A Mann iterative regularization method for elliptic Cauchy problems

Abstract

We investigate the Cauchy problem for linear 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 manifold $\Gamma \subset \partial \Omega$ and our goal is to reconstruct the trace of the $H^1(\Omega )$ solution of an elliptic equation at $\partial \Omega /\Gamma$. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

25 pages 5 figures

References in corpus (2)

• On iterative methods for solving ill-posed problems modeled by PDE's
• An iterative method for solving elliptic Cauchy problems

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