A numerical study of variable coefficient elliptic Cauchy problem via projection method

Abstract

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.

Keywords:

• elliptic Cauchy problem
• regularization method
• inverse problem
• balancing principle
• ill-posed problems
• elliptic partial differential equations

2010 AMS Subject Classifications :

• 47A52
• 35R30
• 65F22
• 65J22

Acknowledgements

The author wishes to express his gratitude to Professor Phoolan Prasad for initiating research in this direction. The work for this paper was supported by the research support grant of the Raja Ramanna fellowship offered to Prof. Prasad by DAE, Government of India.