Numerical analysis of a discontinuous Galerkin method for the Borrvall-Petersson topology optimization problem

Divergence-free discontinuous Galerkin (DG) finite element methods offer a suitable discretization for the pointwise divergence-free numerical solution of Borrvall and Petersson's model for the topology optimization of fluids in Stokes flow [Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77--107]. The convergence results currently found in literature only consider $H^1$-conforming discretizations for the velocity. In this work, we extend the numerical analysis of Papadopoulos and S\"uli to divergence-free DG methods with an interior penalty [I. P. A. Papadopoulos and E. S\"uli, Numerical analysis of a topology optimization problem for Stokes flow, arXiv preprint arXiv:2102.10408, (2021)]. We show that, given an isolated minimizer to the analytical problem, there exists a sequence of DG finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to the minimizer.