<p>This work introduces a numerical scheme designed to solve convection-diffusion complementarity problems. The proposed method integrates a Lagrange–Galerkin approach, which discretizes the material derivative along characteristic paths, with a mixed finite element method. To solve the resulting complementarity saddle point system at the discrete level, an effective primal-dual active set solver is employed. The effectiveness and performance of this approach are validated through rigorous benchmarking against known analytical solutions from the literature.</p>