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Finite element approximation of a problem with a nonlinear Newton boundary condition. (English) Zbl 0888.65118

This paper is concerned with the study of the finite element approximation of an elliptic equation equipped with a nonlinear Newton boundary condition (the nonlinear term has a “polynomial” behaviour). The nonlinearity in the Newton condition causes a difficulty in proving the coercivity of the problem.
The authors pay special attention to the effect of numerical integration in the nonlinear boundary condition. The integration formula is exact for (at least) first degree polynomials and leads, together with the uniform \(W^{1,2}(\Omega)\)-boundedness of approximate solutions (which is our only “regularity” information), to the order of numerical integration \(O(h^{1/2-\varepsilon})\) (\(\varepsilon>0\) arbitrarily small), instead of the expected order \(O(h)\).
So, the numerical integration and the nonlinearity in the Newton boundary condition lead to a loss of order of accuracy, if strong monotonicity of the problem and sufficient regularity of the exact solution are missing.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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