Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations.

*(English)*Zbl 1233.65065When linear evolutionary convection-diffusion-reaction partial differential equations are discretized by standard finite element methods, the corresponding approximations typically exhibit spurious oscillations. In consequence, stabilization techniques are applied to recover the correct physical behavior. The paper under review deals with one such finite element stabilization, namely the streamline-upwind Petrov–Galerkin (or SUPG for short) technique, in particular by analyzing different ways to get error estimates and conditions on the parameters appearing in the procedure.

As a first step, the backward Euler scheme is considered as the temporal discretization. This simple setting already exhibits some of the main difficulties involved and allows to get error estimates under conditions that couple the choice of the stabilization parameters to the length of the time step in such a way that the SUPG stabilization vanishes in the time-continuous limit. Numerical experiments suggest, however, that this behavior is not correct. This fact leads naturally to try to derive estimates in which the stabilization parameters do not depend on the length of the step size. The authors show that this is indeed possible when the convection and reaction terms are both independent of time. Thus, error estimates are derived for the time-continuous case and the fully discrete method with both the backward Euler and Crank–Nicolson schemes. The collected numerical examples illustrate the theoretical results.

As a first step, the backward Euler scheme is considered as the temporal discretization. This simple setting already exhibits some of the main difficulties involved and allows to get error estimates under conditions that couple the choice of the stabilization parameters to the length of the time step in such a way that the SUPG stabilization vanishes in the time-continuous limit. Numerical experiments suggest, however, that this behavior is not correct. This fact leads naturally to try to derive estimates in which the stabilization parameters do not depend on the length of the step size. The authors show that this is indeed possible when the convection and reaction terms are both independent of time. Thus, error estimates are derived for the time-continuous case and the fully discrete method with both the backward Euler and Crank–Nicolson schemes. The collected numerical examples illustrate the theoretical results.

Reviewer: Fernando Casas (Castellon)

##### MSC:

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35K57 | Reaction-diffusion equations |