Computer Methods in Applied Mechanics and Engineering
Computer Methods in Applied Mechanics and Engineering - Special edition on the 20th Anniversary
Stabilized finite element methods for singularly perturbed parabolic problems
Applied Numerical Mathematics
High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
Finite element methods of an operator splitting applied to population balance equations
Journal of Computational and Applied Mathematics
On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
An a posteriori error estimator for an unsteady advection-diffusion-reaction problem
Computers & Mathematics with Applications
Journal of Computational and Applied Mathematics
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Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline-upwind Petrov-Galerkin (SUPG) finite element stabilization of time-dependent convection-diffusion-reaction equations. Exemplarily, it is shown for the SUPG method combined with the backward Euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step. The stabilization vanishes in the time-continuous limit. However, based on numerical experience, this seems not to be the correct behavior. For this reason, the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step. It is shown that such estimates can be obtained in the case of time-independent convection and reaction. An error estimate for the time-continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady-state problem. Analogous estimates are obtained for the fully discrete case using the backward Euler method and the Crank-Nicolson scheme. Numerical studies support the analytical results.