On the stability and convergence of Discontinuous Galerkin schemes for incompressible flow

TL;DR

This paper proves the stability and convergence of Discontinuous Galerkin schemes for non-Newtonian incompressible fluid models, ensuring solutions remain bounded and approximate true solutions under specific conditions.

Contribution

It establishes uniform stability and weak convergence of DG discretizations for non-Newtonian models with p-structure, including new Gagliardo-Nirenberg inequalities on DG spaces.

Findings

DG schemes are stable in the L-infinity in time and L2 in space norm.

The numerical scheme converges weakly to the true weak solution.

New Gagliardo-Nirenberg inequalities for DG spaces are derived.

Abstract

The property that the velocity $\boldsymbol{u}$ belongs to $L^\infty(0,T;L^2(\Omega)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\infty(0,T;L^2(\Omega)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to the RadauIIA Implicit Runge-Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest.