear discretization theory for meshfree collocation methods applied to quasilinear elliptic equations

Abstract

This paper generalizes earlier results concerning meshfree collocation methods for semilinear elliptic second order problems to the quasilinear case. These results require a strong form of well-posedness and a stability analysis in the ∞ norm for the meshfree discretization. Since quasilinear problems are usually treated in Hölder norms, both aspects have to be transformed to the Hölder situation. This finally leads to error bounds and convergence rates, the latter being dependent on the smoothness of the true solution. The paper focuses on Dirichlet conditions and postpones other boundary conditions to a forthcoming paper. A numerical example is provided.