L^∞-Error Estimate of a Finite Element Schwarz Alternating Method for Non-linear Elliptic PDES

In this paper, we provide a maximum norm analysis of a standard finite element overlapping Schwarz method on non-matching grids for non-linear elliptic PDEs. We consider a domain which is the union of tow overlapping subdomains where each subdomain has its own independently generated triangulation. The grid points on the subdomain boundaries need not match the grid points from the other subdomain. Under a discrete maximum principle, we show that the discretization on each subdomain converges quasi-optimally in the L^∞ norm.