Title:      Discontinuous Galerkin methods: An overview

Abstract:
      In this talk, we present the discontinuous Galerkin methods
and describe and discuss their main features. Since the methods use
completely discontinuous approximations, they produce mass matrices
that
are block-diagonal. This renders the methods highly parallelizable
when applied to hyperbolic problems. Another consequence of the use of
discontinuous approximations is that these methods  can easily handle
irregular meshes with hanging nodes and approximations that
have polynomials of different degrees in different elements.
They are thus ideal for use with adaptive algorithms. Moreover,
the methods are locally conservative (a property highly valued by the
computational fluid dynamics community) and, in spite of
providing discontinuous approximations, stable, and high-order
accurate. Even more, when applied to non-linear hyperbolic problems,
the discontinuous Galerkin methods are able to capture highly complex
solutions presenting discontinuities with high resolution.
In this talk, we concentrate on the exposition
of the ideas behind the devising of these methods as well as
on the mechanisms that allow them to perform so well in such a
variety of problems.