论文标题
针对非本地问题的有限元方法的食谱，包括正交规则和近似欧几里得球
A cookbook for finite element methods for nonlocal problems, including quadrature rules and approximate Euclidean balls
论文作者
论文摘要
针对有限互动范围的非局部模型的有限元方法（FEM）的实施构成了偏微分方程（PDE）设置中未面临的挑战。例如，必须处理涉及双重积分的弱形式，这些形式导致离散系统具有较高的组装和解决成本，因为与PDE的FEM相比，稀疏性可能要低得多。此外，可能会遇到非平滑积分。在许多非局部模型中，非局部相互作用仅限于无处不在的邻域，这些社区被无处不在地被选为欧几里得球，从而挑战了处理此类球与有限元素的相交。我们专注于开发有效组装FEM刚度矩阵的配方，并为有助于组装效率的双积分的正交规则选择，并具有足够的准确性。我们食谱的一个主要特征是使用近似球，例如，欧几里得球的几个多边形近似值，除其他优势外，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例证。
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 non-smooth 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.
