Neumann-neumann Methods for a Dg Discretization of Elliptic Problems with Discontinuous Coefficients on Geometrically Nonconforming Substructures

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2-D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ωi, a conforming finite element space associated to a triangulation Thi (Ωi) is introduced. To handle the nonmatching meshes across ∂Ωi, 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 ∂Ωi, a condition number estimate C(1 + maxi log Hi hi )2 is established with C independent of hi, Hi, hi/hj , 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.