Gradient-based nodal limiters for artificial diffusion operators in finite element schemes for transport equations

This paper presents new linearity-preserving nodal limiters for enforcing discrete maximum principles in continuous (linear or bilinear) finite element approximations to transport problems with steep fronts. In the process of algebraic flux correction, the oscillatory antidiffusive part of a high-order base discretization is decomposed into a set of internodal fluxes and constrained to be local extremum diminishing. The proposed nodal limiter functions are designed to be continuous and satisfy the principle of linearity preservation which implies the preservation of second-order accuracy in smooth regions. The use of limited nodal gradients makes it possible to circumvent angle conditions and guarantee that the discrete maximum principle holds on arbitrary meshes. A numerical study is performed for linear convection and anisotropic diffusion problems on uniform and distorted meshes in two space dimensions.