Graduate Thesis Or Dissertation
 

Finite element transport using Wachspress rational basis functions on quadrilaterals in diffusive regions

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  • Wachspress rational functions are ratios of polynomials having certain properties which make them good candidates for basis functions in finite element methods. In addition. it had been theorized that Wachspress rational weight functions should Provide a full-resolution discretization of the neutral particle transport equation in thick diffusive regions, but this had not been confirmed numerically. In this study we derive a discontinuous finite element discretization on quadrilaterals. We discuss (different methods for constructing and integrating Wachspress rational functions. We perform an asymptotic analysis of the transport discretization in the interior of thick diffusive regions and derive an asymptotic-P1 diffusion synthetic acceleration preconditioner. We then provide numerical results demonstrating Wachspress rational functions in the thick diffusive limit, and we provide comparisons with isoparametric bilinear basis functions, which are commonly used on quadrilaterals. Analytic and computational results indicate that Wachspress rational functions provide a full-resolution spatial discretization in the thick diffusive limit.
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