Graduate Thesis Or Dissertation


Virtual Element Methods for Magnetohydrodynamics on General Polygonal and Polyhedral Meshes Public Deposited

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  • The aim of this dissertation is to construct a virtual element method (VEM) for models in magneto-hydrodynamics (MHD), an area that studies the behavior and properties of electrically conducting fluids such as a plasma. MHD models are a coupling of the Maxwell’s equations for electromagnetics and models for fluid flow. First we consider a simplified resistive MHD sub-model where we assume that the fluid flow is prescribed, along with a resistive term in Ohm’s law. This approach is called Kinematics of MHD, and we use it to predict the evolution of the electric and magnetic fields. Then we consider the full coupled MHD system in two spatial dimensions (2D) where the flow is not prescribed and design another novel VEM for the discretization of Maxwell’s and Stokes’ equations. We present variational formulations for each of these models. These formulations reveal two chains of spaces where the exact solutions lie. Our study focuses on developing discrete versions of these chains in both two (2D) and three (3D) spatial dimensions for MHD Kinematics and in 2D for the full MHD system. By defining a series of computable projectors, each of the terms in the continuous problem are approximated. In all our studies we present analysis of the stability of the VEM method by exploiting well-known techniques from the theory of saddle-point problems. The VEMs developed can be implemented on a very general class of polygonal/polyhedral meshes. Moreover, these methods are guaranteed to preserve the divergence of the magnetic field at the discrete level. In the last chapter, we present a study of opinion dynamics applied specifically to debates between legislators, which forms the topic for an interdisciplinary chapter requirement for the NRT program in ”Risk and Uncertainty Quantification in the Marine Sciences”. The context of the study is the preservation of cultural keystone species (CKS) that are part of the core of indigeneous peoples culture. In this chapter, we explore how we can use mathematical modeling to design strategies to influence legislation that supports the protection of CKS.
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