Preprint A670/2010
Additive average Schwarz methods for discretization of elliptic problems with highly discontinuous coefficients

Marcus Sarkis | Dryja , Maksymilian

**Keywords: **
domain decomposition methods | additive Schwarz method | finite element method | discontinuous Galerkin method | elliptic problems with highly discontinuous coefficients | heterogeneous coefficients

The second order elliptic problem with highly discontinuous coefficients is considered. The problem is discretized by two methods: 1) a continuous finite element method (FEM) and 2) a composite discretization given by a continuous
FEM inside the substructures and a discontinuous Galerkin method (DG)
across the boundaries of these substructures. The
main goal of this paper is to design and analyze parallel algorithms for the resulting discretizations. These algorithms are additive Schwarz methods (ASMs) with special coarse spaces spanned by functions that are
almost piecewise constant with respect to the substructures for the first discretization and by piecewise constant functions for the
second discretization. It is established that the condition number of the preconditioned
systems does not depend on the jumps of the coefficients across
the substructure boundaries and outside of a thin layer along the
substructure boundaries. The algorithms are very well suited for parallel computations.