A domain embedding preconditioner for the Lagrange multiplier system

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