A nonlinear consistent penalty method weakly enforcing positivity in the finite element approximation of the transport equation

作者：Burman Erik^*; Ern Alexandre

来源：Computer Methods in Applied Mechanics and Engineering, 2017, 320: 122-132.

DOI：10.1016/j.cma.2017.03.019

• Stabilized finite element method
• Consistent penalty
• Positivity preserving
• Transport equation
• Discrete maximum principle

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.

• 出版日期2017-6-15