Preprint A540/2007
BDDC methods for discontinuous Galerkin discretization of elliptic
problems

Marcus Sarkis | Dryja , Maksymilian | Galvis , Juan

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

A discontinuous Galerkin (DG) discretization of
Dirichlet problem for second-order elliptic equations with
discontinuous coefficients in 2-D is considered.
For this discretization, Balancing Domain Decomposition with
Constraints (BDDC) algorithms are designed and analyzed as an additive
Schwarz method (ASM). The coarse and local problems are defined using
special partitions of unity and edge constraints. Under certain assumptions on
the coefficients and the mesh sizes
across $\partial \Omega_i$,
where the $\Omega_i$ are disjoint subregions of the original region $\Omega$,
a condition number estimate $ C(1 + \max_i\log (H_i/ h_i))^2$ is
established with $C$ independent of $h_i$, $H_i$ and the jumps
of the coefficients. The
algorithms are well suited for parallel computations and can be
straightforwardly extended to the 3-D problems. Results of
numerical tests are included which confirm the theoretical results and
the necessity of the imposed assumptions.