DOMAIN EMBEDDING AND THE DIRICHLET PROBLEM EINAR HAUG, TORGEIR RUSTEN, PANAYOT S. VASSILEVSKI, AND RAGNAR WINTHER Abstract. In this paper we study domain embedding preconditioners for discrete linear systems approximating the Dirichlet problem associated a second order el- liptic equation. We observe that if a mixed finite element discretization is used, then such a preconditioner can be constructed in a straightforward manner from the H(div)–inner product. We also use the H(div)–inner product to construct a new preconditioner for the Lagrange multiplier system. 1. Introduction The purpose of this paper is to review some new results on the construction of domain embedding preconditioners for discrete approximations of the Dirichlet problem associated a second order elliptic equation. A more detailed description of these results are given in the papers [10] and [14]. We consider boundary value problems of the form − div(a grad p) = f in Ω, p = g on ∂Ω. (1.1) The domain Ω ⊂ R 2 is a bounded polygonal domain, and ∂Ω is the boundary. The coefficient matrix a is assumed to be bounded and uniformly positive definite on Ω. In a domain embedding approach we utilize an embedding of the original domain Ω into an extended domain Ω e to construct a proper preconditioner for the discrete system. For problems with natural boundary conditions the construction of such preconditioners is rather straightforward, cf. [3]. However, for essential, or Dirichlet, boundary conditions the domain embedding approach is less obvious. In the discussion below we will assume that the problem (1.1) is discretized either by a mixed or a standard finite element method. For the mixed system the Dirichlet boundary conditions are natural, and therefore the construction of a domain em- bedding preconditioner is rather straightforward. For the standard finite element method we will introduce Lagrange multipliers to “remove” the essential bound- ary conditions. Hence, in both cases we are facing the problem of constructing a preconditioner for an indefinite saddle point system. Several authors have discussed the use of the Lagrange multiplier system in order to construct domain embedding preconditioners for the Dirichlet problem, cf. for example [7] and [13]. However, the contribution here is to propose an alternative construction of the block of the preconditioner which corresponds to the multiplier, 1991 Mathematics Subject Classification. 65F10, 65N20, 65N30. Key words and phrases. second order elliptic problems, Dirichlet boundary conditions, precon- ditioning, domain embedding, mixed finite elements, Lagrange multipliers. This work was partially supported by the Research Council of Norway (NFR), programs 100998/420, 107547/410 and STP.29643. The work of the third author was also partially sup- ported by the Bulgarian Ministry for Education, Science and Technology under grant I–95 # 504. 1