Adaptive Quadrilateral and Hexahedral Finite Element Methods with Hanging Nodes and Convergence Analysis (Journal of Computational Mathematics, 2019)
In this paper we study the convergence of adaptive finite element methods for the general non-affine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ingredients, such as quasi-orthogonality, estimator reduction and Döfler marking strategy, convergence of the adaptive finite element methods for the general second-order elliptic partial equations is proved. Our analysis is effective for all conforming $Q_m$ elements which covers both the two- and three-dimensional cases in a unified fashion.