A novel full-wave method to perform mode analysis on waveguides is introduced in this paper. The core of the method is based on an Interior Penalty Discontinuous Galerkin discretization of the vector wave equation. With an appropriate penalty function a spurious-free and high accuracy method is achieved. We prove the efficiency of the proposed method in several waveguides, including intricate optical waveguides with leaky modes and also on a plasmonic waveguide. We compare our results with the state-of-the-art mode solvers described in the literature. We also discuss the importance of this new approach.