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

Allendes, Alejandro and Barrenechea, Gabriel R. and Rankin, Richard (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. ISSN 1064-8275

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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.