Convergence of a standard adaptive nonconforming finite element method with optimal complexity

In this paper, we analyze the convergence and optimal complexity of the usual simple adaptive nonconforming finite element method by using Dörfler collective marking strategy. Based on several basic ingredients, such as the estimator reduction, quasi-orthogonality, local upper bound and so on, we eventually show the convergence of the adaptive algorithm by establishing the reduction of some total error and the quasi-optimal convergence rate. Our analysis does not need the relation between the nonconforming P1 element and the mixed Raviart–Thomas element. The results of numerical experiments confirm that our adaptive algorithm is optimal.