Donatella Marini

Discontinuous Galerkin Approximations for Elliptic Problems

In this article, we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability and error estimates in various norms are proven.

Discontinuous Galerkin Methods for Elliptic Problems

We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cockburn and Shu, and the method of Baumann and Oden. It also includes the so-called interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others.

Applications of the pseudo residual-free bubbles to the stabilization of convection - diffusion problems

Abstract Residual-free bubbles have been recently introduced in order to compute optimal values for the stabilization methods a la Hughes-Franca. However, unless in very special situations (one-dimensional problems, limit cases, etc.) they require the actual solution of PDE problems (the bubble problems) in each element. Thus, they are very difficult to be used in practice. In this paper we present, for the special case of convection-dominated elliptic problems, a cheap way to compute approximately the solution of the bubble problem in each element. This provides, as a consequence, a cheap way to compute good approximations for the optimal values of the stabilization parameters.

Modeling Subgrid Viscosity for Advection--Diffusion Problems

We analyse the effect of the subgrid viscosity on a finite element discretisation, with piecewise linear elements, of a linear advection-diffusion scalar equation. We point out the importance of a proper tune-up of the viscosity coefficient, and we propose a heuristic method for obtaining reasonable values for it. The extension to more general problems is then hinted in the last section.

Subgrid Phenomena and Numerical Schemes

In recent times, several attempts have been mad e to recover some information from the subgrid scales and transfer them to the computational scales. Many stabilizing techniques can also be considered as part of this effort. We discuss here a framework in which some of these attempts can be set and analyzed.

A monotonic scheme for advection-diffusion problems

Abstract A stable finite element scheme for advection dominated problems is presented. The method is based on a classical piecewise linear continuous approximation of the solution and is proved to verify the discrete maximum principle whenever the triangulation is of weakly acute type. Several numerical tests confirm robustness of the method.