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Structure-preserving Variational Multiscale Stabilization of the Incompressible Navier-Stokes Equations

Kevin Dijkstra, Deepesh Toshniwal

TL;DR

This work develops a structure-preserving, FEEC-based variational multiscale stabilization for the incompressible Navier–Stokes equations in a vorticity–velocity–pressure formulation. By employing elementwise bubble-based fine scales that form a discrete de Rham complex and a Stokes projector coupling to coarse FEEC spaces, the method achieves residual-based stabilization with energetic stability and optimal convergence, while enabling parallelizable fine-scale solves. Theoretical results establish stability, uniqueness, and convergence for the Oseen (linearised) problem under CFL-like conditions, and numerical experiments in 2D validate energy dissipation through fine scales, optimal rates, and improved behavior on under-resolved meshes. The approach supports both low-regularity and high-regularity discretizations and shows potential for extensions to MHD and adaptive refinement using the fine-scale solution as an a posteriori indicator.

Abstract

This paper introduces a Variational Multiscale Stabilization (VMS) formulation of the incompressible Navier--Stokes equations that utilizes the Finite Element Exterior Calculus (FEEC) framework. The FEEC framework preserves the geometric and topological structure of continuous spaces and PDEs in the discrete spaces and model, and helps build stable and convergent discretizations. For the Navier-Stokes equations, this structure is encoded in the de Rham complex. In this work, we consider the vorticity-velocity-pressure formulation discretized within the FEEC framework. We model the effect of the unresolved scales on the finite-dimensional solution by introducing appropriate fine-scale governing equations, which we also discretize using the FEEC approach. This preserves the structure of the continuous problem in both the coarse- and fine-scale solutions; for instance, both the coarse- and fine-scale velocities are pointwise incompressible. We demonstrate that the resulting formulation is residual-based, energetically stable, and optimally convergent. Moreover, our fine-scale model provides an efficient computational approach: by decoupling fine-scale problems across elements, they can be solved in parallel. In fact, the fine-scale equations can be eliminated during matrix assembly, leading to a VMS formulation in which the problem size is governed solely by the coarse-scale discretization. Finally, the proposed formulation applies to both the lowest regularity discretizations of the de Rham complex and high-regularity isogeometric discretizations. We validate our theoretical results through numerical experiments, simulating both steady-, unsteady-, viscous-, and inviscid-flow problems. These tests show that the stabilized solutions are qualitatively better than the unstabilized ones, converge at optimal rates, and, as the mesh is refined, the stabilization is asymptotically turned off.

Structure-preserving Variational Multiscale Stabilization of the Incompressible Navier-Stokes Equations

TL;DR

This work develops a structure-preserving, FEEC-based variational multiscale stabilization for the incompressible Navier–Stokes equations in a vorticity–velocity–pressure formulation. By employing elementwise bubble-based fine scales that form a discrete de Rham complex and a Stokes projector coupling to coarse FEEC spaces, the method achieves residual-based stabilization with energetic stability and optimal convergence, while enabling parallelizable fine-scale solves. Theoretical results establish stability, uniqueness, and convergence for the Oseen (linearised) problem under CFL-like conditions, and numerical experiments in 2D validate energy dissipation through fine scales, optimal rates, and improved behavior on under-resolved meshes. The approach supports both low-regularity and high-regularity discretizations and shows potential for extensions to MHD and adaptive refinement using the fine-scale solution as an a posteriori indicator.

Abstract

This paper introduces a Variational Multiscale Stabilization (VMS) formulation of the incompressible Navier--Stokes equations that utilizes the Finite Element Exterior Calculus (FEEC) framework. The FEEC framework preserves the geometric and topological structure of continuous spaces and PDEs in the discrete spaces and model, and helps build stable and convergent discretizations. For the Navier-Stokes equations, this structure is encoded in the de Rham complex. In this work, we consider the vorticity-velocity-pressure formulation discretized within the FEEC framework. We model the effect of the unresolved scales on the finite-dimensional solution by introducing appropriate fine-scale governing equations, which we also discretize using the FEEC approach. This preserves the structure of the continuous problem in both the coarse- and fine-scale solutions; for instance, both the coarse- and fine-scale velocities are pointwise incompressible. We demonstrate that the resulting formulation is residual-based, energetically stable, and optimally convergent. Moreover, our fine-scale model provides an efficient computational approach: by decoupling fine-scale problems across elements, they can be solved in parallel. In fact, the fine-scale equations can be eliminated during matrix assembly, leading to a VMS formulation in which the problem size is governed solely by the coarse-scale discretization. Finally, the proposed formulation applies to both the lowest regularity discretizations of the de Rham complex and high-regularity isogeometric discretizations. We validate our theoretical results through numerical experiments, simulating both steady-, unsteady-, viscous-, and inviscid-flow problems. These tests show that the stabilized solutions are qualitatively better than the unstabilized ones, converge at optimal rates, and, as the mesh is refined, the stabilization is asymptotically turned off.

Paper Structure

This paper contains 28 sections, 10 theorems, 106 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structure-preserving discretizations for incompressible Navier--Stokes
  3. Function spaces
  4. Continuous weak formulation
  5. The de Rham complex
  6. Variational Multiscale Stabilization via discontinuous fine-scales
  7. Modelling assumptions
  8. Stabilized semi-discrete formulation
  9. Energetically-stable fully-discrete formulation
  10. General boundary conditions
  11. Stability, uniqueness and convergence of the Oseen equations
  12. Stability and uniqueness
  13. Convergence analysis
  14. Numerical Results
  15. Choices of stabilization parameter
  16. ...and 13 more sections

Key Result

Lemma 3.1

The solutions to Method method:ns-stabilization, evolve over time as:

Figures (6)

  • Figure 1: The $L^2$ error convergence rates of the steady unregularized lid-driven cavity flow are shown above. This is performed for coarse-scale degrees $\boldsymbol{k} = (k,k)$ where $k=1,2,3$, with different degrees indicated by different colours. The different choices of fine-scale degree $\boldsymbol{k}' = (k',k')$ are instead indicated by different markers. Additionally, results without stabilization are shown with dashed lines. For a given coarse-scale degree, the lines overlap, indicating that the stabilized solution does not spoil the accuracy or the convergence rate. As later tests show, the stabilized solution does, however, improve the quality of the solution for under-resolved meshes.
  • Figure 2: For the inviscid case, the above plots show the changes in the coarse scale, fine scale, and total kinetic energies. The results for the mesh sizes $n=8,12,16,24,32,48$ with time step $\Delta t = 0.001$, with and without stabilization, are shown for degrees $k=2$ and $k'=3$.
  • Figure 3: The vorticity reference solution for the viscid shear layer roll up ($\mathrm{Re} = 1600$), solved with cubic B-splines on a mesh of $128\times128$ elements. The contour lines are given by $-6,-5,\dots,5,6$.
  • Figure 4: The differences of the kinetic energies compared to the reference ($n=128$) in the viscous case for $\mathrm{Re}=1600$ are shown. The results are for the mesh sizes $n=8,16,24,32$ with time step $\Delta t = 0.01$, degrees $k=3$ and $k'=4$, and for full-CN and semi-CN stabilization choices. Note that the semi-CN and full-CN stabilization results are so close that they overlap.
  • Figure 5: The vorticity at $T=8.0$ for the viscous case ($\mathrm{Re}=1600$). Observe that the stabilized solutions are a better qualitative approximation of the high-resolution solution. The contour levels are given by $-6, -5, \dots, 5, 6$.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • Remark 4.1
  • Lemma 4.2
  • ...and 16 more