Maximum norm a posteriori error estimates for convection-diffusion problems

TL;DR

This paper develops residual-type a posteriori error estimates in the maximum norm for convection-diffusion problems, including singularly perturbed cases, enabling effective adaptive refinement and error control.

Contribution

It introduces a maximum-norm residual estimator incorporating a mesh-dependent weighted seminorm, extending energy norm analysis to the maximum norm for convection-diffusion problems.

Findings

Estimator accurately captures maximum-norm errors in singularly perturbed problems.

Numerical experiments validate the effectiveness of the estimator in adaptive refinement.

Constant independence from perturbation parameters ensures robustness.

Abstract

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 Verf\"{u}rth 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.