We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the adaptive mesh-refinement as well as the inexact solution of the arising discrete systems. We prove that the proposed strategy leads to linear convergence with optimal algebraic rates. Unlike prior works, however, we focus on convergence rates with respect to the overall computational costs. In explicit terms, the proposed adaptive strategy thus guarantees quasi-optimal computational time. In particular, our analysis covers linear problems, where the linear systems are solved by an optimally preconditioned CG method as well as nonlinear problems with strongly monotone nonlinearity which are linearized by the so-called Zarantonello iteration.
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 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）
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）
Instance optimality of the adaptive maximum strategy
Mathematisches Institut der Universität München（慕尼黑大学数学研究所）；Fakultät für Mathematik, Ruhr-Universität Bochum（波鸿鲁尔大学数学系）；Korteweg-de Vries Institute (KdVI) for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
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）
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）
The quasi-optimality criterion in the linear functional strategy
Industrial Mathematics Institute, Johannes Kepler University of Linz（林茨约翰·开普勒大学工业数学研究所）；Department of Mathematics, University of Innsbruck（因斯布鲁克大学数学系）；Institute of Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）；Igor Sikorsky Kyiv Polytechnic Institute（伊戈尔·西科尔斯基基辅理工学院）
