  We present a new formulation based on the classical Dirichlet-Neumann
formulation for interface coupling problems in linearized elasticity. By using
Taylor series expansions, we derive a new set of interface conditions that
allow our formulation to pass the linear consistency test. In addition, we
propose an iterative method to determine the solution of our formulation. We
demonstrate in our numerical results that we may achieve the desired piecewise
linear finite element error bounds for both nonoverlapping domain decomposition
problems as well as for interface coupling problems where the Lam\'e parameters
of the structures differ.
