Approximating Volumetric Shape Gradients for Shape Optimization with Curved Boundaries Constrained by Parabolic PDEs

.We quantify the accuracy of the approximate shape gradient for a shape optimization problem constrained by parabolic PDEs. The focus is on the volume form of the shape gradient, which is discretized using the finite element method and the implicit Euler scheme. Our estimate goes beyond previous work done in the elliptic setting and considers the error introduced by polygonal approximation of curved domains. Numerical experiments support the theoretical findings, and the code is made publicly available.