WIAS Preprint No. 1916, (2014)

Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in 1D



Authors

  • Barrenechea, Gabriel R.
  • John, Volker
    ORCID: 0000-0002-2711-4409
  • Knobloch, Petr

2010 Mathematics Subject Classification

  • 65N06 65N30

Keywords

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

DOI

10.20347/WIAS.PREPRINT.1916

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.

Appeared in

  • IMA J. Numer. Anal., 35:4 (2015), pp. 1729--1756, changed title: Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension.

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