Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation

We investigate stabilized Galerkin approximations of linear and nonlinear convection-diffusion-reaction equations. We derive nonlinear streamline and cross-wind diffusion methods that guarantee a discrete maximum principle for strictly acute meshes and first order polynomial interpolation. For pure convection-diffusion problems, the discrete maximum principle is achieved using a nonlinear cross-wind diffusion factor that depends on the angle between the discrete solution and the flow velocity. For convection-diffusion-reaction problems, two methods are considered: residual based, isotropic diffusion and the previous nonlinear cross-wind diffusion factor supplemented by additional isotropic diffusion scaling as the square of the mesh size. Practical versions of the present methods suitable for numerical implementation are compared to previous discontinuity capturing schemes lacking theoretical justification. Numerical results are investigated in terms of both solution quality (violation of maximum principle, smearing of internal layers) and computational costs.