The paper discusses membrane instability phenomena by a Fourier-related double scale approach that was introduced recently. This leads to a reduced-order model that is able to capture the main features of the wrinkles with few degrees of freedom. The paper focuses on the corresponding finite element procedure, its implementation and evaluation and applications to various cases of loading and boundary condition. The finite element model has first been implemented in a home-made code, the nonlinear system being solved by the Asymptotic Numerical Method (ANM), which has advantages of efficiency and reliability for stability analyses. It has also been implemented as a user element in a commercial software to evaluate the effectiveness of the reduction technique. Various loading cases were considered and the numerical tests establish that the reduced model can predict the wrinkling patterns, even when there are few wrinkles. The numerical results highlight the strong influence of a dimensionless parameter for wrinkling initiation.