Fictitious domains methods with distributed Lagrange multipliers Part I: Application to the solution of elliptic state problems

Fictitious domain approaches are an attractive way to overcome the usual dr awbacks of the so-called boundary variation technique for solving optimal s hape problems. In the first part of this paper, we introduce fictitious dom ain methods with distributed Lagrange multipliers for elliptic state proble ms. Continuous formulations as well as their finite element discretizations are discussed. The second part of the paper will be devoted to the applica tion of these methods to shape optimization.