Identification of Robin Coefficient in Elliptic Problem by a Coupled Complex Boundary Method

In this paper, we study a Robin inverse problem governed by an elliptic partial differential equations. The unknown Robin coefficient is to be determined by additional boundary measurements, including both Dirichlet and Neumann boundary conditions. To solve this problem, we apply the coupled complex boundary method (CCBM) originally proposed in []. With CCBM, we introduce a complex elliptic partial differential equation with a boundary condition coupling the Dirichlet and Neumann boundary data, and we optimize the imaginary part of the solution in the domain to determine the Robin coefficient. Afterward, we use Tikhonov regularization to obtain a stable approximate Robin coefficient where we compute the gradient and the second derivative of our cost function. The latter expression allows us to show the existence of the optimum. Numerically, we use the gradient method and the finite element method for discretization. Finally, we give some numerical examples which show the feasibility and the stability of the proposed method.