Mixing Finite Elements and Finite Differences in Nonlinear Schwarz Iterations for Nonlinear Elliptic Pdes

In this paper, we are concerned with a nonmatching grid mixed finite-elements–finite-differences approximation (FEM-FD) method of overlapping nonlinear multiplicative Schwarz iterations for nonlinear elliptic PDEs. By means of a geometric convergence result in L^∞ for the nonlinear Schwarz iterations and a Lipschitz property with respect to the data of both the FEM and FD solutions of the corresponding linear PDE problems, we derive an L^∞ error estimate on each subdomain between the discrete nth Schwarz iterate and the true solution of the nonlinear PDE.