Kernel-Based Meshless Methods




Corrigan, Andrew

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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.



Meshless Methods, Sampling Inequality, Graphics Hardware, Kernel Method, Volume Rendering, Computational Fluid Dynamics