报 告 人：阴小波 教授
Many nonlocal models have adopted a finite and radially symmetric nonlocal interaction domains. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such interaction domains. A crucial question is, to what extent such approximations affect the nonlocal operators and the corresponding nonlocal solutions. While recent works have analyzed this issue for nonlocal operators in the case of 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 report, we are interested in addressing this issue and establishing the convergence of new nonlocal solutions by polygonal approximations 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 shown. These results may be used to guide future computational studies of nonlocal problems.
阴小波，本科毕业于南开大学数学科学学院，博士毕业于中国科学院数学与系统科学研究院，现为华中师范大学数学与统计学学院教授、博士生导师。主要研究方向为有限元高精度算法、移动网格方法和非局部问题的数值分析。已在Journal of Computational Physics, Journal of Scientific Computing, Communications in Mathematical Sciences, Advance in Computational Mathematics等杂志上发表多篇文章。主持三项国家自然科学基金项目，作为主要成员参与一项国家自然科学基金重大研究计划重点支持项目。