The Chow-Yorke algorithm, i.e., a homotopy method that has been proved globally convergent j problems, certain classes of zero finding and nonlinear programming problems, and two-point, boundary based on shooting, finite differences, and spline collocation. The method is numerically stable and has been applied to a wide range of practical engineering problems. Here the Chow-Yorke algorithm is proved globally c Galerkin approximations to nonlinear two-point boundary value problems. Several numerical implement are briefly described, and computational results are presented for a fairly difficult, magnet-hydrodynamic problem. Key words. homotopy method, Chow-Yorke algorithm, globally convergent,, two-point boundary value method, finite element method, nonlinear equations