Hybrid Adaptive Dual Reciprocity Method for Efficient Solution of Large-Scale Non-Linear Boundary Conditions

Abstract

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.

Figures (3)

• Figure 1: Comparison of convergence between the methods H-DRM, Gauss-Seidel, DRM, and the Dual Reciprocity Method.
• Figure 2: 3D comparative graph of errors between numerical methods
• Figure 3: Comparison of numerical methods in 3D, showing the error as a function of $x$ and $y$.