非局部Helmholtz问题的PML和数值离散方法

非局部问题是近些年来的新型热点问题，在数学、物理和工程领域都有广泛应用，受到广泛关注。非局部Helmholtz问题可从非局部波传播问题得到，且当非局部效应的视界趋于零时等价于经典的Helmholtz问题。PML是电磁波散射问题的吸收边界层，具有有效、结构简单和易于程序实现的特点。本项目主要研究如下问题： .（1）将PML技术推广到非局部Helmholtz方程，从数值试验和理论上分析PML的有效性，即截断的PML吸收层的指数收敛性，以及截断误差与PML介质参数和宽度的依赖关系；.（2）分析有限元方法的渐近相容性和（预渐近）误差，并最终设计改良算法。.本项目的目标是提供无界域上非局部Helmholtz问题的的高效计算方法。对这些问题的研究，不仅具有针对该问题本身的理论和应用意义，同时对非局部波传播问题和经典Helmholtz问题的研究具有重要的参考意义。

Nonlocal models have been paid much attention in various research areas recently. The nonlocal Helmholtz problem may come from the nonlocal wave propagation, and is equivalent to the standard Helmholtz problem as the horizon of nonlocal effects goes to zero. PML (perfect matched layer) is an absorbing boundary layer for the Maxwell’s radiation problem, which is effect, simple in structure and easy to implement. The project is devoted to.(1) extending the PML technique to the nonlocal Helmholtz equation, studying the effectiveness by numerical tests and theoretical analysis, i.e. the exponentially convergence of the PML truncation error, and its dependence on the PML medium parameters and width of the boundary layers;.(2) analyzing the asymptotic compatibility and the preasymptotic error, which are used to design better numerical methods..The aim of the project is to obtain effective methods solving the nonlocal Helmholtz problem in unbounded domain. Studying the problem not only is significant for the problem itself in theory and application, but also has important reference significance for the study of nonlocal wave propagation and standard Helmholtz problems.