We study the Hermite collocation solution of the one-dimensional-steady-state convection-diffusion equation with Dirichlet boundary conditions. The diffusion coefficient is constant while the convection coefficient is piecewise contant. A uniform mesh is imposed on each portion of the domain on which the convection coefficient is constant, but each portion may have a different uniform mesh. Formulas are derived for the exact solution of the matrix equation that arises from the collocation discretization. These formulas possess free "upstreaming/downstreaming" parameters, the values of which may be chosen to yield numerical solutions of great accuracy.