Computational Optimization and Applications, 26, 231–251, 2003 c  2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Shape Optimization and Fictitious Domain Approach for Solving Free Boundary Problems of Bernoulli Type J. HASLINGER haslin@apollo.karlov.mff.cuni.cz Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Praha 2, Czech Republic T. KOZUBEK tomas.kozubek@vsb.cz Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic K. KUNISCH karl.kunisch@uni-graz.at G. PEICHL gunther.peichl@uni-graz.at Department for Mathematics, University Graz, Heinrichstrasse 36, A-8010 Graz, Austria Abstract. This contribution deals with an efficient method for the numerical realization of the exterior and interior Bernoulli free boundary problems. It is based on a shape optimization approach. The state problems are solved by a fictitious domain solver using boundary Lagrange multipliers. Keywords: fictitious domain method, shape optimization, Bernoulli’s free boundary problems Introduction Fictitious domain methods (FDM) represent an efficient tool for realizing complicated problems in practice. The idea of any FDM is simple: the original problem which is defined in a domain  with a complicated geometry is transformed into a new problem defined and solved in a domain ˆ  with simple geometry (e.g. an n-cube), containing . In ˆ  one can use fairly structured meshes which enable us to utilize fast solvers and special preconditioning techniques. There are different ways to construct the new problem in ˆ  from the original one defined in . In this paper we use the variant of the FDM which is based on Lagrange multipliers defined on the boundary ∂. This approach combined with shape optimization will be used for the numerical realization of free boundary problems of Bernoulli type. Free boundary problems of Bernoulli type arise for example in ideal fluid dynamics, optimal insulation and electro-chemistry (see [1, 7, 9]). In terms of electrostatics one wants to design a condenser with one component of the boundary given ` a-priori and another one to be determined from the condition that the electrostatic field is constant along it. A standard way of solving such problems is the use of the first or the second order trial free boundary methods based on the explicit or implicit Neumann scheme (see [8, 24]). Another way of solving free boundary problems is to transform them into shape optimization problems. Every optimal domain of this problem is a solution to the original free boundary problem