On accuracy of approximate boundary and distributed H¹ shape gradient flows for eigenvalue optimization

The boundary and distributed shape gradients of elliptic eigenvalues in shape optimization are approximated by the finite element method. We show a priori error estimates for the two approximate shape gradients in H¹ shape gradient flows. The convergence analysis shows that the volume integral formula converges faster and offers higher accuracy when the finite element method is used for discretization. Numerical results verify the theory for the Dirichlet case. Shape optimization examples solved by algorithms illustrate the more effectiveness of distributed shape gradients for the Dirichlet case. For optimizing a Neumann eigenvalue, the boundary and volume H¹ flows have the same efficiency. Moreover, we observe that the distributed H¹ shape gradient flow is more efficient than the boundary L² shape gradient flow in literature.