AbstractThe numerical solution of the compressible Euler and Navier‐Stokes equations in primitive variables form requires the use of artificial viscosity or upwinding. Methods that are first‐order‐accurate are too dissipative and reduce the effective Reynolds number substantially unless a very fine grid is used. A first‐order finite element method for the solution of the Euler and Navier‐Stokes equations can be constructed by adding Laplacians of the primitive variables to the governing equations. Second‐order schemes may require a fourth‐order dissipation and higher‐order elements. A finite element approach is proposed in which the fourth‐order dissipation is recast as the difference of two Laplacian operators, allowing the use of bilinear elements. The Laplacians of the primitive variables of the first‐order scheme are thus balanced by additional terms obtained from the governing equations themselves, tensor identities or other forms of nodal averaging. To demonstrate formally the accuracy of this scheme, an exact solution is introduced which satisfies the continuity equation identically and the momentum equations through forcing functions. The solutions of several transonic and supersonic inviscid and laminar viscous test cases are also presented and compared to other avail