Weak formulation of Hadamard variation applied to the filtration problem

Quantities defined using a solution of an elliptic boundary value problem may vary when the boundary of the domain is perturbed. Such a variation with respect to domain perturbation is called Hadamard variation. We present a weak formulation of Hadamard variation and apply it to the filtration (or dam) problem. We obtain first Hadamard variations of quantities arising in the variational principle of the filtration problem. The correctness of the obtained first variations is confirmed by numerical experiments. Using the first variations, we propose a numerical iterative scheme for the filtration problem. Numerical examples show good performance of the presented scheme.