Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems

In this article, we are interested in the standard finite element approximation method of linear additive Schwarz iterations for a class of semi-linear elliptic problems, for two subdomains, in the context of non-matching grids. More precisely, by means of a uniform convergence result from the study by Lui and a fundamental lemma consisting of estimating, at each iteration, the gap between the continuous and the finite element Schwarz iterates, we prove that the discrete Schwarz sequences converge, in the maximum norm, to the true solution. Moreover, we also give numerical results to support the theoretical findings.