As a specific type of shape gradient descent algorithm, shape gradient flow is widely used for shape optimization problems constrained by partial differential equations. In this approach, the constraint partial differential equations could be solved by finite element methods on a domain with a solution-driven evolving boundary. Rigorous analysis for the stability and convergence of such finite element approximations is still missing from the literature due to the complex nonlinear dependence of the boundary evolution on the solution. In this article, rigorous analysis of numerical approximations to the evolution of the boundary in a prototypical shape gradient flow is addressed. First-order convergence in time and $k$th order convergence in space for finite elements of degree $k\geqslant 2$ are proved for a linearly semi-implicit evolving finite element algorithm up to a given time. The theoretical analysis is consistent with the numerical experiments, which also illustrate the effectiveness of the proposed method in simulating two- and three-dimensional boundary evolution under shape gradient flow. The extension of the formulation, algorithm and analysis to more general shape density functions and constraint partial differential equations is also discussed.

中文翻译：

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形状梯度流下边界演化的收敛演化有限元近似

作为形状梯度下降算法的一种具体类型，形状梯度流广泛应用于偏微分方程约束的形状优化问题。在这种方法中，约束偏微分方程可以通过有限元方法在具有解驱动的演化边界的域上求解。由于边界演化对解的复杂非线性依赖性，文献中仍然缺少对此类有限元近似的稳定性和收敛性的严格分析。在本文中，对原型形状梯度流中边界演化的数值近似进行了严格分析。对于给定时间的线性半隐式演化有限元算法，证明了 $k\geqslant 2$ 度有限元的时间一阶收敛性和空间 $k$ 阶收敛性。理论分析与数值实验结果一致，也说明了该方法在模拟形状梯度流下二维和三维边界演化方面的有效性。还讨论了将公式、算法和分析扩展到更一般的形状密度函数和约束偏微分方程。