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

TL;DR

This paper provides a rigorous numerical analysis of divergence-free discontinuous Galerkin methods for the Borrvall-Petersson topology optimization problem, demonstrating convergence to the true minimizer under certain conditions.

Contribution

It extends existing convergence results to divergence-free DG methods with interior penalty, a novel discretization approach for this problem.

Findings

Proves strong convergence of DG solutions to the minimizer.

Establishes that solutions satisfying optimality conditions approximate the true minimizer.

Extends theoretical analysis to divergence-free DG methods with interior penalty.

Abstract

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 Suli to divergence-free DG methods with an interior penalty [I. P. A. Papadopoulos and E. Suli, Numerical analysis of a topology optimization problem for Stokes flow, arXiv preprint arXiv:2102.10408, (2021)]. 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.