A Local Discontinuous Galerkin approximation for the $p$-Navier-Stokes system, Part I: Convergence analysis

TL;DR

This paper introduces a Local Discontinuous Galerkin method for the p-Navier-Stokes system, providing convergence analysis, stability, and well-posedness proofs, along with a novel discretization of the convective term.

Contribution

It develops a new LDG approximation for p-Navier-Stokes equations, including a novel DG discretization and a non-conforming pseudo-monotonicity theory, with convergence and stability results.

Findings

Proved well-posedness and stability of the LDG method.

Established weak convergence of the approximation.

Developed a new DG discretization for the convective term.

Abstract

In the present paper, we propose a Local Discontinuous Galerkin (LDG) approximation for fully non-homogeneous systems of $p$-Navier-Stokes type. On the basis of the primal formulation, we prove well-posedness, stability (a priori estimates), and weak convergence of the method. To this end, we propose a new DG discretization of the convective term and develop an abstract non-conforming theory of pseudo-monotonicity, which is applied to our problem. We also use our approach to treat the $p$-Stokes problem.