Jichun Li

Numerical comparisons of two meshless methods using radial basis functions

The recent advance in the development of various kinds of meshless methods for solving partial differential equations has drawn attention of many researchers in science and engineering. One of the domain-type meshless methods is obtained by simply applying the radial basis functions (RBFs) as a direct collocation, which has shown to be effective in solving complicated physical problems with irregular domains. More recently, a boundary-type meshless method that combines the method of fundamental solutions and the dual reciprocity method with the RBFs has been developed. In this paper, the performances of these two meshless methods are compared and evaluated. Numerical results indicate that these two methods provide a similar optimal accuracy in solving both 2D Poissons and parabolic equations.

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]).

Overlapping domain decomposition method by radial basis functions

In this paper, overlapping domain decomposition with both multiplicative and additive Schwarz iterative techniques are incorporated into the radial basis functions for solving partial differential equations. These decomposition techniques circumvent the ill-conditioning problem resulted from using the radial basis functions as a global interpolant. Both the multiplicative and additive Schwarz iterative techniques achieve high performances even without the Krylov subspace accelerators. The effectiveness of the algorithms are demonstrated by performing numerical experiments for both a regular elliptic problem and a singularly perturbed elliptic problem respectively.

A comparison of efficiency and error convergence of multiquadric collocation method and finite element method

In this paper, we demonstrate the efficiency and accuracy of the multiquadric collocation method as compared to the finite element method. When used as interpolants, the multiquadric (MQ) radial basis function has the property of exponential convergence with respect to a shape parameter, according to a proof by Madych and Nelson. Although the optimal choice of shape parameter is still an unsettled issue, we demonstrate by three examples that the accuracy achieved by the MQ solution cannot be rivaled by the FEM.

Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems I: Reaction-diffusion Type☆

We consider the bilinear finite element method on a Shishkin mesh for the singularly perturbed elliptic boundary value problem −ϵ2(ι2uιx2 + ι2uιy2) + a(x,y)u = f(x,y) in two space dimensions. By using a very sophisticated asymptotic expansion of Han et al. [1] and the technique we used in [2], we prove that our method achieves almost second-order uniform convergence rate in L2-norm. Numerical results confirm our theoretical analysis.

High-Order Compact ADI Methods for Parabolic Equations

In this paper we develop a sixth-order compact scheme coupled with Alternating Direction Implicit (ADI) methods and apply it to parabolic equations in both 2-D and 3-D. Unconditional stability is proved for linear diffusion problems with periodic boundary conditions. Numerical examples supporting our theoretical analysis are provided.

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.

Finite Element Analysis for Wave Propagation in Double Negative Metamaterials

In this paper, we develop both semi-discrete and fully-discrete mixed finite element methods for modeling wave propagation in three-dimensional double negative metamaterials. Here the model is formed as a time-dependent linear system involving four dependent vector variables: the electric and magnetic fields, and the induced electric and magnetic currents. Optimal error estimates for all four variables are proved for Nédélec tetrahedral elements. To our best knowledge, this is the first error analysis obtained for Maxwell’s equations when metamaterials are involved.

Uniform Convergence and Superconvergence of Mixed Finite Element Methods on Anisotropically Refined Grids

The lowest order Raviart--Thomas rectangular element is considered for solving the singular perturbation problem

Superconvergence analysis for Maxwell’s equations in dispersive media

-\mbox{div}(a\nabla p)+bp=f,