Stable Numerical Solution of the Cauchy Problem for the Laplace Equation in Irregular Annular Regions

This article is mainly concerned with the numerical study of the Cauchy problem for the Laplace equation in a bounded annular region. To solve this ill-posed problem, we follow a variational approach based on its reformulation as a boundary control problem, for which the cost function incorporates a penalized term with the input data. The cost function is minimized by a conjugate gradient method in combination with a finite element discretization. In the case where the input data is noisy, some preliminary error estimates, show that the penalization parameter may be chosen like the inverse of the level of noise. Numerical solutions in simple and complex domains show that this methodology produces stable and accurate solutions. (c) 2017 Wiley Periodicals, Inc.