BDDC methods for discontinuous Galerkin discretization of elliptic problems
Marcus Sarkis | Dryja , Maksymilian | Galvis , Juan
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.