Various natural and physical phenomena can be described by partial differential equations (PDEs). To solve these equations, numerical methods based on the Finite Element (FE) and Discontinuous Galerkin (DG) approaches have been widely used and developed. In this talk, we focus on FE and DG methods for solving second-order elliptic equations. We discuss their theoretical foundations, numerical implementation, and key differences, particularly in terms of accuracy, stability, and conservation properties. While these methods allow for high-order accuracy, their convergence behavior is significantly influenced by the regularity of the solution. When the solution is not sufficiently smooth, adaptive algorithms play a crucial role in improving efficiency and accuracy. Here, we introduce reliable and fully computable a posteriori error estimators, which guide adaptive mesh refinement to enhance solution accuracy while reducing computational cost. Finally, numerical results will be presented to validate the theoretical findings and demonstrate the effectiveness of adaptive algorithms.