Robust error estimates in weak norms for advection dominated transport problems with rough data

TL;DR

This paper develops robust error estimates in weak norms for convection-diffusion problems with multiscale, rough velocity data, ensuring stability and accuracy independent of Péclet number and solution regularity.

Contribution

It introduces a multiscale decomposition approach for deriving a posteriori and a priori error estimates that are robust for advection-dominated problems with rough data.

Findings

Error estimates are independent of Péclet number.

Estimates are robust with respect to solution regularity.

Method improves stability for multiscale, rough velocity fields.

Abstract

We consider mixing problems in the form of transient convection--diffusion equations with a velocity vector field with multiscale character and rough data. We assume that the velocity field has two scales, a coarse scale with slow spatial variation, which is responsible for advective transport and a fine scale with small amplitude that contributes to the mixing. For this problem we consider the estimation of filtered error quantities for solutions computed using a finite element method with symmetric stabilization. A posteriori error estimates and a priori error estimates are derived using the multiscale decomposition of the advective velocity to improve stability. All estimates are independent both of the P\'eclet number and of the regularity of the exact solution.