Convergence analysis of mixed finite element approximations to shape gradients in the Stokes equation

Eulerian derivatives of shape functionals in shape optimization can be written in two formulations of boundary and volume integrals. The former is widely used in shape gradient descent algorithms. The latter holds more generally, although rarely being used numerically in literature. For shape functionals governed by the Stokes equation, we consider the mixed finite element approximations to the two types of shape gradients from corresponding Eulerian derivatives. The standard MINI and Taylor-Hood elements are employed to discretize the state equation, its adjoint and the resulting shape gradients. We present thorough convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the volume integral formula has superconvergence property. Numerical results are presented to verify the theory and show that the volume formulation is more accurate.