A new upwind finite element formulation is presented for the first-order hyperbolic systems of conservation laws with particular emphasis on the two dimensional compressible Euler equations. The basis of formulation is the exponential weighting function derived from variational principles by Shen. A new perturbation parameter of the weighting function is defined in each direction for the quadrilateral element using Jacobian matrix of flux vectors and the typical element length. And a discontinuity capturing term is included in the optimal weighting function enabling non-oscillatory yet crisp shock profiles to be obtained. The accuracy of present method for the steady compressible flow is demonstrated on several numerical examples. Present results exhibit almost equal effects in discontinuity capturing for supersonic flows compared with SU/PG's and are in good agreement with the exact solutions. Comprehensive computational results for the internal flow over a bicircular are are presented and compared with other results in order to show that this formulation works as well in describing the subsonic flow.