Convergence to suitable weak solutions for a finite element approximation of the Navier-Stokes equations with numerical subgrid scale modeling

TL;DR

This paper proves that a variational multiscale finite element method for the Navier-Stokes equations produces suitable weak solutions, even with pressure stabilization and equal-order elements, advancing numerical analysis of fluid dynamics.

Contribution

It demonstrates the suitability of weak solutions obtained via a specific subgrid scale model with orthogonality and time-tracking, addressing pressure stabilization issues.

Findings

Weak solutions are suitable in the sense of Scheffer.

The method works with equal-order velocity and pressure elements.

Pressure stabilization does not hinder the suitability of solutions.

Abstract

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.