Numerical Methods for Solving Partial Differential Equations

As a fundamental area of mathematical research, partial differential equations (PDEs) play a critical role in economics, scientific studies, and engineering applications. With the deepening of research into many natural phenomena, the PDEs we formulate have become increasingly complex, often leading to scenarios where analytical solutions do not exist, are unattainable with existing mathematical methods, or are prohibitively expensive to compute, rendering them impractical. Consequently, analytical solutions are not always suitable for solving complex PDEs. This has made numerical methods for solving PDEs an essential research focus, attracting growing attention from scholars. In this context, this paper provides a comprehensive overview of the development and current research status of numerical methods for PDEs. It also delves into three commonly used numerical approaches: the finite element method, the finite difference method, and neural network-based methods. The advantages and disadvantages of the three methods are surmised in the paper. These methods have been widely applied in fields such as fluid mechanics, electromagnetics, and financial mathematics, making significant contributions to the advancement of various disciplines.