The Cauchy problem for Laplace's equation via the conjugate gradient method

A variational formulation of the Cauchy problem for the Laplace equation is studied. An efficient conjugate gradient method based on an optimal-order stopping criterion is presented together with its numerical implementation based on the boundary-element method. Several numerical examples involving smooth or non-smooth geometries and over-, equally, or under-specified Cauchy data are discussed. The numerical results show that the numerical solution is convergent with respect to increasing the number of boundary elements and stable with respect to decreasing the amount of noise included in the input Cauchy data.