Mixing finite-element and finite-difference discretizations in linear monotone Schwarz iterations for semilinear elliptic PDEs

In this paper, we are concerned with a mixed finite-element and finite-difference (FEM-FDM) approximation method of linear Schwarz alternating iterations for a class of semilinear elliptic PDEs.More precisely, we consider two overlapping sub-domains $ \Omega_{1} $ and $ \Omega_{2} $, discretized by FEM-FDM, respectively, in the context of non-matching grids in the overlapping region. In this situation, combining a convergence result due to S.H. Lui [18] and a fundamental lemma consisting of estimating the gap between the continuous and discrete Schwarz iterations, we prove that the corresponding discrete Schwarz iterations converge uniformly to the true solution on $ \Omega_{1} $ and $ \Omega_{2} $, respectively. We also give some numerical results to support the theory.