The main purpose of this paper is to study the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with multi-step implicit-explicit (IMEX) time discretization schemes, for solving time-dependent incompressible fluid flows. We will give theoretical analysis for the Oseen equation, and assess the performance of the schemes for incompressible Navier-Stokes equations numerically. For the Oseen equation, using first order IMEX time discretization as an example, we show that the IMEX-LDG scheme is unconditionally stable for $\Qcal_k$ elements on cartesian meshes, in the sense that the time-step $\dt$ is only required to be {bounded from above} by a positive constant independent of the spatial mesh size $h$. Furthermore, by the aid of the \emph{Stokes projection} and an elaborate energy analysis, we obtain the $L^{\infty}(L^2)$ optimal error estimates for both the velocity and the stress (gradient of velocity), in both space and time. By the \emph{inf-sup} argument, we also obtain the $L^{\infty}(L^2)$ optimal error estimates for the pressure. Numerical experiments are given to validate our main results.

METHODS AND APPLICATIONS OF ANALYSIS. 2000 International Press Vol. 7, No. 3, pp. 591-604, September 2000 012

In this paper, we propose and analyze a nodal-continuous and $H^1$-conforming finite element method for the numerical computation of Maxwell's equations, with singular solution in a fractional order Sobolev space $H^r(\Omega)$, where $r$ may take any value in the most interesting interval $(0, 1)$. The key feature of the method is that mass-lumping linear finite element $L^2$ projections act on the curl and divergence partial differential operators so that the singular solution can be sought in a setting of $L^2(\Omega)$ space. We shall use the nodal-continuous linear finite elements, enriched with one element bubble in each element, to approximate the singular and non-$H^1$ solution. Discontinuous and nonhomogeneous media are allowed in the method. Some error estimates are given and a number of numerical experiments for source problems as well as eigenvalue problems are presented to illustrate the superior performance of the proposed method.

Image restoration is a common problem in visual process. In this paper, a modified minimization model is presented, which combines the L1 and L2 fidelity terms with a combined quadratic L2 and TV regularizer just as the regularizer of Cai et al. (2013). The combined regularizer has the priorities of preserving desirable edges and ensuring several kinds of noises can be removed clearly. Split-Bregman algorithm is effi- ciently employed to solve this model and convergence analysis is also discussed. Moreover, we extend the proposed model and algorithm for image restoration involving blurry images and color images. Experimental results show that our proposed model and algorithm have good performance both in visual and ISNR values for different kinds of blurs and noises including mixed noise.

In this paper we propose a finite element time-domain method for modeling the optical black holes (OBHs) coupled with the perfectly matched layer (PML) technique. Stability analysis is carried out for the proposed scheme. Simulations of cylindrical, elliptical and square black holes demonstrate that our method is quite effective in modeling OBHs in time domain. To our best knowledge, this is the first OBHs simulation realized by the finite element time-domain method.