This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second order elliptic PDEs and dimension d>1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs and beyond coercive problems to inf-sup stable AFEMs.

翻译：本文综述了自适应有限元方法（AFEMs）的理论基础。自适应有限元方法是现代计算科学与工程中的核心工具，但其数学分析仍面临严峻挑战。我们针对线性二阶椭圆型偏微分方程（定义域维度d>1）提供了自包含且前沿的自适应有限元方法讨论，重点聚焦基础理论问题。在简要回顾泛函分析与基本有限元理论（包括渐变网格中的分片多项式逼近）后，系统阐述了强制性问题（coercive problems）的核心内容。首先提出一种适用于粗糙数据的后验误差分析方法，该方法可生成与解误差完全等价的估计量。基于这些估计量，根据数据结构差异设计并研究了三类自适应有限元算法。我们证明了这些算法的线性收敛性，并在解与数据属于适当逼近类的前提下验证了其速率最优性。此外，探讨了逼近类与正则性类之间的关联。最后，将该理论拓展至非协调自适应有限元范例——间断Galerkin方法，并进一步推广至超越强制性问题的inf-sup稳定自适应有限元方法（inf-sup stable AFEMs）。