All finite element methods, as well much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems type: find u?V such a(v,u)=l(v) each v?L, where V,L are Sobolev spaces. However, systems Friedrichs type, there is a sharp disparity between established well-posedness theories, which not variational, and very successful discontinuous Galer...

This paper is the second part of a work attempting to give a unified analysis of Discontinuous Galerkin methods. The setting under scrutiny is that of Friedrichs’ systems endowed with a particular 2×2 structure in which some of the unknowns can be eliminated to yield a system of second-order elliptic-like PDE’s for the remaining unknowns. For such systems, a general Discontinuous Galerkin metho...

In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by the second author and D. Bochiş, who showed that they cannot have more than 80 facets, except perhaps for crystallographic space groups in the cubic system. Taking advantage of the recent, simpler, classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and ...

In this paper we present the exact solution of the Riemann Problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system obtained by adding an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satis es the p...

This work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant-Friedrichs-Levy condition. The method can be naturally completed by adaptive local mesh refinements and it tur...

Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space-time finite element scheme of the tent pitching type, with spaces that conform to the ...