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

Unconditionally energy stable IEQ-FEMs for the Cahn-Hilliard equation and Allen-Cahn equation

In this paper, we present several unconditionally energy-stable invariant energy quadratization (IEQ) finite element methods (FEMs) with linear, first- and second-order accuracy for solving both the Cahn-Hilliard equation and the Allen-Cahn equation. For time discretization, we compare three distinct IEQ-FEM schemes that position the intermediate function introduced by the IEQ approach in different function spaces: finite element space, continuous function space, or a combination of these spaces. Rigorous proofs establishing the existence and uniqueness of the numerical solution, along with analyses of energy dissipation for both equations and mass conservation for the Cahn-Hilliard equation, are provided. The proposed schemes' accuracy, efficiency, and solution properties are demonstrated through numerical experiments.Comment: 33 pages, 15 figure

A C0 Finite Element Method For The Biharmonic Problem In A Polygonal Domain

This dissertation studies the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. The biharmonic problem appears in various real-world applications, for example in plate problems, human face recognition, radar imaging, and hydrodynamics problems. There are three classical approaches to discretizing the biharmonic equation in the literature: conforming finite element methods, nonconforming finite element methods, and mixed finite element methods. We propose a mixed finite element method that effectively decouples the fourth-order problem into a system of one steady-state Stokes equation and one Poisson equation. As a generalization to the above-decoupled formulation, we propose another decoupled formulation using a system of two Poison equations and one steady-state Stokes equation. We solve Poisson equations using linear and quadratic Lagrange\u27s elements and the Stokes equation using Hood-Taylor elements and Mini finite elements. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are, in turn, proposed to solve the decoupled systems. Solving this problem in a non-convex domain is challenging due to the singularity occurring near re-entrant corners. We introduce a weighted Sobolev space and a graded mesh refine Algorithm to attack the singularity near re-entrant corners. We show the regularity results of each decoupled system in both Sobolev space and weighted Sobolev space. We derive the $H^1$ and $L^2$ error estimates for the numerical solutions on quasi-uniform and graded meshes. We present various numerical test results to justify the theoretical findings. Given the availability of fast Poisson solvers and Stokes solvers, our Algorithm is a relatively easy and cost-effective alternative to existing algorithms for solving the biharmonic equation

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal