Happy 25th Anniversary DDM! ... But How Fast Can the Schwarz Method Solve Your Logo?

摘要

"Vous n'avez vraiment rien a faire"! This was the smiling reaction of Laurence Halpern when the first author told her about our wish to accurately estimate the convergence rate of the Schwarz method for the solution of the ddm logo, see Figure 1 (left). Anyway, here we are: to honor the 25th anniversary of the domain decomposition conference, we study the convergence rate of the alternating Schwarz method for the solution of Laplace's equation defined on the ddm logo. This method was invented by H.A. Schwarz in 1870 [12] for the solution of the Laplace problem Δu=0 in Ω, u = g on ∂Ω.(1) Here g is a sufficiently regular function and Ω is the ddm logo, obtained from the union of a disc Ω_1 and a rectangle Ω_2. as historically considered by Schwarz [12]; see Figure 1 (center). In this paper, we assume that Ω_1 is a unit disc, and Ω_2 has length δ+ L and height 2 cos α. Here, δ, L and α are used to parametrize Ω; see Figure 1 (right). In particular, δ and L measure the overlapping and non-overlapping parts of Ω_2, and α is the angle that parametrizes the interface Γ_1:= ∂Ω_1∩Ω_2 The other interface Γ_2 := ∂Ω_2∩Ω_1) ) is clearly parametrized by δ and α, and it is composed by three segments whose vertices are (δ,0), (0,0), (0,2sinα), and (δ,2 sin α). To avoid meaningless geometries (e.g., Ω_2 Ω_1 becomes a disjoint set), we assume that δ and α are non-negative and satisfy δ < 2 cos α.