A nonlinear consistent penalty method for positivity preservation in the finite element approximation of the transport equationReport as inadecuate




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1 UCL - University College of London London 2 CERMICS - Centre d-Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Abstract : 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.

Keywords : discrete maximum principle transport equation stabilized finite element method consistent penalty positivity preserving





Author: Erik Burman - Alexandre Ern -

Source: https://hal.archives-ouvertes.fr/



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