This report outlines the formulation and implementation of the Petrov-Galerkin finite element model under development at The Universe of Arizona into the adaptive computational mesh algorithms of the International Center for Numerical Methods in Engineering, Barcelona, Spain. This work included the formulation of linear triangular elements, their use on the unstructured grids generated by the mesh generator of the ICNME, and the subsequent numerical error estimation of the results and remeshing. The implementation yielded good results for the Petrov-Galerkin weighting of the convective terms. Additional algorithmic modifications were made in order to investigate the effects of alternate weighting schemes, higher order spatial and temporal integration schemes, more consistent treatment of the mass matrices, and a formulation based on the use of the internal energy. In general, oscillatory results were obtained for schemes in which all terms were weighted with the Petrov-Galerkin functions, but additional work in this area and an investigation of the use of the other modifications are needed. The computer codes generated or modified for use in this study are described herein. A copy of the flow solvers themselves is also included.