The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop.

中文翻译：

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用于逼近欧几里得球和非局部问题的有限元方法中的求积规则的食谱

对于具有有限相互作用范围的非局部模型，有限元方法 (FEM) 的实施带来了偏微分方程 (PDE) 设置中未面临的挑战。例如，必须处理涉及双积分的弱形式，这导致离散系统具有更高的组装和求解成本，因为与 PDE 的 FEM 相比，稀疏性可能要低得多。此外，可能会遇到不光滑的被积函数。在许多非局部模型中，非局部相互作用仅限于有界邻域，这些邻域普遍被选为欧几里得球，从而导致处理此类球与有限元的交集的挑战。我们专注于开发有效组装 FEM 刚度矩阵的方法，以及为有助于组装效率和足够精度的双积分选择正交规则。我们的方法的一个主要特点是使用近似球，例如欧几里得球的几个多边形近似，除其他优点外，它还减轻了处理球元素交叉点的挑战。我们提供了我们开发的几种方法的相对准确性和效率的数值说明。