L^∞-Convergence Analysis of a Finite Element Linear Schwarz Alternating Method for a Class of Semi-Linear Elliptic PDEs

In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for a class of semi-linear elliptic partial differential equations, in the context of linear iterations and non-matching grids. More precisely, making use of the subsolution-based concept, we prove that finite element Schwarz iterations converge, in the maximum norm, to the true solution of the PDE. We also give numerical results to validate the theory. This work introduces a new approach and generalizes the one in [14] as it encompasses a larger class of problems.