A new numerical method based on fictitious domain methods for shape
optimization problems governed by the Poisson equation is proposed.
The basic idea is to combine the boundary variation technique, in which
the mesh is moving during the optimization, and efficient fictitious
domain preconditioning in the solution of the (adjoint) state equations.
Neumann boundary value problems are solved using an algebraic fictitious
domain method. A mixed formulation based on boundary Lagrange
multipliers is used for Dirichlet boundary problems and the resulting
saddle-point problems are preconditioned with block diagonal fictitious
domain preconditioners. Under given assumptions on the meshes, these
preconditioners are shown to be optimal with respect to the condition
number. The numerical experiments demonstrate the efficiency of
the proposed approaches.