The Origins of the Alternating Schwarz Method

Schwarz methods are nowadays known as parallel solvers, and there are many variants: alternating and parallel Schwarz methods at the continuous level, additive and multiplicative Schwarz methods at the discrete level, also with restricted variants, which in the additive case build the important bridge between discrete and continuous Schwarz methods, see [4]. But where did these methods come from? Why were they invented in the first place? We explain in this paper that Hermann Amandus Schwarz invented the alternating Schwarz method in [18] to close an important gap in the proof of the Riemann mapping theorem, which was based on the Dirichlet principle. The Dirichlet principle itself addresses the important question of existence and uniqueness of solutions of Laplace’s equation on a bounded domain with Dirichlet boundary conditions, and in the 19th century, this equation appeared independently in many different areas. It was therefore of fundamental importance to put the Dirichlet principle on firm mathematical grounds, and this is one of the major achievements of Schwarz.