Adaptive Grids in the Context of Algebraic Stabilizations for Convection-Diffusion-Reaction Equations.

Three algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations are studied on adaptively refined grids. These schemes are the algebraic flux correction (AFC) scheme with the Kuzmin limiter, the AFC scheme with the Barrenechea-John-Knobloch limiter, and the recently proposed monotone upwind-type algebraically stabilized method. Both conforming closure of the refined grids and grids with hanging vertices are considered. A nonstandard algorithmic step becomes necessary before these schemes can be applied on grids with hanging vertices. The assessment of the schemes is performed with respect to the satisfaction of the global discrete maximum principle, the accuracy, e.g., smearing of layers, and the efficiency in solving the corresponding nonlinear problems.