Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization
This work provides theoretical convergence guarantees for shape gradient approximations, which is important for practitioners in shape optimization who rely on gradient-based methods.
The paper analyzes the convergence of Galerkin finite element approximations for two types of shape gradients (boundary and volume) in eigenvalue optimization. It shows that the volume integral formula converges faster and offers better accuracy for Dirichlet boundary conditions, while both formulations converge similarly for Neumann conditions.
Numerical computation of shape gradients from Eulerian derivatives is essential to wildly used gradient type methods in shape optimization. Boundary type Eulerian derivatives are popularly used in literature. The volume type Eulerian derivatives hold more generally, but are rarely noticed and used numerically. We investigate thoroughly the accuracy of Galerkin finite element approximations of the two type shape gradients for optimization of elliptic eigenvalues. Under certain regularity assumptions on domains, we show \emph{a priori} error estimates for the two approximate shape gradients. The convergence analysis shows that the volume integral formula converges faster and generally offers better accuracy. Numerical experiments verify theoretical results for the Dirichlet case. For the Neumann case, however, the boundary formulation surprisingly converges as fast as the volume one. Numerical results are presented.