Error estimates for the patch bubble method for convection-dominated channel flow problem

TL;DR

This paper derives error estimates for the BMZ residual-free bubble method applied to convection-diffusion equations in convection-dominated regimes, supported by numerical validation.

Contribution

It provides the first error estimates for the BMZ residual-free bubble method in convection-dominated flows, including element and residual-free bubbles on patches.

Findings

Error estimates remain valid as diffusion approaches zero.

Numerical experiments confirm theoretical predictions.

The method shows competitive performance in simulations.

Abstract

We present error estimates for the BMZ (Bubble Mesh Zoom) residual-free bubble method applied to a convection-diffusion equation in the convection-dominated regime. The method incorporates both element bubbles and residual-free bubbles supported on patches of two adjacent elements. We focus on the case of a parallel flow in a square domain and derive error estimates in an energy norm that remain valid as diffusion becomes small. The theoretical findings are corroborated by numerical experiments, which also exhibit a competitive performance of the method.