Non-overlapping Domain Decomposition Applied to Incompressible Flow Problems

A non-overlapping domain decomposition method with Robin-type transmission conditions which is known for scalar advection-diffusion-reaction problems [2],[5] is generalized to cover the Oseen equations. The presented method, which is later referred to as DDM, is an additive iteration-by-subdomains algorithm. Hence parallelism is given in a very natural way. The formulation is based on the continuous level to study the DDM without dealing with a special discretization. A convergence result for the “continuous” algorithm is presented. To treat incompressible Navier-Stokes problems, the A parallel implementation based on a finite element discretization has been done. Numerical results indicating linear convergence with a rate independent of the mesh size are presented for both the (linear) Oseen equations and the (nonlinear) Navier-Stokes equations. We denote by L(Ω) the space of square integrable functions with norm ‖ · ‖0,Ω and inner product (·, ·)Ω. H(Ω) denotes the usual Sobolev space with norm ‖·‖s,Ω. For Γ ⊂ ∂Ω we write 〈·, ·〉Γ for the inner product in L(Γ) (or, if needed, for the duality product between H 1 2 00(Γ) and H − 2 00 (Γ)). The space H 1 2 00(Γ) consists of functions u ∈ H 2 (Γ) with d− 12u ∈ L(Γ) where d(x) = dist(x, ∂Γ) [3, Chap 1., Sec. 11.4]. We explain the DDM for the Oseen equations in Section 2 and look into its analysis in Section 3. Then we explain how to discretize the method (Section 4) and apply it to the Navier-Stokes equations (Section 5). Numerical results are presented in Section 6.