History

Schwarz methods have their origins in an alternating algorithm devised by H.A. Schwarz to demonstrate the existence of harmonic functions (i.e. twice continuously derivable solutions of the Laplace equation Δu = 0) in a composite domain, with a prescribed edge value g. He began by considering a domain composed of a disk T ₁ and a square T ₂ (see figure 1 a), domains for which an explicit solution by means of Fourier series exists.

The respective boundaries of these two domains are denoted ∂T...