Nonconforming mixed finite element methods for linear elasticity

We consider mixed finite element methods for linear elasticity based on the Hellinger-Reissner variational formulation (stress-displacement formulation). We have developed two mixed finite element methods using rectangular elements. First, we construct a nonconforming mixed finite element method for 2 dimensions. A modification has been found for boundary elements with a (possiblely) curved edge so that a domain with curved boundary can be treated. We obtain the convergence rates of [special characters omitted](h) and [special characters omitted](h2) for the stress and displacement, respectively. This element extends to 3 dimensions with the same convergence rate for both stress and displacement. We confirm our theoretical result numerically for the 2-dimensional version of the method. Then, we construct a simpler mixed finite element method. The convergence analysis involves an estimate of the consistency error term and makes use of the properties of the Adini-Clough-Melosh rectangle. We establish error estimates for the pure traction boundary problem and pure displacement boundary problem separately. We prove an optimal (suboptimal) rate of [special characters omitted](h2) ([special characters omitted]([special characters omitted])) for the stress in the pure traction (displacement) boundary problem and an optimal rate of [special characters omitted](h) for the displacement in L 2 norm in both problems. However, numerical experiments have yielded optimal-order convergence rates of [special characters omitted](h2) for the stress in both problems and [special characters omitted](h) for the displacement, as expected, and have shown a superconvergence rate of [special characters omitted](h2) for the displacement at the midpoint of each element in both problems. Other numerical experiments have shown the same convergence rates for large values of the Lamé constant λ