Locking-free finite element methods for shells

We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer $k$. The methods are based on a nonstandard mixed formulation, and the $k$th method employs triangular Lagrange finite elements of degree $k+2$ augmented by bubble functions of degree $k+3$ for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree $k$ for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone, as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the $H^{1}$ norm of the displacement and rotation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order $k+1$ uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.