Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization

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.