We design and analyze an adaptive $hp$-finite element method (hp-AFEM) in dimensions $n=1,2$. The algorithm consists of iterating two routines: hp-NEARBEST finds a near-best $hp$-approximation of the current discrete solution and data to a desired accuracy, and REDUCE improves the discrete solution to a finer but comparable accuracy. The former hinges on a recent algorithm by Binev for adaptive $hp$-approximation, and acts as a coarsening step. We prove convergence and instance optimality.
High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates
Texas A&M University（德克萨斯农工大学）；Universidad de Salamanca（萨拉曼卡大学）；Chulalongkorn University（朱拉隆功大学）；CONICET（国家科学研究委员会）；Universidad Nacional del Litoral（国立里奥斯特拉大学）；University of Maryland（马里兰大学）
Adaptive finite element method for elliptic optimal control problems: convergence and optimality
Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院系统科学研究所）
Convergence of $L^2$-norm based adaptive finite element method for elliptic optimal control problems
NCMIS, LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；LSEC, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院系统科学研究所）；School of Mathematical Sciences, Shandong Normal University（山东师范大学数学科学学院）
Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity
Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions（索邦大学，巴黎 Cité 大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；TU Wien, Institute of Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）
Consejo Nacional de Investigaciones Científicas y Técnicas（国家科学技术研究委员会）；Universidad Nacional del Litoral（利托拉尔国立大学）；IMAL - Güemes 3450（IMAL - 盖勒斯街3450号）
Local Multilevel Preconditioned Jacobi-Davidson Method for Elliptic Eigenvalue Problems on Adaptive Meshes
School of Mathematical Science, Tongji University（同济大学数学科学学院）；Key Laboratory of Intelligent Computing and Applications, Tongji University, Ministry of Education（同济大学教育部智能计算与应用重点实验室）
Convergence and optimality of the adaptive Morley element method
LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Institute of Computational Mathematics, Chinese Academy of Sciences（中国科学院计算数学研究所）；The School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Department of Mathematics, Pennsylvania State University（宾夕法尼亚州立大学数学系）
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院）；The School of Mathematical Sciences and the Beijing International Center for Mathematical Research, Peking University（北京大学数学科学学院和北京国际数学研究中心）；Department of Mathematics, Pennsylvania State University（宾夕法尼亚州立大学数学系）
Convergence and Optimality of Adaptive Mixed Finite Element Methods
University of California at Irvine（加州大学尔湾分校）；University of California at San Diego（加州大学圣地亚哥分校）；Peking University（北京大学）；Pennsylvania State University（宾夕法尼亚州立大学）
Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity
Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions, LJLL（索邦大学，巴黎西岱大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；TU Wien, Institute of Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）
On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators
Inria（法国国家数字与工程科学研究所）；Univ. Lille（里尔大学）；Laboratoire Paul Painlevé（保罗·庞加莱实验室）；Inria Paris（法国国家数字与工程科学研究所巴黎分部）；Institute for Numerical Simulation, University of Bonn（波恩大学数值模拟研究所）；University of Twente（特文特大学）；Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）
