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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8 Citations (Scopus)

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.
LanguageEnglish
Number of pages28
JournalIMA Journal of Numerical Analysis
Early online date17 Oct 2014
DOIs
Publication statusPublished - 2014

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Maximum principle
Convection-diffusion Equation
One Dimension
Fluxes
Discrete Maximum Principle
Picard Iteration
State Equation
Nonexistence
Convection
Solvability
Counterexample
Discretization

Keywords

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

Cite this

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title = "Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension",
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.",
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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T1 - Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension

AU - Barrenechea, Gabriel

AU - Volker, John

AU - Knobloch, Petr

PY - 2014

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N2 - 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.

AB - 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.

KW - finite element method

KW - solvability of nonlinear problem

KW - solvability of linear subproblems

KW - fixed point iteration

KW - discrete maximum principle

KW - algebraic flux correction

KW - convection-diffusion equation

U2 - 10.1093/imanum/dru041

DO - 10.1093/imanum/dru041

M3 - Article

JO - IMA Journal of Numerical Analysis

T2 - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

ER -