We describe a novel method for modeling the global, steady solar wind using observed photospheric magnetic fields as a driving boundary condition. The Field Line Universal relaXer (FLUX) numerical code models the solar corona as a collection of magnetic domains, represented by a quasi-Lagrangian grid of discrete field lines (fluxons). Each fluxon represents a quantized bundle of magnetic flux and responds to computed magnetic tension and pressure forces from neighboring fluxons. The model relaxes a collection of fluxons to solve the nonlinear force-free field with a prescribed boundary and topology. Synoptic magnetogram data are used to drive initial fluxon placement and topology, providing the output of an observationally-driven relaxed coronal magnetic field. The FLUX model provides an intermediate approach between rapid heuristic methods and intensive 3D magnetohydrodynamic models, providing the best of both worlds. The FLUX model has the distinct advantages of being computationally efficient (scaling with the magnetic complexity of the two-dimensional photospheric boundary) and preserving connectivity to allow for tracking the history of a bundle of magnetic flux. Open fluxons extending from the photospheric boundary are used to compute a set of modified one-dimensional isothermal Parker solar wind solutions, with transonic solutions interpolated to an outer spherical boundary uniform grid at 21.5 solar radii for comparison with and distribution to other heliospheric models. We describe the method, the simulation code, and solar wind model results.