Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation

Alejandro Allendes, Gabriel R. Barrenechea, Richard Rankin

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Abstract

This work is devoted to the proposal, analysis, and numerical testing of a fully computable a posteriori error bound for a class of nonlinear discretizations of the convection-diffusion-reaction equation. The type of discretization we consider is nonlinear, since it has been built with the aim of preserving the discrete maximum principle. Under mild assumptions on the stabilizing term, we obtain an a posteriori error estimator that provides a certified upper bound on the norm of the error. Under the additional assumption that the stabilizing term is both Lipschitz continuous and linearity preserving, the estimator is shown to be locally efficient. We present examples of discretizations that satisfy these two requirements, and the theory is illustrated by several numerical experiments in two and three space dimensions.
Original languageEnglish
Pages (from-to)A1903–A1927
Number of pages25
JournalSIAM Journal on Scientific Computing
Volume39
Issue number5
DOIs
Publication statusPublished - 12 Sep 2017

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Keywords

  • posteriori error estimation
  • shock-capturing method
  • fully computable error bound
  • algebraic flux correction

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