Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition

Slodička, Marian

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 691-711.

We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain $\Omega \subset {ℝ}^{dim}$ with a nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant, accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part ${\Gamma }_n$. The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization algorithm are derived in $L_2\left(\Omega \right),H^1\left(\Omega \right)$ and $L_{\infty }\left(\Omega \right)$ spaces.

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