This paper concerns the accuracy of Galerkin finite element approximations to two types of shape gradients for eigenvalue optimization. Under certain regularity assumptions on domains, a priori error estimates are obtained for the two approximate shape gradients. Our convergence analysis shows that the volume integral formula converges faster and offers higher accuracy than the boundary integral formula. Numerical experiments validate the theoretical results for the problem with a pure Dirichlet boundary condition. For the problem with a pure Neumann boundary condition, the boundary formulation numerically converges as fast as the distributed type.

中文翻译：

---

特征值优化中形状梯度的伽辽金有限元近似的收敛分析

本文关注用于特征值优化的两种形状梯度的伽辽金有限元近似的准确性。在域的某些规律性假设下，获得了两个近似形状梯度的先验误差估计。我们的收敛分析表明，体积积分公式比边界积分公式收敛速度更快，精度更高。数值实验验证了纯狄利克雷边界条件问题的理论结果。对于具有纯 Neumann 边界条件的问题，边界公式在数值上收敛的速度与分布式类型一样快。