The primary goal of this study is to develop a finite element method for the calculation of an attached, two-dimensional, laminar, compressible boundary layer about an air-foil in a subsonic free stream. An introduction to the subsequent viscous-inviscid interaction model is also given. The two-dimensional partial differential equations are reduced to integral equations that are independent of density and resemble the weak form of the two dimensional incompressible boundary layer equations. A Galerkin finite element method is applied to this Dorodnitsyn formulation and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of first order ordinary differential equations, solved by an implicit, non-iterative finite difference marching scheme in the stream-wise direction. Results presented include the coefficient of friction and displacement thickness about a circular cylinder in an incompressible freestream, the compressible boundary layer about a NACA 0012 airfoil, and a validation of the linear thermal equation.