Mean value iterations for nonlinear elliptic Cauchy problems

11/17/2020

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by   P. Kügler, et al.

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We investigate the Cauchy problem for a class of nonlinear elliptic operators with C^∞-coefficients at a regular set Ω⊂ R^n. The Cauchy data are given at a manifold Γ⊂∂Ω and our goal is to reconstruct the trace of the H^1(Ω) solution of a nonlinear elliptic equation at ∂Ω / Γ. 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.