Maximum norm a posteriori error estimates for convection-diffusion problems

Research output: Contribution to journalResearch articleContributedpeer-review



We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection–diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the convective derivative of the error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error, which functions as a maximum-norm counterpart to the dual norm used in the energy norm setting. The total error is then defined as the sum of this seminorm, the maximum norm of the error and data oscillation. The natural maximum norm residual error estimator is shown to be equivalent to this total error notion, with constant independent of singular perturbation parameters. These estimates are proved under the assumption that certain natural estimates hold for the Green’s function for the problem at hand. Numerical experiments confirm that our estimators effectively capture the maximum-norm error behavior for singularly perturbed problems, and can effectively drive adaptive refinement in order to capture layer phenomena.


Original languageEnglish
Pages (from-to)2562-2584
Number of pages23
JournalIMA journal of numerical analysis
Issue number5
Publication statusPublished - 20 Feb 2023

External IDs

Mendeley e3ece1e8-44e8-3db7-a69f-e19015ed36ab
unpaywall 10.1093/imanum/drad001
ORCID /0000-0002-2458-1597/work/142239743
Scopus 85174540669


DFG Classification of Subject Areas according to Review Boards


  • A posteriori error estimate, Convection-diffusion, Maximum norm, Singular perturbation, a posteriori error estimate, singular perturbation, maximum norm, convection-diffusion

Library keywords