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

Abstract

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 subdomains $\Omega_{1}$ and $\Omega_{2}$, discretized by FEM-FDM, respectively, in the context of nonmatching grids in the overlap region. Then, combining a convergence result due to S. H. Lui 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. Numerical results are also provided to support the theory.