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A Nonlinear Discretization Theory with Applications to Meshfree Methods for Nonlinear PDEs

Klaus Boehmer
Philipps-University Marburg


This lecture is an appetizer for my two books in OUP: Numerical Methods for Nonlinear Elliptic Differential Equations, A Synopsis, and Numerical Methods for Bifurcation and Center Manifolds in Nonlinear Elliptic and Parabolic Differential Equations, 2010 and 2011. We extend for the first time the linear discretization theory of Schaback, developed for meshfree methods, to nonlinear operator equations, relying heavily on methods of Boehmer, Vol I. There is no restriction to elliptic problems nor to symmetric numerical methods like Galerkin techniques. Trial spaces can be arbitrary, but have to approximate the solution well, and testing can be weak or strong. We present Galerkin techniques as an example. 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. As an example we present the meshless method for some nonlinear elliptic problems of order 2. Some numerical examples are added for illustration.

Tuesday, October 16, 2012
3:00PM AP&M 2402