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.