AI 中文总结
研究基于自适应局部核方法逼近泊松方程解，通过开发误差局部估计和自适应程序，利用局部德劳内三角剖分提供节点集局部几何信息，经实验验证该方法能有效解决解的局部特征，避免全局均匀细化的难题。
AI 中文摘要
基于近期自适应局部核方法逼近线性算子作用的发展，开发了误差的局部估计和逼近泊松方程解的自适应程序。误差估计用于自适应过程中，以确定解域中减小节点间距可降低解误差的位置。该方法本质上是“无网格”的，仅利用局部德劳内三角剖分来提供节点集局部几何信息，使用后即丢弃。实验表明误差估计与近似解中的实际绝对前向误差密切一致，并与现有细化指标进行了比较。自适应程序和误差估计的结合提供了一种自动技术，可解决解的局部特征，而无需在整个解域进行通常难以处理的均匀细化。
英文摘要
Expanding on the recent development of adaptive local kernel methods for approximating the action of linear operators, a local estimate of the error and an adaptive procedure for approximating solutions to the Poisson equation is developed. The error estimate is used in the midst of the adaptive procedure to determine locations in the solution domain where decreasing the spacing between nodes can decrease the error in the solution. The approach described here is essentially "meshless", with only local Delaunay triangulations leveraged as a convenience to provide information on the local geometry of the node set. Once this information is utilized, it is discarded. The experiments performed show close agreement between the error estimate and actual absolute forward error in the approximate solution along with a comparison to existing indicators for refinement. The combination of the adaptive procedure and error estimate provides an automated technique to resolve localized features of a solution without the, often intractable, expense of uniform refinement across the entire solution domain.
