Numerical simulation of rotating devices (e.g., stirred tanks, spinning disk mixers) in the Finite Element Method (FEM) context requires strategies to handle complex geometry and accurate flow representation. The Mortar Element Method (MEM) [1] is a domain decomposition technique that imposes field continuity by penalizing its jump at the generated rotor-stator interface. Common formulations involve a Continuous Galerkin (CG) framework with Lagrange multipliers [2], or a Discontinuous Galerkin (DG) formulation with an interior penalty method [3]. Due to the necessity of reconstructing mortar elements, computational efficiency is deemed fundamental. Exploring the MEM accuracy through robust, efficient, and stable numerical implementation is an ongoing challenge in Computational Fluid Dynamics (CFD). Thus, we present a novel MEM-CFD model to simulate rotating devices based on a coupled CG-DG approach. The model is inserted within a matrix-free solver, a high-performance computing technique that allows high-order approximation while providing order of magnitude reduction in simulation computational time [4]. The numerical implementation is integrated in Lethe [5], an open-source CFD software that solves the incompressible Navier-Stokes equations. The discretized domain is solved by the CG method, while the interface discontinuity is modeled with a DG-like formulation. The model is employed in two- and three-dimensional problems in which curved mortar elements connects the rotor-stator domains. Optimal convergence rates (p ≥ 2, p is the polynomial order) are obtained, and errors are constant regardless of rotor rotating angles. Stability is ensured through the Streamline-Upwind and Pressure-Stabilizing Petrov-Galerkin (SUPG-PSPG) methods. We discuss the compatibility of such stabilization with the weakly imposed continuity at the mortar interface, and explore enrichment functions and grad-jump stabilization alternatives to recover optimal convergence for linear (p = 1) approximation. Scalability of the MEM implementation highlights its low computational cost. The derived model is a robust tool for simulating rotating devices using implicit Large-Eddy Simulations (LES).