The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving one-dimensional convection-diffusion equations with a nonlinear convection. Both Runge-Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step $\dt$ is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size $h$. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.

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Abstract BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance T O L. Upper bounds for condition numbers of the preconditioned systems are shown to be C T O L, with the constant C depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.

In this paper, a mathematical analysis of the popular carpet cloak model is carried out. The well-posedness of the model is first proved, and then a finite element time-domain method is developed for solving this model. Three carpet cloak simulations are demonstrated to show the effectiveness of our model and the developed algorithm. To the best of our knowledge, this is the first mathematical analysis carried out for the carpet cloak model. The numerical simulation using edge elements for the carpet cloak model is also original.

In this paper we present an {\em a priori} error estimate of the Runge-Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge-Kutta method and the finite element space is made up of piecewise polynomials of degree $k\geq 2$. Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition $\tau\leq\gamma h$, where $h$ is the element length and $\tau$ is time step, and $\gamma$ is a positive constant independent of $h$ and $\tau$. Optimal estimates are also considered when the upwind numerical flux is used.