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.