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 topology optimization fluids in Stokes flow [Topology flow, International Journal Numerical Methods Fluids 41 (1) (2003) 77--107]. The convergence results currently found literature only consider $H^1$-conforming discretizations velocity. In this work, we extend analysis Papadopoulos Suli to DG with an interior penalty [I. P. A. E. Suli, problem arXiv preprint arXiv:2102.10408, (2021)]. We show that, given isolated minimizer analytical problem, there exists sequence solutions, satisfying necessary first-order optimality conditions, that strongly converges minimizer.