Sváček, Petr; Najzar, Karel Numerical solution of problems with non-linear boundary conditions. (English) Zbl 1020.65091 Math. Comput. Simul. 61, No. 3-6, 219-228 (2003). 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 M. Feistauer and K. Najzar [Numer. Math. 78, 403-425 (1998; Zbl 0888.65118)], M. Feistauer, K. Najzar and V. Sobotíková [Numer. Funct. Anal. Optimization 20, 835-851 (1999; Zbl 0947.65116)] and M. Feistauer, K. Najzar, P. Sváček and 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. Reviewer: Dinh Nho Hao (Brussel) MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation Keywords:nonlinear boundary conditions; Poisson equation; finite element method; numerical examples Citations:Zbl 0888.65118; Zbl 0947.65116; Zbl 0972.65095 PDFBibTeX XMLCite \textit{P. Sváček} and \textit{K. Najzar}, Math. Comput. Simul. 61, No. 3--6, 219--228 (2003; Zbl 1020.65091) Full Text: DOI References: [2] Feistauer, M.; Najzar, K., Finite element approximation of a problem with a lnon-inear Newton boundary condition, Num. Math., 78, 403-425 (1998) · Zbl 0888.65118 [3] Feistauer, M.; Najzar, K.; Sobotíková, V., Error estimates for the finite element solution of elliptic problems with non-linear Newton boundary conditions, Num. Funct. Anal. Optimiz., 20, 835-851 (1999) · Zbl 0947.65116 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.