Stability analysis of polytopic Discontinuous Galerkin approximations of the Stokes problem with applications to fluid-structure interaction problems

TL;DR

This paper conducts a stability analysis of the PolyDG method for the Stokes problem on complex meshes, establishing stability conditions, error estimates, and applying findings to fluid-structure interaction simulations.

Contribution

It provides a novel stability analysis of PolyDG for Stokes equations, including inf-sup conditions, error estimates, and application to fluid-structure interaction problems.

Findings

Validated stability of PolyDG on polygonal meshes

Derived a priori hp-error estimates for the method

Demonstrated stability in fluid-structure interaction simulations

Abstract

We present a stability analysis of the Discontinuous Galerkin method on polygonal and polyhedral meshes (PolyDG) for the Stokes problem. In particular, we analyze the discrete inf-sup condition for different choices of the polynomial approximation order of the velocity and pressure approximation spaces. To this aim, we employ a generalized inf-sup condition with a pressure stabilization term. We also prove a priori hp-version error estimates in suitable norms. We numerically check the behaviour of the inf-sup constant and the order of convergence with respect to the mesh configuration, the mesh-size, and the polynomial degree. Finally, as a relevant application of our analysis, we consider the PolyDG approximation for a fluid-structure interaction problem and we numerically explore the stability properties of the method.