An error estimate for the finite element solution of an elliptic problem with a nonlinear Newton boundary condition in nonpolygonal domains

The article is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional domain with a curved boundary. The existence and uniqueness of the weak solution of the continuous problem is a consequence of the monotone operator theory. The problem is discretized with the use of the finite element method. The main attention is paid to the effect of the approximation of the curved boundary by a piecewise linear boundary and of the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlamal's ideal triangulation and interpolation, the error estimate for the solution of the discrete problem is derived.