Abstract:
In order to improve their applicability as a tool for solving partial differential equations
in computational science, we equip kernel-based meshless methods with a number
of new capabilities. First, we provide kernel-based meshless methods with the first
wellposed, general technique which allows for adaptively-scaled trial functions. This is
done by constructing an adaptively-scaled kernel which maintains positive definiteness.
We extend sampling inequalities to optimally bound fractional order Sobolev norms in
terms of possibly higher order data. This sampling inequality is then applied to obtain
more optimal error bounds in a reformulation of Schaback’s framework for unsymmetric
meshless methods. We provide kernel-based meshless methods with a direct
visualization technique, by adapting Fourier volume rendering to deal directly with
meshless data, which was previously only used directly for grid-based data. Modern
graphics hardware has emerged as a powerful architecture for scientific computing. We
implement an unstructured grid-based inviscid, compressible flow solver on modern
graphics hardware, and obtain an order of magnitude speed-up in comparison to an
equivalent code running on a quad-core CPU.
