Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension

Gabriel Barrenechea, John Volker, Petr Knobloch

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Abstract

Algebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection–diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method.
Original languageEnglish
Number of pages28
JournalIMA Journal of Numerical Analysis
Early online date17 Oct 2014
DOIs
Publication statusPublished - 2014

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Keywords

  • finite element method
  • solvability of nonlinear problem
  • solvability of linear subproblems
  • fixed point iteration
  • discrete maximum principle
  • algebraic flux correction
  • convection-diffusion equation

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