In this paper, a fully discrete stabilized discontinuous Galerkin method is proposed to solve the Biot's consolidation problem. The existence and uniqueness of the finite element solution are obtained. The stability of the fully discrete solution is discussed. The corresponding error estimates for the approximation of displacement and pressure in a mesh dependent norm are obtained. The error es...

In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on [H 0 (Ω)] -conforming velocity reconstruction and H(div, Ω)-conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given ...

The present paper deals with the continuous work of extending multi-dimensional limiting process (MLP) onto correction procedure via reconstruction (CPR). MLP, which has been originally developed in finite volume method (FVM), provides an accurate, robust and efficient oscillation-control mechanism in multiple dimensions for linear reconstruction. Recently, MLP has been extended into higher-ord...

In this report, we address several aspects of the approximation of the MHD equations by a Galerkin Discontinuous finite volume schemes. This work has been initiated during a CEMRACS project in July and August 2008 in Luminy. The project was entitled GADMHD (for GAlerkin Discontinuous approximation for the Magneto-Hydro-Dynamics). It has been supported by the INRIA CALVI project. 1 Some properti...

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy [10]. First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes [6] also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations...

In this paper, we present a class of finite volume trigonometric weighted essentially non-oscillatory (TWENO) schemes and use them as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods based on trigonometric polynomial spaces to solve hyperbolic conservation laws and highly oscillatory problems. As usual, the goal is to obtain a robust and high order limiting procedure for such a RK...

In [20], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two dimensional problems on unstructured meshes, with the goal of obtaining a r...

Abstract In [22], two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in [23]. The extension...

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization...

We discuss the matrix-free implementation of Discontinuous Galerkin methods for compressible flow problems, i.e. the compressible Navier-Stokes equations. For the spatial discretization the CDG2 method and for temporal discretization an explicit Runge-Kutta method is used. For the presented matrix-free approach we discuss asynchronous communication, shared memory parallelization, and automated ...