Neumann‐Neumann methods for a DG discretization on geometrically nonconforming substructures

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ω_i of size O(H_i). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ω_i, a conforming finite element space associated to a triangulation begin{align*} {mathcal{T}}_{h_i}(Omega_i)end{align*} is introduced. To handle the nonmatching meshes across ?Ω_i, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ?Ω_i, a condition number estimate begin{align*} C(1 + max_ilog frac{H_i}{h_i})^2end{align*} 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. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012