The goal of this project is to apply the inexact Newton framework to the solution of large nonlinear problems arising from various disciplines in the physical sciences. In each case, the basic iterative algorithm will require the solution of a substantial subproblem at each iteration. Guided by the inexact Newton theory, each subproblem will be solved only to the accuracy needed to ensure the rapid convergence of the iterative method. The following problems will be considered: 1) The large constrained optimization problem arising from the finite element discretization of Skyrme's model for meson field. 2) The implementation of a domain decomposition algorithm for elliptic boundary value problems. 3) An effective method for the solution of the time dependent Navier-Stokes equation. 4) Newton's method for solving continuous nonlinear boundary value problems. During the preliminary research activities the P.I. will try to; a) determine how the accuracy requirements for solving the two linear systems in Tapia's Lagrange multiplier formula, b) complete the implementation of the augmented Lagrangian algorithm for the domain decomposition problem in preparation for the inexact iteration modification, c) will apply a preconditioned conjugate gradient algorithm to a symmetric positive definite linear system, and d) will try a simple finite element method for inexact Newton's method.
