A new numerical approach to the Cauchy problem for the Laplace equation

In this paper, we present a variational method coupled with Tikhonov regularization for a Cauchy problem for the Laplace equation, where the Cauchy data are considered in the energy spaces. The novel idea is to use the Tikhonov regularization term in H1/2-norm, rather than in L2-norm. As an equivalent norm in H1/2, we use the energy norm induced by the so-called Steklov-Poincaré operator, which realizes the Dirichlet-to-Neumann map. The optimality condition is derived in terms of the Steklov-Poincaré operators and then discretized by the standard finite element method. The unknown Dirichlet datum is approximated by a piecewise linear and continuous function. We derive the error estimates for noisy data based on the finite element approximations of the Steklov-Poincaré operators. Some numerical examples are presented to show the efficiency of our approach.