An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain (Communications in Computational Physics, 2011)
Detecting corrosion by electrical field can be modeled by a Cauchy problemof Laplace equation in annulus domain under the assumption that the thickness of thepipe is relatively small compared with the radius of the pipe. The interior surface of thepipe is inaccessible and the nondestructive detection is solely based on measurementsfrom the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posedproblem whose solution does not depend continuously on the boundary data. In thiswork, we assume that the measurements are available on the whole outer boundary onan annulus domain. By imposing reasonable assumptions, the theoretical goal here isto derive the stabilities of the Cauchy solutions and an energy regularization method.Relationship between the proposed energy regularization method and the Tikhonovregularization with Morozov principle is also given. A novel numerical algorithm isproposed and numerical examples are given.