A domain embedding preconditioner for the Lagrange multiplier system

Finite element approximations for the Dirichlet problem associated to a sec ond-order elliptic differential equation are studied. The purpose of this p aper is to discuss domain embedding preconditioners for discrete systems. T he essential boundary condition on the interior interface is removed by int roducing Lagrange multipliers. The associated discrete system, with a saddl e point structure, is preconditioned by a block diagonal preconditioner. Th e main contribution of this paper is to propose a new operator, constructed from the N(div)-inner product, for the block of the preconditioner corresp onding to the multipliers.