Abstract: When the high-order discretization scheme is applied for solving the hyperbolic conservation laws, the Gibbs oscillations emerging around the discontinuities may cause the fields to take unphysical values, which could result in the undesired interruption of calculation process. This problem is especially severe for the discontinuous Galerkin scheme. Therefore, suitable limiting calculators are supposed to be designed in order to damp the appearance of unphysical oscillations. However, the existed limiters for the discontinuous Galerkin scheme originate mostly from the finite volume methods, thus for the unstructured grids they can only limit the spatial solutions of the solutions, namely the lowerorder derivatives, rather than offer an general criterion for the higher order derivatives. Due to this background, the discontinuous finite element solutions are firstly processed into general Fourier expansion series for the realization of spectral decomposition in different frequency domains. Afterwards, a modal coordinate system is originally proposed, describing the changing patterns of the directional derivatives in different orders in a new form. Finally, after the combination of the information of the current mode and its adjacent mode, and the limitation of the directional derivatives in different orders for different levels, the unphysical solutions are successfully damped. Based on this method, the Euler equations are solved for the numerical simulations of the 2-D Riemann problems, the flow field of the windtunnel with a front step, the reflection of shockwaves in the inner field of scramjet engines as well as the subsonic and transonic flow surrounding airfoils. The obtained results demonstrate that the new limiter is relatively reliable and it can be generalized to highorder schemes.