A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data
Provides theoretical convergence guarantees for meshless methods in fractional order Sobolev spaces, benefiting numerical analysts working on boundary value problems.
The paper extends a Sobolev bound to incorporate discrete samples of arbitrary order derivatives and to optimally bound fractional order Sobolev semi-norms, enabling higher order convergence rates for unsymmetric meshless methods, particularly for inhomogeneous boundary value problems.
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.