A nonlinear discretization theory

摘要

This paper extends for the first time Schaback’s linear discretization theory to nonlinear operator equations, relying heavily on the methods in Böhmer’s 2010 book. There is no restriction to elliptic problems or to symmetric numerical methods like Galerkin techniques. Trial spaces can be arbitrary, including spectral and meshless methods, but have to approximate the solution well, and testing can be weak or strong. On the downside, stability is not easy to prove for special applications, and numerical methods have to be formulated as optimization problems. Results of this discretization theory cover error bounds and convergence rates. Some numerical examples are added for illustration.