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

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.
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Submitted on : Friday, November 24, 2017 - 4:59:25 PM
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Erik Burman, Alexandre Ern. A nonlinear consistent penalty method weakly enforcing positivity in the finite element approximation of the transport equation. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2017, 320, pp.122-132. ⟨10.1016/j.cma.2017.03.019⟩. ⟨hal-01383295v2⟩

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