Elliptic partial differential equations (PDEs) are ubiquitous in various fields, and their solutions play a crucial role in understanding the behavior of physical systems. However, solving these equations can be challenging, especially when the solution contains singularities. In this article, we focus on the characterization of the singular term method for solving Elliptic PDEs with singularities. We provide a detailed overview of the method and its applications, highlighting its effectiveness in capturing the behavior of the solution near singular points.
Methodology
The singular term method is based on the idea of decomposing the solution into two parts: one containing the regular part and the other containing the singular part. The regular part is obtained by solving a linear equation, while the singular part is handled using a special technique that involves the concept of moment tensors. By combining these two parts, we can obtain an accurate approximation of the solution near singular points.
Results
We demonstrate the effectiveness of the characterization of the singular term method through several numerical examples. In each example, we show how to compute the stress concentration factor by solving the touching problem, and how to use the moment tensors to capture the behavior of the solution near singular points. Our results show that the method is able to accurately capture the behavior of the solution even when the number of grid points is relatively small.
Conclusion
In conclusion, this article provides a comprehensive overview of the characterization of the singular term method for solving Elliptic PDEs with singularities. We demonstrated its effectiveness through several numerical examples and highlighted its potential in capturing the behavior of the solution near singular points. The method is simple to implement and can be useful in various applications, including damage analysis of fiber composites and electrostatic problems in engineering. By using this method, researchers and practitioners can gain a better understanding of the behavior of solutions to Elliptic PDEs with singularities, which can lead to more accurate predictions and better decision-making.
