Mean value iterations for nonlinear elliptic Cauchy problems

TL;DR

This paper develops and analyzes two iterative methods for solving nonlinear elliptic Cauchy problems, focusing on reconstructing boundary traces from partial data, with convergence proofs and numerical insights.

Contribution

It introduces two novel iterative algorithms based on Mann iteration for nonlinear elliptic Cauchy problems, including convergence analysis and preliminary numerical results.

Findings

Linear fixed point iteration converges with proven rates.

Nonlinear iteration shows promising preliminary convergence.

Numerical analysis supports the effectiveness of the methods.

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.