Quasi-optimal approximation of surface based lagrange multipliers in finite element methods

We show quasi-optimal a priori convergence results in the L ²- and H ^-1/2-norm for the approximation of surface based Lagrange multipliers such as those employed in the mortar finite element method. We improve on the estimates obtained in the standard saddle point theory, where error estimates for both the primal and dual variables are obtained simultaneously and thus only suboptimal a priori estimates for the dual variable are reached. For the lowest order case, i.e., k = 1, an additional factor of √h| ln h| and for higher order cases, i.e., k > 1, an additional factor of √h in the a priori bound for the dual variable can be recovered. The proof is based on the use of new estimates for the primal variable in strips of width O(h) near these surfaces.