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Abstract. We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h), where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345-357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.Mathematics Subject Classification. 65N30, 65N55.

Abstract.A postprocessing technique for mixed finite-element methods for the incompressible Navier-Stokes equations is studied. The technique was earlier developed for spectral and standard finite-element methods for dissipative partial differential equations. The postprocessing amounts to solving a Stokes problem on a finer grid (or higher-order space) once the time integration on the coarser mesh is completed. The analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations. Numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency.Key words. Navier-Stokes equations, mixed finite-element methods AMS subject classifications. 65M60, 65M20, 65M15, 65M12 DOI. 10.1137/0406028211. Introduction. This paper in a sense culminates the development of a postprocessing technique to increase the accuracy and computational efficiency of Galerkin methods for dissipative partial differential equations introduced in [18]. We turn to the equations which gave rise to this postprocessing technique, the incompressible Navier-Stokes equations, and we address those Galerkin methods for these equations which, when complex-shaped bodies are present, are acknowledged to be of wider applicability, mixed finite-element (MFE) methods.The postprocessing technique we study here was originally developed for spectral methods [18], [19]. At that moment, either its analysis and understanding or its development seemed to depend heavily on the properties of the Fourier modes, although this was not a shortcoming to prove its usefulness in the study of nonlinear shell vibrations [27]. In later works [13], [14], the dependence on the Fourier modes was overcome. Of particular importance to the present work, besides [14], has been the development of the postprocessing technique for finite-element methods in [20], [15]. In [20], it was devised how to carry out the postprocessing without the help of an approximate inertial manifold [11], [12], a concept more suited to spectral methods and eigenfunction expansion. In [15], it is shown what gains can be expected when postprocessing low-order elements.As is usually the case with MFE methods, it is the experience and understanding gained in previous works (see [14] [20], and the references cited therein) with simpler equations and methods which has allowed the present one to be written. Furthermore, although for simplicity we focus on Hood-Taylor [26] elements, the postprocessing technique can be easily adapted to other kinds of mixed

We present a computational study for a family of discontinuous Galerkin methods for the one dimensional Vlasov-Poisson system, recently introduced in [4]. We introduce a slight modification of the methods to allow for feasible computations while preserving the properties of the original methods. We study numerically the verification of the theoretical and convergence analysis, discussing also the conservation properties of the schemes. The methods are validated through their application to some of the benchmarks in the simulation of plasma physics.Numerical simulation has become a major tool for understanding the complex behavior of a plasma or a particle beam in many situations. This is due not only to the large number of physical applications and technological implications of the behavior of plasmas, but also to the intrinsic difficulties of the models used to describe such behavior. In fact, it was recognized long time ago that there does not exist any fully satisfactory macroscopic model (fluid equations) which can be used to describe the particle interaction in laser-fusion problems. In contrast, microscopic models (kinetic equations) can provide a more accurate description of the plasmas.One of the simplest model problems that is currently used in the simulation of plasmas is the Vlasov-Poisson system. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. The unknown, typically denoted by f (x, v, t) (with x standing for position, v for velocity and t for time), represents the distribution function of particles (ions, electrons, etc.) in the phase space. The coupling with a self-consistent electrostatic field (neglecting magnetic effects) is taken into account through the Poisson equation. The nonlinear structure of the system prevents from obtaining analytical solutions, except for a few academic cases (see the surveys [35,13,26] for a good description on the state of the art of the mathematical analysis of the problem). Therefore, numerical simulations have to be performed to study realistic physical phenomena.At the present time, there can be distinguished two main classes of numerical methods for simulating plasmas; Lagrangian (or probabilistic) and Eulerian (or deterministic) methods. The former class include all different types of particle methods [12,22,47,34,41,32,7] and has been a preferred choice since the beginnings of numerical simulations in plasma physics 1991 Mathematics Subject Classification. 82C80, 65M60, 65M12, 82A70.

Abstract. In this paper we introduce and analyze some non-overlapping multiplicative Schwarz methods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods. For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion on the issue of preconditioning non-symmetric DG approximations of elliptic problems is also included. Extensive numerical experiments to confirm the theoretical results and to assess the robustness and the efficiency of the proposed preconditioners are provided.Mathematics Subject Classification. 65N30, 65N55.