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arXiv · 2505.13776

Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow

Bangti Jin·Jing Li·Yifeng Xu·Shengfeng Zhu
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Abstract

In this work, we develop an adaptive nonconforming finite element algorithm for the numerical approximation of phase-field parameterized topology optimization governed by the Stokes system. We employ the conforming linear finite element space to approximate the phase field, and the nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants to approximate the velocity field and the pressure field, respectively. We establish the convergence of the adaptive method, i.e., the sequence of minimizers contains a subsequence that converges to a solution of the first-order optimality system, and the associated subsequence of discrete pressure fields also converges. The analysis relies crucially on a new discrete compactness result of nonconforming linear finite elements over a sequence of adaptively generated meshes. We present numerical results for several examples to illustrate the performance of the algorithm, including a comparison with the uniform refinement strategy.

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Bangti Jin, Jing Li, Yifeng Xu, Shengfeng Zhu. 2025-05-19. Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow. https://doi.org/10.1090/mcom%2F4192

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