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.