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Numerical Analysis of the Virtual Element Approximation for the Smagorinsky Turbulence Model

Karol L. Cascavita, Francesca Marcon, Maria Strazzullo

arXiv:2609.02433v1math.NA

TL;DR

The paper studies the Smagorinsky model for incompressible Navier–Stokes equations within a virtual element framework, addressing unavailable theoretical analysis. It develops divergence-free VEM theory, proves well-posedness and convergence results, and validates the estimates numerically. The analysis obtains classical h rates and sharper convergence under weaker regularity assumptions.

  • Problem

    The paper addresses the lack of theoretical analysis for the Smagorinsky model within the Virtual Element Method.

  • Method

    The paper combines the Navier–Stokes–Smagorinsky equations with divergence-free VEM discretization and analyzes its discrete problem and error estimates.

  • Results

    The analysis proves well-posedness under small data, recovers classical h convergence rates, and obtains sharper convergence under weaker regularity assumptions, with numerical tests supporting the theory.

  • Takeaways & Limitations

    The work extends virtual-element analysis to the classical Smagorinsky turbulence model and provides a theoretical foundation for its divergence-free discretization.

  • Takeaways & Limitations

    The main treatment is two-dimensional, while the three-dimensional estimates scale with h1/2; the mesh assumptions can be relaxed but are imposed in the presented analysis.

Abstract

from arXiv · show

In this paper, we consider the Smagorinsky model for the Navier-Stokes equations within a virtual element framework. Under the standard assumption of small data, we prove the existence and uniqueness of a solution. Assuming more regularity to the solutions, we derive the known convergence rates $h$ for the a priori error estimates of the Smagorinsky model in two dimensional domains. We additionally prove that divergence-free virtual discretizations provide improved convergence orders, with weaker regularity assumptions than in the finite element literature. We conclude the paper with numerical results that corroborate the theory.

1. Introduction

Turbulence is computationally demanding because resolving multiscale flow features requires fine meshes, motivating turbulence models such as Smagorinsky. This paper supplies missing VEM theory for the model, including well-posedness and error estimates.

  • Motivation: Turbulent-flow simulation is challenging because turbulence is multiscale and chaotic, while mesh sizes must resolve the Kolmogorov scale.The resulting computational cost motivates turbulence-modeling strategies.
  • Motivation: Large Eddy Simulation resolves large-scale eddies while modeling unresolved subgrid-scale effects.Smagorinsky is an early and widely used LES closure.
  • Smagorinsky model: The Smagorinsky model augments Navier–Stokes equations with nonnegative turbulent eddy viscosity that prevents kinetic energy from growing.The term also acts as stabilization for convection-dominated flows or models unresolved subgrid stresses.
  • Research gap: The paper addresses the previously unavailable theoretical analysis of Smagorinsky approximation within the Virtual Element Method.It establishes discrete well-posedness and a priori error estimates.
  • Virtual element framework: Divergence-free VEM enforces incompressibility point-wise by construction and yields pressure-robust velocity error estimates.This framework is designed for incompressible flows on general polytopal meshes.
  • Contributions: The main analysis proves discrete well-posedness under small data, recovers rate h estimates, and obtains H1-error order h2 for k = 2 under weaker regularity assumptions.The h2 result is identified as optimal in the supplied contribution statement.

2. Problem formulation

The paper formulates incompressible Navier–Stokes flow with velocity, pressure, forcing, viscosity, and homogeneous boundary conditions. It also gives divergence-free and skew-symmetric weak formulations and recalls small-data well-posedness.

