Maximum Norm Analysis of Overlapping Nonmatching Grid Discretizations of Elliptic Equations

In this paper, we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations. We consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from adjacent subdomains. To obtain a global finite difference discretization of the elliptic problem, we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain, by interpolating the discrete solution on adjacent subdomains. If the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large, we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations. Our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains. We describe also how the resulting linear system can be solved iteratively by a parallel Schwarz alternating method or a Schwarz preconditioned Krylov subspace iterative method. Several numerical results are included to support the theory.