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

• ID: 2713, RIV: 10318467
• ISSN: 0272-4979, ISBN: not specified
• source: IMA Journal of Numerical Analysis
• keywords: finite element method; convection-diffusion equation; algebraic flux correction; discrete maximum principle; fixed-point iteration; solvability of linear subproblems; solvability of nonlinear pr
• authors: Gabriel R. Barrenechea, Volker John, Petr Knobloch
• authors from KNM: Knobloch Petr

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 nonexistence 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.