Numerical simulation of the hydrodynamic semiconductor device equations requires powerful numerical schemes. A Space-time Galerkin/Least-Squares finite element formulation, that has been successfully applied to problems of fluid dynamic, is proposed for the solution of the hydrodynamic device equations. Similarity between the equations of fluid dynamic and semiconductor devices is discussed. The robustness and accuracy of the numerical scheme are demonstrated with the example of a single electron carrier submicron silicon MESFET device.