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

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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