Numerical solution of problems with non-linear boundary conditions

In this paper, we are concerned with an elliptic problem in a bounded two-dimensional domain equipped with a non-linear Newton boundary condition. This problem appears, e.g. in the modelling of electrolysis procedures. We assume that the non-linearity has a polynomial behaviour. The problem is discretized by the finite element (FE) method with conforming piecewise linear or polynomial approximations. This problem has been investigated in [Num. Math. 78 (1998) 403; Num. Funct. Anal. Optimiz. 20 (1999) 835] in the case of a polygonal domain, where the convergence and error estimates are established. In [Feistauer et al., On the Finite Element Analysis of Problems with Non-linear Newton Boundary Conditions in Non-polygonal Domains, in press] the convergence of the FE approximations to the exact solution is proved in the case of a nonpolygonal domain with curved boundary. The analysis of the error estimates leads to interesting results. The non-linearity in boundary condition causes the decreas of the approximation error. Further decreas is caused by the application of the numerical integration in the computation of boundary integrals containing the non-linear terms. In [Feistauer et al., Numerical analysis of problems with non-linear Newton boundary conditions, in: Proceedings of the Third Conference of ENUMATH’99, p. 486], numerical experiments prove that this decreas is not the result of a poor analysis, but it really appears.