Parallel Newton-Krylov-Schwarz Algorithms for the Transonic Full Potential Equation

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear nite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact nite-di erence Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and economical for this class of mixed elliptic-hyperbolic nonlinear partial di erential equations, with proper speci cation of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of ll-in in the incomplete factorization, and report their e ect on numerical convergence rate, overall execution time, and parallel e ciency on a distributed-memory parallel computer.