Energy Space Approaches to the Cauchy Problem for Poisson’s Equation

Variational methods coupled with Tikhonov’s regularization for solving the Cauchy problem for Poisson’s equation are suggested and studied. The novel idea is to use the Tikhonov regularization term in H1/2 norm rather than in L2 norm. The penalty term is evaluated by some appropriate boundary integral operators. The optimality condition in the form of boundary integral equations is derived and then discretized by the Galerkin boundary element method. The error estimates for the discretized problems are proved for noisy data. Some numerical examples and comparisons with the L2 setting are presented for showing the efficiency of our approaches.