AbstractA finite element formulation and algorithm for calculation of shock‐free compressible flows and design of shockless airfoils is developed. The formulation is conceptually similar to current finite difference methods in that it combines the use of a fictitious gas for a regularized compressible potential flow calculation, together with the method of characteristics in the supersonic pocket. The redesigned shock‐free airfoil surface is determined from a subsidiary streamline calculation. In the present analysis, a variational finite element method is used and optimal control techniques are employed to compute the solution to the discrete problem. Numerical results and comparison studies with finite difference results for a representative case are gi