Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell\u27s Equations

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell\u27s equations. We begin with the introduction of Maxwell\u27s equations and a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell\u27s equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments

A unified description for dipoles of the fine-structure constant and SnIa Hubble diagram in Finslerian universe

We propose a Finsler spacetime scenario of the anisotropic universe. The Finslerian universe requires both the fine-structure constant and accelerating cosmic expansion have dipole structure, and the directions of these two dipoles are the same. Our numerical results show that the dipole direction of SnIa Hubble diagram locates at $(l,b)=(314.6^\circ\pm20.3^\circ,-11.5^\circ\pm12.1^\circ)$ with magnitude $B=(-3.60\pm1.66)\times10^{-2}$. And the dipole direction of the fine-structure constant locates at $(l,b)=(333.2^\circ\pm8.8^\circ,-12.7^\circ\pm6.3^\circ)$ with magnitude $B=(0.97\pm0.21)\times10^{-5}$. The angular separation between the two dipole directions is about $18.2^\circ$.Comment: 10 pages, 1 figur