This is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to 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.

中文翻译：

---

自适应有限元方法


这是对自适应有限元方法 (AFEM) 理论的综述，该方法是现代计算科学和工程的基础，但其数学评估是一项艰巨的挑战。我们对线性二阶椭圆偏微分方程和维度 d > 1 的 AFEM 进行了独立且最新的讨论，重点是基础问题。在简要回顾泛函分析和基本有限元理论（包括分级网格中的分段多项式近似）之后，我们提出了强制问题的核心材料。我们从一种适用于粗略数据的新颖的后验误差分析开始，它提供了与解误差完全等效的估计量。它们根据数据结构用于三个 AFEM 的设计和研究。如果解和数据属于合适的近似类，我们证明了这些算法的线性收敛和速率最优性。我们还讨论了近似类和正则类之间的关系。我们最终将该理论扩展到不连续伽辽金方法，作为非一致性 AFEM 的原型，并超越了强制问题，扩展到 inf-sup 稳定 AFEM。