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

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 Süli to divergence-free DG methods with an interior penalty [I. P. A. Papadopoulos and E. Süli, 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.

Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization

“…Building on previous work [48], it was shown by Papadopoulos [46] that, for every isolated minimizer (u, ρ, p) of (BP), there exists a sequence of discretized solutions ), and p − p h L 2 (Ω) → 0. By extrapolating known results of a BDM discretization for the Stokes and Stokes-Brinkman equations [14,15,41] we expect a first-order BDM 1 discretization to converge at a rate of O(h) in the broken H 1 -norm for the velocity, the L 2 -norm for the pressure, and the L 2 -norm for the material distribution.…”