Summary: The discretization by finite element methods of a new variational formulation of crack problems is considered. The new formulation, called the smooth domain method, is derived for crack problems in the case of an elastic membrane. Inequality type boundary conditions are prescribed at the crack faces. The resulting model takes the form of a unilateral contact problem on the crack. We study and implement various mixed finite element methods for the numerical approximation of the model. A priori error estimates are derived, and results of computations are provided. The convergence rates obtained from the numerical simulations are in agreement with the theoretical estimates.