Recent Results on Meshless Symmetric

Meshless collocation methods for the numerical solution of partial differen-tial equations have recently become more and more popular. They provide a greater flexibility when it comes to adaptivity and time-dependent changes of the underlying region. Radial basis functions or, more generally, (conditionally) positive definite kernels are one of the main stream methods in the field of meshless collocation. In this talk, I will give a survey of well-known and recent results on mesh-less, symmetric collocation for boundary value problems using positive definite kernels. In particular, I will address the following topics 1. Well-posedness of the problem, particularly for differential operators with non-constant coefficients. 2. Error analysis in Sobolev spaces. 3. Stability analysis of the collocation matrix. 4. Stabilization by smoothing. 5. Examples. I will refer to the previous results in [1, 3, 2, 5]. However, this talk i