Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity

Novel Methods for the Time-Dependent Maxwell’s Equations and their Applications

This dissertation investigates three different mathematical models based on the time domain Maxwell\u27s equations using three different numerical methods: a Yee scheme using a non-uniform grid, a nodal discontinuous Galerkin (nDG) method, and a newly developed discontinuous Galerkin method named the weak Galerkin (WG) method. The non-uniform Yee scheme is first applied to an electromagnetic metamaterial model. Stability and superconvergence error results are proved for the method, which are then confirmed through numerical results. Additionally, a numerical simulation of backwards wave propagation through a negative-index metamaterial is given using the presented method. Next, the nDG method is used to simulate signal propagation through a corrugated coaxial cable through the use of axisymmetric Maxwell\u27s equations. Stability and error analysis are performed for the semi-discrete method, and are verified through numerical results. The nDG method is then used to simulate signal propagation through coaxial cables with a number of different corrugations. Finally, the WG method is developed for the standard time-domain Maxwell\u27s equations. Similar to the other methods, stability and error analysis are performed on the method and are verified through a number of numerical experiments

FINITE ELEMENT METHODS FOR AXISYMMETRIC PDES AND DIVERGENCE FREE FINITE ELEMENT PAIRS ON PARTICULAR MESH REFINEMENTS

This dissertation discusses the following two main topics. 1) Finite element approximation for Partial Differential Equations (PDEs) defined on axisymmetric domains: We introduce the Darcy equations on axisymmetric domains and we show the stability of a low--order Raviart-Thomas element pair. We provide numerical experiments to support our theoretical results. Also, we introduce the Stokes equations on axisymmetric domains and show that the axisymmetric Stokes equations can fit within a commutative de Rham complex. 2) Connection between the grad-div stabilized and divergence-free Stokes finite element pairs and low--order divergence-free elements on particular mesh refinements: We introduce the most recent results that connect the grad-div stabilized Taylor--Hood (TH) finite element pair and divergence-free Scott--Vogelius (SV) finite element pairs, and we use these results to extend and generalize this connection to other Stokes finite element pairs. Finally, we provide numerical examples for low order divergence-free Stokes finite element pairs defined on particular mesh refinements. This research is focused on the numerical implementation aspects of these finite element pairs

MULTIGRID IN A WEIGHTED SPACE ARISING FROM AXISYMMETRIC ELECTROMAGNETICS

Abstract. Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that the multigrid V-cycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a dual mixed method in weighted spaces. The uniformity of the multigrid convergence rate with respect to meshsize is then established theoretically and illustrated through numerical experiments. 1