SIAM Review, Volume 62, Issue 1, Page 199-227, January 2020.
Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations, which can be highly desirable in practice. The asymptotically compatible schemes studied in an earlier published version of this paper meet such objectives for a class of parametrized problems. An extended version of the abstract mathematical framework is established rigorously here, together with applications to the numerical solution of both nonlocal models and their local limits. In particular, the framework can be applied to nonlocal models of diffusion and a general state-based peridynamic system parametrized by the horizon radius. Recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter. Thus, it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical discretizations to problems involving nonlocal interactions on multiple scales. This work provides new insight in this regard through a careful analysis of related conforming finite element discretizations, and the finding is valid under minimal regularity assumptions on exact solutions. It reveals that for the nonlocal models under consideration and their local limit, as long as the finite element space contains continuous piecewise linear functions, the Galerkin finite element approximation is always asymptotically compatible. For piecewise constant finite elements, whenever applicable, it is shown that a correct local limit solution can also be obtained as long as the discretization (mesh) parameter decreases faster than the modeling (horizon) parameter does. These results can be used to guide future computational studies of nonlocal problems. Some other applications, such as the fractional PDE limit of nonlocal models, and open questions are also presented.

中文翻译：

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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

SIAM评论，第62卷，第1期，第199-227页，2020年1月。
具有参数特征的自然界中的许多问题都具有固定的参数值和接近渐近极限的参数。在两种情况下都收敛的数值方案提供了可靠的离散化，这在实践中可能是非常需要的。本文较早发行的版本中研究的渐近兼容方案满足了针对一类参数化问题的此类目标。此处严格建立了抽象数学框架的扩展版本，并将其应用于非局部模型及其局部极限的数值解。特别地，该框架可以应用于扩散的非局部模型和由视界半径参数化的基于状态的一般基于动力学的系统。最近的发现暴露了当视线半径与离散参数成比例时，与非局部模型离散化相关的风险。因此，期望为此类模型开发渐近兼容方案，以便为涉及多个尺度上的非局部相互作用的问题提供鲁棒的数值离散化。通过对相关一致的有限元离散化的仔细分析，这项工作为这方面提供了新的见解，并且该发现在精确解的最小规则假设下是有效的。结果表明，对于所考虑的非局部模型及其局部极限，只要有限元空间包含连续的分段线性函数，Galerkin有限元逼近始终是渐近兼容的。对于分段常量有限元，只要适用，表明只要离散化（网格）参数的下降速度比建模（水平）参数的下降速度快，就可以获得正确的局部极限解。这些结果可用于指导非局部问题的未来计算研究。还介绍了其他一些应用程序，例如非局部模型的分数PDE限制和开放性问题。