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.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1799–1822, 2017

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This paper presents a stable method for the identification of sources located on the separation interface of two homogeneous media (where one of them is contained by the other one), from measurement yielded by those sources on the exterior boundary of the media. This is an ill-posed problem because numerical instability is presented, i.e., minimal errors in the measurement can result in significant changes in the solution. To obtain the proposed stable method the identification problem is categorized into three subproblems, two of which present numerical instability and regularization methods must be applied to obtain their solution in a stable form. To manage the numerical instability due to the ill-posedness of these subproblems, the Tikhonov regularization and sequential smoothing methods are used. We illustrate this methodology in a circular and irregular region to demonstrate the feasibility of the proposed method, which yields convergent and stable solutions for input data with and without noise.