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K-State Today

December 6, 2012

Mathematics colloquium today

By Reta McDermott

At 2:30 p.m. today Prince Chidyagwai from Temple University will present "Finite Element Approximation of Computational Flows" in 122 Cardwell Hall.

Abstract: Numerical approximation of flow is common in many applications in science and engineering. We consider two applications of computational flow. The fi rst is a problem that arises in modelling the interaction between free flow and porous media flow in geosciences. We consider a coupled model of the Navier-Stokes and Darcy equations to model this phenomena. The Navier-Stokes equations are used to model flow in the free flow region and Darcy's law models flow in the porous medium.

We show existence and uniqueness of a weak solution to the coupled problem and a numerical solution based on a multi-numerics scheme combining the fi nite element and discontinuous Galerkin method. Numerical results to illustrate the nature of the coupled flows are presented. The second application arises in modelling the flow of radiative particles — such as photons — modelled by the radiative transport equation with applications to electron radio-therapy. The radiative transport equation describes the distribution of particles in space and time assuming no interaction between them. Our goal is to determine the distribution of the radiative particles in space and time. The full radiative transfer equation is computationally expensive to solve. We consider hyperbolic systems derived using the method of moments to approximate the radiative transport equation. We present numerical results from benchmark radiative therapy test cases from a third order discontinuous Galerkin scheme.