  The space mapping technique is used to efficiently solve complex optimization
problems. It combines the accuracy of fine model simulations with the speed of
coarse model optimizations to approximate the solution of the fine model
optimization problem. In this paper, we propose novel space mapping methods for
solving shape optimization problems constrained by partial differential
equations (PDEs). We present the methods in a Riemannian setting based on
Steklov-Poincar\'e-type metrics and discuss their numerical discretization and
implementation. We investigate the numerical performance of the space mapping
methods on several model problems. Our numerical results highlight the methods'
great efficiency for solving complex shape optimization problems.
