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Finite element approximations for the Dirichlet problem associ- ated to a second{order elliptic dierential equation are studied. The purpose of this paper is to discuss domain embedding preconditioners for discrete sys- tems. The essential boundary condition on the interior interface is removed by introducing Lagrange multipliers. The associated discrete system, with a sad- dle point structure, is preconditioned by a block diagonal preconditioner. The main contribution of this paper is to propose a new operator, constructed from the (div){inner product, for the block of the preconditioner corresponding to the multipliers.

A domain embedding preconditioner for the Lagrange multiplier system