{'fraction_non_alphanumeric': 0.02737382378100941, 'fraction_numerical': 0.005988023952095809, 'mean_word_length': 5.573033707865169, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Adaptive Finite Element Methods on Quadrilateral Meshes without Hanging Nodes

Hanging nodes have some disadvantages in the implementation of adaptive finite element methods on quadrilateral meshes, which usually need further techniques to treat them. In this paper, we present a shape regular local refinement algorithm on quadrilateral meshes without hanging nodes, which can be viewed as an extension of the original red-green refinement proposed by Bank, Sherman, and Weiser [Refinement algorithms and data structures for regular local mesh refinement, in Scientific Computing: Applications of Mathematics and Computing to the Physical Sciences, R. S. Stepleman, ed., North-Holland, Amsterdam, 1983, pp. 3-17]. We also prove this quadrilateral red-green refinement can hold the shape regularity and conformity of quadrilateral meshes. Furthermore, we have successfully accomplished the adaptive finite element computation on quadrilateral meshes without hanging nodes. Numerical results show that the adaptive algorithm with this red-green refinement is quasi-optimal. Some properties of this local refinement on quadrilateral meshes are also covered in this paper.