Numerical solution of problems with non-linear boundary conditions (Q1861343)

Let \(\Omega \subset R^2\) be a bounded domain with Lipschitz continuous boundary \(\partial \Omega\). The authors consider the boundary value problem: \(-\Delta u = f\) in \(\Omega\), \(\partial u/ \partial n +\kappa |u|^\alpha u = \varphi\) on \(\partial \Omega\), where \(f \in L^2(\Omega)\), \(\varphi \in L^2(\partial \Omega)\) are given functions and \(\kappa > 0, \alpha \geq 0\) are given constants. The problem is discretized by the finite element method with conforming piecewise linear or polynomial approximations. They give a review on previous results of \textit{M. Feistauer} and \textit{K. Najzar} [Numer. Math. 78, 403-425 (1998; Zbl 0888.65118)], \textit{M. Feistauer, K. Najzar} and \textit{V. Sobotíková} [Numer. Funct. Anal. Optimization 20, 835-851 (1999; Zbl 0947.65116)] and \textit{M. Feistauer, K. Najzar, P. Sváček} and \textit{V. Sobotíková} [ENUMATH 99. Numerical mathematics and advanced applications. Proceedings of the 3rd European conference, Jyväskylä, Finland, July 26-30, 1999. Singapore: World Scientific. 486-493 (2000; Zbl 0972.65095)] and then prove some error estimates for a higher-order finite element method. Numerical examples are given to compare them with these estimates.