Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier--Stokes equations

28 pagesInternational audienceTwo discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier--Stokes equations. Two discrete convective trilinear forms are proposed, a non-conservative one relying on Temam's device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure

Error analysis of discontinuous Galerkin methods for Stokes problem under minimal regularity

In this article, we analyze several discontinuous Galerkin methods (DG) for the Stokes problem under the minimal regularity on the solution. We assume that the velocity u belongs to [H1 0 (­)]d and the pressure p 2 L2 0 (­). First, we analyze standard DG methods assuming that the right hand side f belongs to [H¡1(­) \ L1(­)]d. A DG method that is well de¯ned for f belonging to [H¡1(­)]d is then investigated. The methods under study include stabilized DG methods using equal order spaces and inf-sup stable ones where the pressure space is one polynomial degree less than the velocity space.Preprin

Energy norm error estimates for averaged discontinuous Galerkin methods: multidimensional case

A mathematical analysis is presented for a class of interior penalty (IP) discontinuous Galerkin approximations of elliptic boundary value problems. In the framework of the present theory one can derive some overpenalized IP bilinear forms in a natural way avoiding any heuristic choice of fluxes and penalty terms. The main idea is to start from bilinear forms for the local average of discontinuous approximations which are rewritten using the theory of distributions. It is pointed out that a class of overpenalized IP bilinear forms can be obtained using a lower order perturbation of these. Also, error estimations can be derived between the local averages of the discontinuous approximations and the analytic solution in the $H^1$-seminorm. Using the local averages, the analysis is performed in a conforming framework without any assumption on extra smoothness for the solution of the original boundary value problem

Energy norm error estimates for averaged discontinuous Galerkin methods in 1 dimension

Numerical solution of one-dimensional elliptic problems is investigated using an averaged discontinuous discretization. The corresponding numerical method can be performed using the favorable properties of the discontinuous Galerkin (dG) approach, while for the average an error estimation is obtained in the i?1-seminorm. We point out that this average can be regarded as a lower order modification of the average of a well-known overpenalized symmetric interior penalty (IP) method. This allows a natural derivation of the overpenalized IP methods. © 2014 Institute for Scientific Computing and Information

On discrete functional inequalities for some finite volume schemes

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the space $BV(\Omega)$ into $L^{N/(N-1)}(\Omega)$ for a Lipschitz domain $\Omega \subset \mathbb{R}^{N}$, with $N \geq 2$. Finally, we give several applications to discrete duality finite volume (DDFV) schemes which are used for the approximation of nonlinear and non isotropic elliptic and parabolic problems