Optimal error estimates of the discrete shape gradients for shape optimizations governed by the Stokes-Brinkman equations

This work aims at investigating the convergence of shape gradients for shape optimizations governed by the Stokes-Brinkman equations. Two types of optimal control problems are considered, the shape inverse problem and the dissipated energy minimization problem, where the distinction between these two problems lies in the difference of the objective functionals. Lagrangian functional is introduced to obtain the adjoint equations and Eulerian derivatives at a fixed domain Ω in the direction V in volume and boundary integral forms are derived by the Piola material derivative approach and function space parametrization technique. Mixed finite element method is applied to discretize both the state and adjoint equations, as well as the corresponding Eulerian derivatives. The optimal error estimates for both forms of the shape derivatives are obtained. We infer that the volume-based shape derivative has a convergence rate of order h², while the boundary-based one has a convergence rate of order h|log⁡h| under the MINI element. Numerical experiments are reported to demonstrate the theoretical analysis and indicate the better accuracy of volume-based expressions.