A nonlinear consistent penalty method for positivity preservation in the finite element approximation of the transport equation

Erik Burman (1) , Alexandre Ern (2, 3)

1 UCL - University College of London [London]
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

We devise and analyze a new stabilized finite element method to solve the first-order transport (or advection-reaction) equation. The method combines the usual Galerkin/Least-Squares approach to achieve stability with a nonlinear consistent penalty term inspired by recent discretizations of contact problems to weakly enforce a positivity condition on the discrete solution. We prove the existence and the uniqueness of the discrete solution. Then we establish quasi-optimal error estimates for smooth solutions bounding the usual error terms in the Galerkin/Least-Squares error analysis together with the violation of the maximum principle by the discrete solution. Numerical examples are presented to illustrate the performances of the method.

Mots clés: consistent penalty; positivity preserving; discrete maximum principle; transport equation; stabilized finite element method

Domaines: Analyse numérique [math.NA]

Erik Burman, Alexandre Ern. A nonlinear consistent penalty method for positivity preservation in the finite element approximation of the transport equation. 2016. ⟨hal-01383295v1⟩