Hybridization of Discontinuous Galerkin Finite Element Methods with Application to Convection Dominated Problems and Domain Decomposition
Hybridization of Discontinuous Galerkin Finite Element Methods with Application to Convection Dominated Problems and Domain Decomposition
Discontinuous Galerkin methods have attracted significant interest
in the past, in particular for convection dominated problems, as they allow
a generalization of stabilized finite volume methods to higher order.
One of the drawbacks of DG methods is that they typically have "enlarged
stencils", i.e., the system matrices resulting from the discretization
are substantially less sparse than that of standard finite element
methods,
and hence the assembling and solution of the resulting linear systems
becomes more expensive. These disadvantages can however be overcome
by "hybridization", which is a well established technique for mixed finite
element methods.
After a short introduction to the interior penalty DG method, we derive a
corresponding hybrid formulation and demonstrate how the resulting
finite element scheme can be applied to convection-dominated problems.
We then show that hybridization can further be utilized for a flexible
discretization of interface problems and domain decomposition. The
theoretical
results are illustrated by numerical experiments.