# Variational Multiscale Modeling with Discontinuous Subscales: Analysis and Application to Scalar Transport

## TL;DR

This paper introduces a variational multiscale method with discontinuous subscales for scalar transport, demonstrating stability, optimal convergence, and robustness across different discretizations and flow regimes.

## Contribution

It develops a novel multiscale approach using discontinuous Galerkin subscales, providing theoretical analysis and practical validation for scalar transport problems.

## Key findings

- Method is stable and accurate in advective regimes.
- Achieves optimal convergence rates in the SUPG norm.
- Robust with respect to Peclet number when using rich subscale spaces.

## Abstract

We examine a variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method. We establish stability and convergence results for the methodology as applied to the scalar transport problem, and we prove that the method exhibits optimal convergence rates in the SUPG norm and is robust with respect to the Peclet number if the discontinuous subscale approximation space is sufficiently rich. We apply the method to isogeometric NURBS discretizations of steady and unsteady transport problems, and the corresponding numerical results demonstrate that the method is stable and accurate in the advective limit even when low-order discontinuous subscale approximations are employed.