Stable numerical solution of the Cauchy problem for the Laplace equation in a bounded annular región

Autor: José Julio Conde Mones
Coautor(es): Dr. Lorenzo Héctor Juárez Valencia *, Dr. José Jacobo Oliveros Oliveros **, Dra. María Monserrat Morín Castillo**. * Departamento de Matemáticas, División de Ciencias Básicas e Ingeniería, UAM-Izt., ** Facultad de Ciencias Físico Matemáticas, BUAP, *** Facultad de Ciencias de la Electrónica, BUAP. * hect@xanum.uam.mx, ** oliveros@fcfm.buap.mx, *** mmorin@ece.buap.mx,
This work presents a numerical study of 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. This functional is equivalent a the Tikhonov functional with parametre of regularitation 1/k, where k is the parametre of penelization. This functional is minimized by a conjugate gradient method in combination with a finite element discretization and where the regularization parameter is choosen using Tikhonov regularization method. Numerical solutions in simple and complex domains show that this methodology produces stable and accurate solutions.