A polytopal discontinuous Galerkin method for the pseudo-stress formulation of the unsteady Stokes problem

TL;DR

This paper develops a polytopal discontinuous Galerkin method combined with an implicit time scheme to solve the pseudo-stress formulation of the unsteady Stokes problem, with stability and convergence analysis.

Contribution

It introduces a novel PolydG method for the pseudo-stress formulation of unsteady Stokes equations, including stability and convergence proofs.

Findings

Method is stable for semi- and fully-discrete problems.

Convergence estimates are derived and verified.

Applicable to engineering problems involving non-Newtonian flows.

Abstract

This work aims to construct and analyze a discontinuous Galerkin method on polytopal grids (PolydG) to solve the pseudo-stress formulation of the unsteady Stokes problem. The pseudo-stress variable is introduced due to the growing interest in non-Newtonian flows and coupled interface problems, where stress assumes a fundamental role. The space-time discretization of the problem is achieved by combining the PolydG approach with the implicit theta-method time integration scheme. For both the semi- and fully-discrete problems we present a detailed stability analysis. Moreover, we derive convergence estimates for the fully discrete space-time discretization. A set of verification tests is presented to verify the theoretical estimates and the application of the method to cases of engineering interest.