Convergence of the Dirichlet–Neumann Alternating Method for Semilinear Elliptic Equations

The Dirichlet–Neumann alternating method is a common domain decomposition method for nonoverlapping domain decompositions without cross-points, and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet–Neumann alternating method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov–Poincaré formulation of the Dirichlet–Neumann alternating method.