Convergence analysis of approximate shape gradients for shape optimization in parabolic problems

Numerical approximations of shape gradients and their applications have recently caused much interest in the shape optimization community. In this paper, existing research results on shape optimization governed by steady problems (e.g., elliptic problems in Hiptmair et al. [BIT Numer. Math. 55 (2015) 459–485]) are extended to those governed by parabolic problems. Convergence analysis is presented for numerical approximations of shape gradients associated with a parabolic problem. Both the backward Euler scheme and the backward differentiation formula are employed for time discretization, and the Galerkin finite element method is used for spatial discretization of the parabolic state and adjoint problems. The error of the distributed shape gradient is shown to have a higher convergence order in the mesh-size than that of the boundary type. A priori error estimates with respect to the time step-size are also presented. Numerical examples are provided to illustrate the theoretical results.