A moving mesh fictitious domain approach for shape optimization problems

A new numerical method based on fictitious domain methods for shape optimiz ation problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is movin g during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value pr oblems are solved using an algebraic fictitious domain method. A mixed form ulation based on boundary Lagrange multipliers is used for Dirichlet bounda ry 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 t o the condition number. The numerical experiments demonstrate the efficienc y of the proposed approaches. Mathematics Subject Classification. 49M29, 65 F10, 65K10, 65N30, 65N55.