The Continuous Galerkin (CG) method, though simple and versatile in its formulation, suffers from instability in convection-dominated problems. This has driven the development of numerous stabilization techniques, specially in the last years in which great strides have been made. Based on the most recent contributions from the literature, we seek for an accurate and efficient methodology that can be applied to a wide range of flow scenarios. Our approach combines a high-order stabilization term with an artificial viscosity. For the high-order term, a Local Projection Stabilization [1] is selected, which acts as a filter removing the spurious oscillations in smooth regions, however it is not sufficient in the vicinity of strong gradients where Gibb’s oscillations show up. To suppress those oscillations a shock capturing is required, we apply an Entropy residual-based viscosity [2] that consists on a nonlinear equation that acts in those problematic locations reducing the risk of over-undershoots. Spectral elements are derived from the Lagrange functions and address certain shortcomings of classical high-order FEM. They requires a stabilization compatible with its formulation and we prove that the proposed method is suitable for this type of elements. The proposed stabilization framework is presented for a generic conservation law and then validated on benchmark problems. First, it is applied to the linear advection problem, and then to Euler equations problems with the presence of shocks. These cases prove the property-preserving capability of our method and its ability to control the damping of over-undershoots, leading to a low-dissipative solution. Finally, the Local Projection Stabilization is applied to the Fractional Step technique to demonstrate its virtues in solving incompressible flow problems.