Preprint A634/2009
Neumann-Neumann methods for a DG discretization of elliptic problems
with discontinuous coefficients on
geometrically nonconforming substructures

Marcus Sarkis | Dryja, Maksymilian | Galvis , Juan

**Keywords: **
interior penalty discretization |
discontinuous Galerkin method | elliptic problems with
discontinuous coefficients | finite element method | Neumann-Neumann algorithms | Schwarz methods | preconditioners | nonconforming decomposition

A discontinuous Galerkin discretization
for second order elliptic equations with
{\it discontinuous coefficients} in 2-D is considered. The domain
of interest $\Omega$ is assumed to be a union of
polygonal substructures $\Omega_i$ of size $O(H_i)$.
We allow this substructure decomposition to be geometrically nonconforming.
Inside each
substructure $\Omega_i$, a conforming finite element space
associated to a triangulation ${\mathcal{T}}_{h_i}(\Omega_i)$
is introduced. To handle the
nonmatching meshes across $\partial \Omega_i$, a
discontinuous Galerkin discretization is considered. In this paper
additive and hybrid Neumann-Neumann Schwarz methods are designed and analyzed.
Under natural assumptions on the coefficients and on the mesh sizes
across $\partial \Omega_i$,
a condition number estimate $ C(1 + \max_i\log \frac{H_i}{h_i})^2$ is
established with $C$ independent of $h_i$, $H_i$, $h_i/h_j$, and the jumps
of the coefficients. The
method is well suited for parallel computations and can be
straightforwardly extended to three dimensional problems. Numerical
results are included.