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

Alejandro Allendes; Gabriel R. Barrenechea; Richard Rankin

doi:10.1137/16M1092763

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

Alejandro Allendes, Gabriel R. Barrenechea, Richard Rankin

Mathematics And Statistics

Research output: Contribution to journal › Article

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.

Language	English
Pages	A1903–A1927
Number of pages	25
Journal	SIAM Journal on Scientific Computing
Volume	39
Issue number	5
DOIs	10.1137/16M1092763
Publication status	Published - 12 Sep 2017

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Convection-diffusion-reaction Equation

Error Estimation

Positivity

Error analysis

Discretization

Discrete Maximum Principle

A Posteriori Error Estimators

Maximum principle

Term

Linearity

Error Bounds

Lipschitz

Numerical Experiment

Upper bound

Estimator

Norm

Testing

Requirements

Convection

Experiments

Keywords

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

Cite this

Allendes, A., Barrenechea, G. R., & Rankin, R. (2017). Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation. SIAM Journal on Scientific Computing, 39(5), A1903–A1927. https://doi.org/10.1137/16M1092763

Allendes, Alejandro ; Barrenechea, Gabriel R. ; Rankin, Richard. / Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation. In: SIAM Journal on Scientific Computing. 2017 ; Vol. 39, No. 5. pp. A1903–A1927.

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Allendes, A, Barrenechea, GR & Rankin, R 2017, 'Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation' SIAM Journal on Scientific Computing, vol. 39, no. 5, pp. A1903–A1927. https://doi.org/10.1137/16M1092763

Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation. / Allendes, Alejandro; Barrenechea, Gabriel R.; Rankin, Richard.

In: SIAM Journal on Scientific Computing, Vol. 39, No. 5, 12.09.2017, p. A1903–A1927.

Research output: Contribution to journal › Article

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