SIAM Review, Volume 62, Issue 1, Page 197-197, January 2020.
The SIGEST article in this issue, “Asymptotically Compatible Schemes for Robust Discretization of Parametrized Problems with Applications to Nonlocal Models,” by Xiaochuan Tian and Qiang Du, concerns differential equation models that (a) involve a physical parameter, and (b) change their nature when this parameter reaches an asymptotic limit. The work is motivated by nonlocal peridynamic models in continuum mechanics, which reduce to classical differential equation models as the horizon parameter tends to zero. In this setting, a numerical method is said to be asymptotically compatible if it produces approximations that are robust in such a limit. This is an important property which allows us to avoid inaccurate or unphysical numerical solutions from inappropriate choices of model and discretization parameters. The authors develop a general framework that relies on minimal assumptions. Their high-level results, provided in Theorems 2.6 to 2.8, are neatly summarized in Figure 2.1. The results are then applied to finite element discretizations of nonlocal diffusion and peridynamic models. Numerical experiments are also provided to validate the results. The original article appeared in the SIAM Journal on Numerical Analysis (SINUM) in 2014. As indicated by its high citation count, this work is relevant to many circumstances where multiscale or nonlocal effects are present; examples include viscosity limits in nonlinear conservation laws, phase field models with sharp interfaces, and smoothed particle hydrodynamics. The original SINUM article presented an asymptotically compatible framework for self-adjoint problems. In converting their article into SIGEST form, the authors have extended the results to cover both self-adjoint and non-self-adjoint problems with parameter-dependent data. Here, some modifications, and new notation, were required to distinguish between operators and their adjoints, between test and trial functions, and between the associated function spaces. The authors also reorganized the derivation of the main results and streamlined the presentation of numerical experiments. In addition, they have discussed subsequent developments in the field, increasing the size of the bibliography from 40 to 63, and highlighted some current challenges. We also note that the results in the original SINUM article formed part of the 2017 Columbia University Ph.D. dissertation of the first author, Xiaochuan Tian, who was subsequently awarded an Association for Woman in Mathematics Dissertation Prize.

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SIGEST

SIAM评论，第62卷，第1期，第197-197页，2020年1月。
肖小田和杜强撰写的本期SIGEST文章“用于参数化问题的鲁棒离散化的渐近兼容方案及其在非局部模型中的应用”，涉及的微分方程模型（a）涉及物理参数，并且（b）更改其物理参数此参数达到渐近极限时的自然状态。这项工作是由连续力学中的非局部绕动力学模型推动的，随着层位参数趋于零，该模型简化为经典的微分方程模型。在这种情况下，如果数值方法产生的近似值在这种极限下很稳健，则可以说它是渐近兼容的。这是一个重要的属性，它使我们能够避免因模型和离散化参数的选择不当而产生不精确或不自然的数值解。作者开发了一个基于最小假设的通用框架。定理2.6至2.8中提供了它们的高级结果，这些结果在图2.1中进行了很好的总结。然后将结果应用于非局部扩散和周动力模型的有限元离散化。还提供了数值实验以验证结果。最初的文章发表在2014年的《 SIAM数值分析期刊》上。正如其高被引数所表明的那样，这项工作与存在多尺度或非局部效应的许多情况有关；示例包括非线性守恒定律中的粘度极限，具有尖锐界面的相场模型以及平滑的粒子流体动力学。SINUM的原始文章提出了一种关于自伴问题的渐近兼容框架。在将他们的文章转换为SIGEST形式时，作者将结果扩展到涵盖参数相关数据的自伴和非自伴问题。在此，需要进行一些修改和新的标记，以区分运算符及其伴随符号，测试功能和试用功能以及关联的功能空间。作者还重新组织了主要结果的推导，并简化了数值实验的表示。此外，他们还讨论了该领域的后续发展，将书目的大小从40增加到63，并强调了当前的挑战。我们还注意到，原始SINUM文章中的结果构成了2017年哥伦比亚大学博士学位的一部分。第一作者田小川的论文