Discrete methods in the study of an inverse problem for Laplace's equation

Let u be harmonic in the interior of a rectangle Ω and satisfy the third-kind boundary condition un + yu = Φ where Φ ⩾ 0, y ⩾ 0 with supports included in the bottom and in the top side of Ω, respectively. Recovering y from a knowledge of Φ and of the trace of u on the bottom Ω is a nonlinear inverse problem of interest in the field of nondestructive evaluation. A convergent Galerkin method for approximating y is proposed and tested in numerical experiments.