T. Borrvall and J. Petersson [Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77--107] developed the first model for topology optimization of fluids in Stokes flow. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this work, we prove novel regularity results and extend their numerical analysis. In particular, given an isolated local minimizer to the analytical problem, we show that there exists a sequence of finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to it. We also provide the first numerical investigation into convergence rates.

翻译：T. Borrvall和J. Petersson[斯托克斯流流流液体的理学优化,《国际流体数字方法杂志》41(1)(2003年)77-107] 开发了斯托克斯流流液体地形优化的第一个模型,这些模型证明在无限维度环境中存在最小化器,并表明一个适当选择的有限元素方法在弱(-*)意义上会汇合到一个不确定的解决方案。在这项工作中,我们证明了新颖的规律性结果,并扩展了其数字分析。特别是,鉴于局部孤立的最小化器对分析问题的影响,我们显示存在一系列有限的元素解决方案,满足了必要的一级最佳性条件,因此非常一致。我们还对趋同率进行了首次数字调查。