On the effect of boundary conditions on the scalability of Schwarz methods

Here, q > 0 and the subscripts ‘b’ and ‘t’ stand for ‘bottom’ and ‘top’. As shown in Fig. 1, the domain Ω is the union of subdomains Ωj , j = 1, . . . , N , defined as Ωj := (aj , bj) × (0, 1), where a1 = 0, aj = L + aj−1 for j = 2, . . . , N + 1 and bj = aj+1 + 2δ for j = 0, . . . , N . Hence, the length of each subdomain is L+ 2δ and the length of the overlap is 2δ with δ ∈ (0, L/2). It is well known that one-level Schwarz methods are not weakly scalable, if the number of subdomains increases and the whole domain Ω is fixed.