Preprint A480/2006
Balancing Domain Decomposition Methods for Discontinuous Galerkin Discretizations

Marcus Sarkis | Dryja, Maksymilian | Galvis , Juan

**Keywords: **
Garlerkin Discontinuous | preconditioners | Schwarz methods | domain decomposition | Finite Element | discontinuous coefficients | nonmatching grids

A discontinuous Galerkin (DG) discretization of
a Dirichlet problem for second order elliptic equations with
discontinuous coefficients in two dimensions is considered. The
problem is considered in a polygonal region $\Omega$ which is a
union of disjoint polygonal
substructures $\Omega_i$ of size $O(H_i)$.
Inside each substructure $\Omega_i$, a
triangulation ${\cal{T}}_{h_i}(\Omega_i)$
with a parameter $h_i$ and a conforming finite element method are
introduced. To handle
nonmatching meshes across $\partial \Omega_i$, a
DG method that uses symmetrized interior penalty terms on
the boundaries $\partial \Omega_i$ is considered. In this paper we
design and analyze Balancing Domain Decomposition (BDD)
algorithms for solving the resulting discrete systems.
Under certain assumptions on the coefficients and the mesh sizes
across $\partial \Omega_i$,
a condition number estimate $ C(1 + \max_i\log^2 \frac{H_i}{h_i})$ is
established with $C$ independent of $h_i$, $H_i$ and the jumps
of the coefficients. The
algorithm is well suited for parallel computations and can be
straightforwardly extended to three-dimensional problems. Results of
numerical tests are included which confirm the theoretical results and
the imposed assumption.