Acceleration of a Domain Decomposition Method for Advection-Diffusion Problems

in a bounded polyhedral domain Ω ⊂ R with a Lipschitz boundary ∂Ω and 0 < ǫ ≤ 1,b ∈ [H(Ω) ∩ L∞(Ω)]d, c ∈ L∞(Ω), f ∈ L(Ω), c− 1 2∇ · b ≥ 0. Let {Ωk} be a non-overlapping macro partition with Ω = ∪k=1Ωk. The goal of the well-known DDM of Robin type (Lions [1990]) is to enforce (in appropriate trace spaces) continuity of the solution u and of the diffusive and advective fluxes ǫ∇u·nkj resp.− 1 2 (b·nkj)u on the interfaces Γkj := ∂Ωk∩∂Ωj . The algorithm reads in strong form: For given uk , n ∈ N0, in each Ωk, find (in parallel) u k , such that Lu k = f in Ωk (3) u k = 0 on ∂Ωk ∩ ∂Ω (4) Φk(u n+1 k ) = Φk(u n j ) on Γkj (5)