Mean value iterations for nonlinear elliptic Cauchy problems

Abstract

We investigate the Cauchy problem for a class of nonlinear elliptic operators with $C^{\infty }$-coefficients at a regular set $\Omega \subset R^n$. The Cauchy data are given at a manifold $\Gamma \subset \partial \Omega$ and our goal is to reconstruct the trace of the $H^1(\Omega )$ solution of a nonlinear elliptic equation at $\partial \Omega /\Gamma$. We propose two iterative methods based on the segmenting Mann iteration applied to fixed point equations, which are closely related to the original problem. The first approach consists in obtaining a corresponding linear Cauchy problem and analyzing a linear fixed point equation; a convergence proof is given and convergence rates are obtained. On the second approach a nonlinear fixed point equation is considered and a fully nonlinear iterative method is investigated; some preliminary convergence results are proven and a numerical analysis is provided.

23 pages, 6 figures

References in corpus (3)

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

Cited by in corpus (1)

• On level set type methods for elliptic Cauchy problems