In science and engineering, partial differential equations (PDEs) are employed for modeling and comprehending an extensive variety of physical phenomena. Solving these equations analytically is complicated and requires a lot of research and time. Mesh-based and meshless techniques are two popular ways to solve PDEs numerically. Mesh-based methods depend on breaking up the computational domain into a structured or unstructured mesh. These methods are accurate and based on well-established theories. However, they often have challenges with complex geometries, flexibility, and the high cost of computation that comes with mesh generation and refinement. On the other hand, meshless methods are a different way to do things that don't require meshing. Instead, these methods use a number of points that are spread out to get close to the solution. It can handle complex geometries, is easy to implement, and is easier to deal with problems that have boundaries or interfaces that change. This paper provides a summary of solving PDEs using both mesh-based and meshless approaches, with a focus on elasticity implementation. In addition to explaining the characteristics of each of the two numerical methods.