Solving Galerkin Approximations to Nonlinear Two-Point Boundary Value Problems by a Globally Convergent Homotopy Method

The Chow-Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, certain classes of zero finding and nonlinear programming problems, and two-point boundary value approximations based on shooting, finite differences, and spline collocation. The method is numerically stable and has been successfully applied to a wide range of practical engineering problems. Here the Chow-Yorke algorithm is proved globally convergent for a class of Galerkin approximations to nonlinear two-point boundary value problems. Several numerical implementations of the algorithm are briefly described, and computational results are presented for a fairly difficult magneto-hydrodynamics boundary value problem.