A Finite Difference Domain Decomposition Method Using Local Corrections for the Solution of Poisson's Equation

Abstract

We present a domain decomposition method for computing finite difference solutions to the Poisson equation with infinite domain boundary conditions. Our method is a finite difference analogue of Anderson's Method of Local Corrections. The solution is computed in three steps. First, fine-grid solutions are computed in parallel using infinite domain boundary conditions on each subdomain. Second, information is transferred globally through a coarse-grid representation of the charge, and a global coarse-grid solution is found. Third, a fine-grid solution is computed on each subdomain using boundary conditions set with the global coarse solution, corrected locally with fine-grid information from nearby subdomains. There are three important features of our algorithm. First, our method requires only a single iteration between the local fine-grid solutions and the global coarse representation. Second, the error introduced by the domain decomposition is small relative to the solution error obtained in a single-grid calculation. Third, the computed solution is second-order accurate and only weakly dependent on the coarse-grid spacing and the number of subdomains. As a result of these features, we are able to compute accurate solutions in parallel with a much smaller ratio of communication to computation than more traditional domain decomposition methods. We present results to verify the overall accuracy, confirm the small communication costs, and demonstrate the parallel scalability of the method.