Neumann-Neumann methods for a DG discretization on geometrically nonconforming substructures (Q2910808)

The authors consider a discontinuous Galerkin method (DG) for a 2D elliptic Dirichlet problem with discontinuous diffusion coefficients, continuing their research in [Lecture Notes in Computational Science and Engineering 60, 271--278 (2008; Zbl 1139.65329)] and using, for the DG, a variant with interior penalty going back to the first author, see [Comput. Methods Appl. Math. 3, No. 1, 76--85 (2003; Zbl 1039.65079)]. The discontinuous coefficients are assumed to be constant on the subdomains of a coarse subdivision which is allowed to be geometrically nonconforming (the edges of the subdomains may consist of several parts of edges of neighbouring subdomains). They use then conforming shape regular triangulations in every coarse subdomain and exploit, on the common inner boundaries (the interfaces -- where the nodes may not match) of neighbouring subdomains, harmonic means of the coefficients and step sizes. Furthermore, an interface assumption is formulated on these harmonic means to decide which (parts of) the edges become masters and which slaves in the definition of prolongation operators.NEWLINENEWLINEThe authors prove for their additive and hybrid Schwarz preconditioners (to be used in parallel solution algorithms) estimates of the condition numbers which are independent of the quotients of the step sizes, and of the jumps in the coefficients, if the interface assumption is satisfied. But these condition numbers depend logarithmically on the step sizes. Extensive numerical experiments show the effectiveness of the algorithms proposed and the interface condition also to be necessary.