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Numerical solution of problems with non-linear boundary conditions. (English) Zbl 1020.65091

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.

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
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[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
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