A Local Discontinuous Galerkin approximation for the $p$-Navier-Stokes system, Part II: Convergence rates for the velocity

TL;DR

This paper establishes optimal convergence rates for the Local Discontinuous Galerkin method applied to $p$-Navier-Stokes and $p$-Stokes systems with $p>2$, supported by numerical validation.

Contribution

It provides the first rigorous proof of convergence rates for LDG approximations of $p$-Navier-Stokes systems, extending previous work to nonlinear, non-Newtonian flows.

Findings

Optimal convergence rates for linear ansatz functions.

Numerical experiments confirm theoretical results.

Applicable to systems with $p$ in (2,∞).

Abstract

In the present paper, we prove convergence rates for the Local Discontinuous Galerkin (LDG) approximation, proposed in Part I of the paper, for systems of $p$-Navier-Stokes type and $p$-Stokes type with $p\in (2,\infty)$. The convergence rates are optimal for linear ansatz functions. The results are supported by numerical experiments.