Stable numerical solution of the Cauchy problem for the Laplace equation in a bounded annular región
Ponente(s): José Julio Conde Mones, 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.