Schwarz alternating method

Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved.

This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907.

It was first formulated by H. A. Schwarz[1] and served as a theoretical tool: its convergence for general second order elliptic partial differential equations was first proved much later, in 1951, by Solomon Mikhlin.

An iterative algorithm is introduced: At convergence, the solution on the overlap is the same when computed on the square or on the circle.

[3] Original papers Conformal mapping and harmonic functions PDEs and numerical analysis