A nonlinear discretization theory for spectral and kernel-based methods

Abstract: 

The talk describes PDE solution techniques like finite element, meshless or spectral methods from the viewpoint of function approximation. In particular, their error is shown to be dominated by the quality of the approximation of the true solution by trial functions. Then, by application of standard results of Approximation Theory, smooth solutions allow fast methods. 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. The results date back to papers in SINUM 2008 and Numer. Math. 2010 focusing on kernel–based techniques, and are extended here to spectral methods and nonlinear problems, the latter part being joint work with Klaus Boehmer of Marburg.