Yunqing Huang

Introduction to Finite Element Methods

The finite element method (FEM) is arguably one of the most robust and popular numerical methods used for solving various partial differential equations (PDEs). Due to the diligent work of many researchers over the past several decades, the fundamental theory and implementation of FEM have been well established as evidenced by many excellent books published in this area (e.g., [4, 20, 21, 39, 51, 54, 65, 78, 158, 163, 243]).

Error Estimates and Superconvergence of Mixed Finite Element Methods for Convex Optimal Control Problems

In this paper, we investigate the discretization of general convex optimal control problem using the mixed finite element method. The state and co-state are discretized by the lowest order Raviart-Thomas element and the control is approximated by piecewise constant functions. We derive error estimates for both the control and the state approximation. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problem. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

The paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The conclusion is that if local approximation properties of nonconforming finite element spaces are better than total errors (sums of global approximation errors and consistency errors) of nonconforming finite element methods, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we analyze abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. We show that this condition hold for most low-order nonconforming finite elements in literature. In addition, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we also propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.

Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials

The purpose of this book is to provide an up-to-date introduction to the time-domain finite element methods for Maxwells equations involving metamaterials. Since the first successful construction of a metamaterial with both negative permittivity and permeability in 2000, the study of metamaterials has attracted significant attention from researchers across many disciplines. Thanks to enormous efforts on the part of engineers and physicists, metamaterials present great potential applications in antenna and radar design, sub-wavelength imaging, and invisibility cloak design. Hence the efficient simulation of electromagnetic phenomena in metamaterials has become a very important issue and is the subject of this book, in which various metamaterial modeling equations are introduced and justified mathematically. The development and practical implementation of edge finite element methods for metamaterial Maxwells equations are the main focus of the book. The book finishes with some interesting simulations such as backward wave propagation and time-domain cloaking with metamaterials.

Developing Finite Element Methods for Maxwell's Equations in a Cole-Cole Dispersive Medium

In this paper, we consider the time-dependent Maxwells equations when Cole-Cole dispersive medium is involved. The Cole-Cole model contains a fractional time derivative term, which couples with the standard Maxwells equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1-26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572-591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533-1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12-20], we propose two fully discrete mixed finite element schemes for the Cole-Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

An Alternating Crank--Nicolson Method for Decoupling the Ginzburg--Landau Equations

The time-dependent Ginzburg--Landau model of superconductors consists of coupled nonlinear partial differential equations, which presents difficulties in the numerical solution. We present an alternating Crank--Nicolson method for this model that leads to two decoupled algebraic subsystems; one is linear and the other is semilinear. Both have nice numerical properties and can be solved by efficient matrix solvers. We show the stability and convergence and derive error estimates for this scheme. Numerical results are also reported.

A conforming finite element method for overlapping and nonmatching grids

In this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity. Both overlapping and nonoverlapping cases are considered. We prove that the new method admits an optimal convergence rate. The error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains. Our results are valid for multiple subdomains and any spatial dimensions.

Superconvergence analysis for the explicit polynomial recovery method

A new recovery technique explicit polynomial recovery (EPR) is analyzed for finite element methods. EPR reconstructs the value at edge centers by solving a local problem. In combination with the finite element solution at the vertex, a quadratic approximation is constructed. Besides improving the accuracy, it can also be applied in building the EPR-based error estimator. For the Poisson equation, the element center is a superconvergent point of the gradient of the EPR recovered function on an equilateral triangulation. Numerical examples are presented to verify the theoretical results and to show the performance of the EPR in the adaptive finite element method.

Interior penalty DG methods for Maxwell's equations in dispersive media

In this paper, we develop a fully-discrete interior penalty discontinuous Galerkin method for solving the time-dependent Maxwells equations in dispersive media. The model is described by a vector integral-differential equation. Our scheme is proved to be unconditionally stable and achieve optimal error estimates in both L^2 norm and energy norm. The scheme is implemented and numerical results supporting our analysis are presented.

Superconvergence of quadratic finite elements on mildly structured grids

Superconvergence estimates are studied in this paper on quadratic finite element discretizations for second order elliptic boundary value problems on mildly structured triangular meshes. For a large class of practically useful grids, the finite element solution u h is proven to be superclose to the inter-polant u I and as a result a postprocessing gradient recovery scheme for u h can be devised. The analysis is based on a number of carefully derived identities. In addition to its own theoretical interests, the result in this paper can be used for deriving asymptotically exact a posteriori error estimators for quadratic finite element methods.