Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow
Provides a theoretical convergence guarantee for adaptive methods in a specific PDE-constrained optimization problem, which is incremental for the field of numerical analysis.
The paper develops an adaptive nonconforming finite element method for phase-field topology optimization in Stokes flow and proves its convergence, showing that the sequence of minimizers contains a convergent subsequence to a solution of the optimality system.
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.