An iterative method for solving elliptic Cauchy problems

TL;DR

This paper presents a generalized iterative method for solving elliptic Cauchy problems, extending previous algorithms to more general operators and providing convergence proof and promising numerical results.

Contribution

It generalizes Maz'ya et al.'s algorithm to linear elliptic operators with smooth coefficients and offers a new convergence proof.

Findings

Convergence of the iterative method is established for smooth elliptic operators.

Numerical experiments show promising results for a nonlinear problem.

The method effectively reconstructs boundary traces from Cauchy data.

Abstract

We investigate the Cauchy problem for elliptic operators with $C^\infty$-coefficients at a regular set $\Omega \subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the subset $\Gamma \subset \partial\Omega$ and our objective is to reconstruct the trace of the $H^1(\Omega)$ solution of an elliptic equation at $\partial \Omega / \Gamma$. The method described here is a generalization of the algorithm developed by Maz'ya et al. [Ma] for the Laplace operator, who proposed a method based on solving successive well-posed mixed boundary value problems (BVP) using the given Cauchy data as part of the boundary data. We give an alternative convergence proof for the algorithm in the case we have a linear elliptic operator with $C^\infty$-coefficients. We also present some numerical experiments for a special non linear problem and the obtained results are very promisive.