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arXiv · 2011.08629

Mean value iterations for nonlinear elliptic Cauchy problems

P. Kügler·A. Leitao
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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.

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P. Kügler, A. Leitao. 2020-11-17. Mean value iterations for nonlinear elliptic Cauchy problems. https://doi.org/10.1007/s00211-003-0477-6

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