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

Paola F. Antonietti, Michele Botti, Alessandra Cancrini, Ilario Mazzieri

TL;DR

The paper develops a polytopal discontinuous Galerkin method for the unsteady Stokes problem in a pseudo-stress formulation, establishing well-posedness of the continuous problem and deriving stability and convergence results for both semi- and fully-discrete schemes. It integrates PolydG spatial discretization on polygonal/polyhedral meshes with a $ heta$-time discretization, providing a priori error estimates in a suitable discrete norm. Numerical tests, including a verification case and a flow-around-a-cylinder application, validate the theoretical results and demonstrate practical applicability and computational efficiency. The work advances polytopal DG techniques for stress-velocity-pressure formulations and suggests future extensions to coupling with Biot systems and non-Newtonian fluid models.

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.

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

TL;DR

The paper develops a polytopal discontinuous Galerkin method for the unsteady Stokes problem in a pseudo-stress formulation, establishing well-posedness of the continuous problem and deriving stability and convergence results for both semi- and fully-discrete schemes. It integrates PolydG spatial discretization on polygonal/polyhedral meshes with a -time discretization, providing a priori error estimates in a suitable discrete norm. Numerical tests, including a verification case and a flow-around-a-cylinder application, validate the theoretical results and demonstrate practical applicability and computational efficiency. The work advances polytopal DG techniques for stress-velocity-pressure formulations and suggests future extensions to coupling with Biot systems and non-Newtonian fluid models.

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.
Paper Structure (21 sections, 9 theorems, 70 equations, 7 figures)

This paper contains 21 sections, 9 theorems, 70 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The model problem
  3. Notation
  4. Unsteady Stokes problem
  5. Pseudo-stress weak formulation
  6. Well-posedness of the continuous problem
  7. Stability analysis of the continuous problem
  8. Numerical discretization
  9. PolydG semi-discrete formulation
  10. Fully-discrete formulation
  11. Stability analysis for the semi- and fully-discrete formulations
  12. Stability analysis of the semi-discrete problem
  13. Stability analysis of the fully-discrete problem
  14. Error analysis of the fully-discrete problem
  15. Numerical results
  16. ...and 6 more sections

Key Result

Proposition 1

Let $V_m$ be a seminorm space obtained from a symmetric and non-negative bilinear form $m(\cdot,\cdot)$ and let $\mathcal{M}:V_m\to V_m'$ be the linear operator defined by $\mathcal{M}x(y) = m(x,y)$ for any $x,y \in V_m$. Let $D$ be a dense subspace of $\ V_m$ and $\mathcal{L}: D \rightarrow V_m'$ b

Figures (7)

  • Figure 1: Test case of Section \ref{['sec:unsteady']}. Considered Polytopal meshes: (a) 100 elements ($h \approx 0.1759$), (b) 200 elements ($h \approx 0.1260$), (c) 400 elements ($h \approx 0.0909$), and (d) 800 elements ($h \approx 0.0637$). Boundary edges are highlighted in blue for Dirichlet conditions, and red for Neumann conditions.
  • Figure 2: Test case of Section \ref{['sec:unsteady']}. Left: log-log plot of the computed error $\| \bm \sigma -\bm \sigma_h \|_{E}$ as a function of the mesh size $h$ for $p = 1,...,4$, by fixing $T=0.25$ and $\Delta t = 1.e-3$. Right: log-log plot of the computed error $\| \bm \sigma -\bm \sigma_h \|_{E}$ as a function of time step $\Delta t$ fixing the polynomial degree $p = 4$, and $h \approx 0.0909$.
  • Figure 3: Left: Test case of Section \ref{['sec:unsteady']}. Semi-logy plot of the computed error $\| \bm \sigma -\bm \sigma_h \|_{E}$ as a function of the polynomial degree $p = 1,...,6$, by fixing $T=0.25$ and $\Delta t = 1.e-3$ and $100$ mesh elements. Right: Test case of Section \ref{['sec-sub:fac']}. Polygonal mesh with $2000$ elements of the rectangular domain $\Omega$ with a circular hole. Dirichlet boundary is highlighted in blue (up, left, bottom), while the Neumann boundary is in red (right).
  • Figure 4: Test case of Section \ref{['sec:recovery']}. Computed velocity field ($u_{x}$ and $u_{y}$) and computed pressure field ($p$) at final time $T = 1$.
  • Figure 5: Test case of Section \ref{['sec-sub:fac']}. Computed stress tensor field $\bm \sigma_h$ at final time $T=1$.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Remark 1: Data regularity
  • Remark 2: Pressure and velocity recovery
  • Proposition 1
  • Lemma 1
  • Theorem 1: Stability estimate
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 10 more