Numerical estimation of the Robin coefficient in a stationary diffusion equation

Abstract

A finite-element method is proposed for the nonlinear inverse problem of estimating the Robin coefficient in a stationary diffusion equation from boundary measurements of the solution and the heat flux. The inverse problem is formulated as an output least squares optimization problem with an appropriate regularization, then the finite-element method is employed to discretize the nonlinear optimization system. Mathematical properties of both the continuous and the discrete optimization problems are investigated. The conjugate gradient method is employed to solve the optimization problem, and an efficient preconditioner via the Sobolev inner product is also suggested. Numerical results for several two-dimensional problems are presented to illustrate the efficiency of the proposed algorithm.