Papers on the mathematical theory of meshless methods

Andrew Corrigan, John Wallin, and Thomas Wanner. A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data. December 2008. BibTeX

Abstract: To improve convergence results obtained using a framework for unsymmetric meshless methods due to Schaback (Preprint Göttingen 2006), we extend, in two directions, the Sobolev bound due to Arcangéli et al. (Numer Math 107, 181-211, 2007), which itself extends two others due to Wendland and Rieger (Numer Math 101, 643-662, 2005) and Madych (J. Approx Theory 142, 116-128, 2006). The first is to incorporate discrete samples of arbitrary order derivatives into the bound, which are used to obtain higher order convergence in higher order Sobolev norms. The second is to optimally bound fractional order Sobolev semi-norms, which are used to obtain more optimal convergence rates when solving problems requiring fractional order Sobolev spaces, notably inhomogeneous boundary value problems.

Note: This supersedes the earlier work A high order test discretization for unsymmetric meshless methods, January 2008, arXiv:0801.4097v1.