On the design of flux limiters for finite element discretizations with a consistent mass matrix

The algebraic flux correction (AFC) paradigm is extended to finite element discretizations with a consistent mass matrix. A nonoscillatory low-order scheme is constructed by resorting to mass lumping and conservative elimination of negative off-diagonal coefficients from the discrete transport operator. In order to recover the high accuracy of the original Galerkin scheme, a limited amount of compensating antidiffusion is added in regions where the solution is sufficiently smooth. The raw antidiffusive fluxes, which include a contribution of the consistent mass matrix, are limited node-by-node so as to satisfy algebraic constraints imposed on the discrete solution. The proposed limiting strategy combines the advantages of multidimensional FEM-FCT and FEM-TVD schemes introduced previously. Its performance is illustrated by application to scalar convection problems in 1D and 2D.