Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in 2d

TL;DR

This paper introduces a stabilized Petrov-Galerkin method for convection-diffusion equations that ensures stability and near-optimal convergence on coarse meshes by leveraging localized multiscale correctors.

Contribution

It develops a variational multiscale stabilization technique with localized correctors for efficient and stable numerical solutions of convection-dominated problems.

Findings

Exponential decay of fine-scale correctors is rigorously justified.

The method guarantees stability and quasi-optimal convergence on coarse meshes.

Localization reduces computational cost while maintaining accuracy.

Abstract

We formulate a stabilized quasi-optimal Petrov-Galerkin method for singularly perturbed convection-diffusion problems based on the variational multiscale method. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, is rigorously justified. Under moderate assumptions, this stabilization guarantees stability and quasi-optimal rate of convergence for arbitrary mesh P\'eclet numbers on fairly coarse meshes at the cost of additional inter-element communication.