On finite element method-flux corrected transport stabilization for advection-diffusion problems in a partial differential-algebraic framework

An extension of the finite element method-flux corrected transport stabilization for hyperbolic problems in the context of partial differential-algebraic equations is proposed. Given a local extremum diminishing property of the spatial discretization, the positivity preservation of the one-step θ-scheme when applied to the time integration of the resulting differential-algebraic equation is shown, under a mild restriction on the time step size. As a crucial tool in the analysis, the Drazin inverse and the corresponding Drazin ordinary differential equation are explicitly derived. Numerical results are presented for non-constant and time-dependent boundary conditions in one space dimension and for a two-dimensional advection problem with a sinusoidal inflow boundary condition and the advection proceeding skew to the mesh.