The discontinuous Galerkin (DG) methods for partial differential equations (PDE) are a family of numerical methods used to solve linear, non-linear multidimensional systems of PDE that arise in many areas of engineering, physics and applied mathematics. In particular these methods are used to solve conservation equations and wave equations in electromagnetism and continuum dynamics. This conference aim is to expose these methods in the local community. We look briefly the fundamentals of the discontinuous Galerkin Finite Element Method (DG-FEM), special attention is taken to introduce high-order approximation of the problem in the smooth part of the solution and to resolve clearly discontinuous parts of the former. As a study case the Euler gas equation is considered to test the most important tools in canonical examples: the Sod’s problem, the Isentropic vortex and the Couette flow between two cylinders. In addition, the fundamentals of the (DG-FEM) for high-order equations is shown and furthermore, the formulation and solution of both the Poisson and advection equation is presented using the FEniCS software.