SIAM Review, Volume 65, Issue 4, Page 963-1028, November 2023.
This article is a review of basic concepts and tools devoted to a posteriori error estimation for problems solved with the finite element method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The main goal of this review is to present in a unified manner a large set of powerful verification methods, centered around the concept of equilibrium. Methods based on that concept provide error bounds that are fully computable and mathematically certified. We discuss recovery methods, residual methods, and duality-based methods for the estimation of the whole solution error (i.e., the error in energy norm), as well as goal-oriented error estimation (to assess the error on specific quantities of interest). We briefly survey the possible extensions to nonconforming numerical methods, as well as more complex (e.g., nonlinear or time-dependent) problems. We also provide some illustrating numerical examples on a linear elasticity problem in three dimensions.

中文翻译：

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有限元计算中后验误差估计的介绍性回顾

SIAM Review，第 65 卷，第 4 期，第 963-1028 页，2023 年 11 月。
本文回顾了专用于用有限元方法解决的问题的后验误差估计的基本概念和工具。为了简单和清晰起见，我们主要关注线性椭圆扩散问题，通过一致的数值离散来近似。本次审查的主要目标是以统一的方式提出一大套强大的验证方法，以均衡概念为中心。基于该概念的方法提供了完全可计算且经过数学认证的误差界限。我们讨论恢复方法、残差方法和基于对偶性的方法来估计整体解误差（即能量范数的误差），以及面向目标的误差估计（评估特定感兴趣量的误差） 。我们简要调查了非一致性数值方法的可能扩展，以及更复杂的（例如非线性或时间相关）问题。我们还提供了一些关于三维线性弹性问题的说明性数值示例。