On the design of higher-order FEM satisfying the discrete maximum principle

A fully algebraic approach to the design of nonlinear high-resolution schemes is revisited and extended to quadratic finite elements. The matrices resulting from a standard Galerkin discretization are modified so as to satisfy sufficient conditions of the discrete maximum principle for nodal values. In order to provide mass conservation, the perturbation terms are assembled from skew-symmetric internodal fluxes which are redefined as a combination of firstand second-order divided differences. The new approach to the construction of artificial diffusion operators is combined with a node-oriented limiting strategy. The resulting algorithm is applied to P1 and P2 approximations of stationary convection-diffusion equations in 1D/2D.