Many nonlocal models have adopted Euclidean balls as the nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such balls. A crucial question is, to what extent such approximations affect the nonlocal operators and the corresponding solutions. While recent works have analyzed this issue for a fixed horizon parameter, the question remains open in the case of a small or vanishing horizon parameter, which happens often in many practical applications and has significant impact on the reliability and robustness of nonlocal modeling and simulations. In this work, we are interested in addressing this issue and establishing the convergence of the nonlocal solutions associated with polygonally approximated interaction neighborhoods to the local limit of the original nonlocal solutions. Our finding reveals that the new nonlocal solution does not converge to the correct local limit when the number of sides of polygons is uniformly bounded. On the other hand, if the number of sides tends to infinity, the desired convergence can be established. These results may be used to guide future computational studies of nonlocal models.
翻译:许多非本地模型已经将欧几里得球作为非本地互动区。 当用数字来解答它们时,有时很容易采用这些球的多边近似法。 一个关键问题是,这种近似法在多大程度上影响到非本地操作员和相应的解决方案。 虽然最近的工作分析了固定地平线参数的这一问题,但对于一个小型或消失的地平线参数来说,问题仍然未解决,这种参数在许多实际应用中经常发生,对非本地建模和模拟的可靠性和稳健性有重大影响。在这项工作中,我们有兴趣解决这一问题,并将与多边近似互动区相关的非本地解决方案与原始非本地解决方案的本地范围相趋同。我们的发现表明,当多边形的边数被统一捆绑时,新的非本地解决方案并没有与正确的本地界限趋同。另一方面,如果方数目趋向无限,则可以确定预期的趋同性。这些结果可用于指导非本地模型的未来计算研究。