An Energy Stable Discontinuous Galerkin Time-Domain Finite Element Method in Optics and Photonics

In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell's equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new capabilities of methods of this type are to efficiently model linear and nonlinear effects, for example of Kerr nonlinearities. Energy stable discretizations both at the semi-discrete and the fully discrete levels are presented. In particular, the proposed semi-discrete scheme is optimally convergent in the spatial variable on Cartesian meshes with $Q_k$-type elements, and the fully discrete scheme is conditionally stable with respect to a specially defined nonlinear electromagnetic energy. The approaches presented prove to be robust and allow the modeling of optical problems and the treatment of complex nonlinearities as well as geometries of various physical systems coupled with electromagnetic fields

An energy stable discontinuous Galerkin time-domain finite element method in optics and photonics

In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell’s equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new capabilities of methods of this type are to efficiently model linear and nonlinear effects, for example of Kerr nonlinearities. Energy stable discretizations both at the semi-discrete and the fully discrete levels are presented. In particular, the proposed semi-discrete scheme is optimally convergent in the spatial variable on Cartesian meshes with Qk-type elements, and the fully discrete scheme is conditionally stable with respect to a specially defined nonlinear electromagnetic energy

C0 Interior Penalty Methods for Cahn-Hilliard Equations

In this work we study C0 interior penalty methods for Cahn-Hilliard equations. In Chapter 1 we introduce Cahn-Hilliard equations and the time discretization that leads to linear fourth order boundary value problems. In Chapter 2 we review related fundamentals of finite element methods and multigrid methods. In Chapter 3 we formulate the discrete problems for linear fourth order boundary value problems with the boundary conditions of the Cahn-Hilliard type, which are called C0 interior penalty methods, and we carry out the convergence analysis. In Chapter 4 we consider multigrid methods for the C0 interior penalty methods. We present two smoothing schemes and compare their performance. In Chapter 5 we apply the C0 interior penalty methods and the time discretization scheme to nonlinear time-dependent Cahn-Hilliard equations. Numerical examples for phase separation and image processing are presented

Efficient discontinuous Galerkin (DG) methods for time-dependent fourth order problems

In this thesis, we design, analyze and implement efficient discontinuous Galerkin (DG) methods for a class of fourth order time-dependent partial differential equations (PDEs). The main advantages of such schemes are their provable unconditional stability, high order accuracy, and their easiness for generalization to multi-dimensions for arbitrarily high order schemes on structured and unstructured meshes. These schemes have been applied to two fourth order gradient flows such as the Swift-Hohenberg (SH) equation and the Cahn-Hilliard (CH) equation, which are well known nonlinear models in modern physics. For fourth order PDEs of the form $\partial_t u= -\mathcal{L}^2 u +f$, where $\mathcal{L}$ is an adjoint elliptic operator, the fully discrete DG schemes are constructed in several steps: (a) rewriting the equation as a system of second order PDEs so that $u_t=\mathcal{L}q +f, \quad q=-\mathcal{L}u$; (b) applying the DG discretization to this mixed formulation with central numerical fluxes on interior interfaces and weakly enforcing the specified boundary conditions; and (c) combining a special class of time discretizations, that allows the method to be unconditionally stable regardless of its accuracy. Main contributions of this thesis are as follows: Firstly, we introduce mixed discontinuous Galerkin methods without interior penalty for the spatial DG discretization, and the semi-discrete schemes are shown $L^2$ stable for linear problems, and unconditionally energy stable for nonlinear gradient flows. For the mixed DG method applied to linear problems with periodic boundary conditions, we establish the optimal $L^2$ error estimate of order $O(h^{k+1} +\Delta t^2)$ for polynomials of degree $k$ with the Crank-Nicolson time discretization. In addition, the resulting DG methods can easily handle different boundary conditions. Secondly, for a class of fourth order gradient flow problems, including the SH equation, we combine the so-called \emph{Invariant Energy Quadratization} (IEQ) approach [X. Yang, J. Comput. Phys., 327:294{316, 2016] as time discretization. Coupled with a projection step for the auxiliary variable, both first and second order EQ-DG schemes are shown unconditionally energy stable. In addition, they are linear and can be efficiently solved without resorting to any iteration method. We present extensive numerical examples that support our theoretical results and illustrate the efficiency, accuracy, and stability of our new algorithms. Benchmark problems are also presented to examine the long time behavior of the numerical solutions. Both the theoretical and algorithmic aspects of these methods have potentially wide applications. Progress is made with the IEQ-DG framework to solve the Cahn-Hilliard equation. With the usual penalty in the DG discretization, the resulting EQ-DG schemes are shown to be able to produce free-energy-decaying, and mass conservative solutions, irrespective of the time step and the mesh size. In addition, the schemes are easy to implement, and test cases for the Cahn-Hilliard equation will be reported