We present an asymptotic numerical algorithm for the computation of elastic shells with large rotations. The theoretical formulation involves a three-field Hu—Washizu functional, which allows us to put the problem into a quadratic framework. The spatial discretization is based on geometrically exact element, recently presented by Büchter et al. [Büchter et al., Three dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, Int. J. Numer. Methods Engrg. 37 (1994) 2551-1568]. Several classical benchmarks are discussed to define the best strategy and to assess the validity and the efficiency of the present method, as compared to more classical iterative algorithms.