Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic Boundary Value Problem −Δu = f(u)

We provide a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear elliptic boundary value problem of the form −Δu = f(u), on two overlapping sub domains with non matching grids. We consider a domain which is the union of two overlapping sub domains where each sub domain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a Lipschitz assumption on the nonlinearity, we establish, on each sub domain, an optimal L∞ error estimate between the discrete Schwarz sequence and the exact solution of the boundary value problem. Keywords—Error estimates, Finite elements, Nonlinear PDEs, Schwarz method.