Quasi-Optimal Convergence of a Family of Adaptive Nonconforming Finite Element Methods

This study is devoted to the quasi-optimal convergence analysis of a family of adaptive nonconforming elements with high-order terms, which preserve weak continuities. In contrast with the nonconforming P1 element (Crouzeix–Raviart element) the gradient of the discrete solution considered in this paper is not a piecewise constant vector. New quasi-orthogonality and a new discrete upper bound are established for the first time, based on which the convergence of the adaptive algorithm with a standard Dörfler collective marking strategy and quasi-optimality results are eventually established. Some other properties are also investigated, for example, the discrete Helmholtz decomposition for this family of nonconforming elements. Numerical experiments confirm the theoretical results.