Numerical Analysis of a Discontinuous Galerkin Method for the Borrvall--Petersson Topology Optimization Problem
Ioannis P. A. Papadopoulos · SIAM Journal on Numerical Analysis · 2022
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 [T. Borrvall and J. Petersson, Internat. J. Numer. Methods Fluids, 41 (2003), pp. 77--107]. The convergence results currently found in the literature only consider $H^1$-conforming discretizations for the velocity. In this work, we extend the numerical analysis of Papadopoulos and Süli to divergence-free DG methods with an interior penalty [I. P. A. Papadopoulos and E. Süli, J. Comput. Appl. Math., 412 (2022), 114295]. We show that, given an isolated minimizer of the infinite-dimensional problem, there exists a sequence of DG finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to the minimizer.