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Extrusion contraction upwind schemes for convection-diffusion problems

by H. Heumann and R. Hiptmair

(Report number 2008-30)

Abstract
The calculus of differential forms allows to state general convection diffusion problems using the notion of Lie derivatives. We apply the Cartan formula for Lie derivatives and the contraction extrusion dualism to propose an upwind discretization procedure based on discrete differential forms. We discuss this procedure in detail for $0$-forms and the scalar convection-diffusion boundary value problem. In the case of linear ansatz spaces one of the stable schemes derived with this procedure coincides with Tabata's upwind scheme. In the case of quadratic ansatz spaces we get a new scheme that enjoys stability properties similar to SUPG.

Keywords:

@Techreport{HH08_35,
  author = {H. Heumann and R. Hiptmair},
  title = {Extrusion contraction upwind schemes for convection-diffusion problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-30.pdf },
  year = {2008}
}

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