This article proposes a hybrid adaptive numerical method based on the Dual Reciprocity Method (DRM) to solve problems with non-linear boundary conditions and large-scale problems, named Hybrid Adaptive Dual Reciprocity Method (H-DRM). The method uses a combination of DRM to handle non-homogeneous terms, iterative techniques to deal with non-linear boundary conditions, and an adaptive multiscale approach for large-scale problems. Additionally, the H-DRM incorporates local finite elements in critical regions of the domain. This method aims to improve computational efficiency and accuracy for problems involving complex geometry and non-linearities at the boundary, offering a robust solution for physical and engineering problems. Demonstrations and computational results are presented, validating the effectiveness of the method compared to other known methods through an iterative process of 7 million iterations.

翻译：本文提出了一种基于对偶互易法（DRM）的混合自适应数值方法，用于求解具有非线性边界条件的问题及大规模问题，该方法被命名为混合自适应对偶互易法（H-DRM）。该方法结合使用DRM处理非齐次项、迭代技术处理非线性边界条件，并采用自适应多尺度方法应对大规模问题。此外，H-DRM在域的关键区域引入了局部有限元。该方法旨在提高涉及复杂几何形状和边界非线性问题的计算效率与精度，为物理和工程问题提供稳健的解决方案。文中给出了方法演示与计算结果，通过700万次迭代过程验证了该方法相较于其他已知方法的有效性。