  • Strong formulation: The strong problem seeks velocity u and pressure p on Ω ⊂ R2 satisfying momentum balance, incompressibility, and homogeneous Dirichlet conditions.The forcing term is f and ν > 0 is the kinematic viscosity.
  • Flow regime: The Reynolds number Re = UL/ν relates inertial and viscous forces, with large values indicating convection-dominated turbulent behavior.U and L denote characteristic velocity and length.
  • Weak formulation: The weak Navier–Stokes formulation couples viscosity, convection, pressure, and forcing through bilinear and trilinear forms.The velocity and pressure spaces are equipped with H1-seminorm and L2 norms, respectively.
  • Well-posedness: Under the small-data assumption, the Navier–Stokes problem has a unique solution (u,p).The velocity additionally satisfies a bound stated in the formulation section.
  • Divergence-free formulation: For divergence-free velocities, the pressure coupling is removed and the weak problem is posed over the divergence-free space Z.The resulting formulation uses νa(u,v) + c(u;u,v) = (f,v)Ω.
  • Skew-symmetric convection: The convective form is replaced by a skew-symmetric form cskew(w;z,v) = 1/2(c(w;z,v) − c(w;v,z)).The resulting equation uses cskew in place of the original convection form.

3. Virtual element setting

The VEM discretization uses polygonal meshes, polynomial projections, enhanced local virtual spaces, and computable degrees of freedom. Its divergence structure enforces exact mass conservation.

  • Virtual element setting: The divergence-free VEM is introduced through mesh assumptions and a discrete setting for polygonal tessellations.Elements T have boundaries and faces, with local diameters hT and hF and a global mesh size.
  • Mesh setting: Each element is assumed star-shaped with respect to a ball and has face diameters comparable to its element diameter.The assumptions use a parameter ρ > 0 and inequalities ρhT ≤ hF ≤ hT.
  • Polynomial setting: The construction uses polynomial spaces Pk(O), scaled monomial bases, and a gradient–orthogonal-complement decomposition of vector polynomials.The orthogonal component is built using x⊥ = (y, −x).
  • Projection operators: The H1 and L2 projection operators map virtual functions and tensors to polynomial spaces needed by the discretization.The H1 projection targets [Pk(T)]2, while the tensor projection is introduced analogously.
  • Local spaces and degrees of freedom: For k ≥ 2, the enhanced local virtual space contains [Pk(T)]2 and is represented through vertex, face, vector-moment, and divergence-moment degrees of freedom.The listed degrees of freedom make the required projectors computable.
  • Global spaces: The global velocity and pressure spaces are defined separately, with conformity imposed on velocity and pressure represented piecewise polynomially.The construction distinguishes the discrete velocity space from the global pressure space.
  • Exact mass conservation: The inclusion div(Vk_h) ⊆ Ph forces exact mass conservation in the discrete velocity space.This is the key divergence property of the formulation.

4. Virtual global forms and weak problem

The section builds the VEM weak formulation for the Navier–Stokes–Smagorinsky equations by combining viscous, coupling, convective, and turbulent forms with their stability properties.

  • 4. Virtual global forms and weak problem: The discrete global problem combines viscous, pressure–velocity coupling, convective, forcing, and Smagorinsky terms.The formulation is introduced after defining the associated virtual global forms.
  • 4.1. Viscous term: The viscous bilinear form is coercive, continuous, and k-consistent, while stabilization ensures coercivity of the discrete system.These properties are stated with constants independent of the meshsize h.
  • 4.2. Velocity-pressure coupling term: The pressure–velocity coupling satisfies an inf-sup condition with a constant independent of h.This condition supports stability of the velocity–pressure discretization.
  • 4.3. Convective term: The convective discretization uses a skew-symmetric form with skew-symmetry and non-dissipativity properties.These properties provide the stability structure used in the discrete analysis.
  • 4.4. Smagorinsky term: The discrete Smagorinsky term is nonnegative, continuous, and Lipschitz-continuous, with viscosity determined by the mesh and Smagorinsky constant.The paper uses the standard choice cs = 0.1.
  • 4.4. Smagorinsky term: The eddy viscosity scales as O(h2) and vanishes as hT → 0, so refined meshes require less stabilization.The model therefore retains stabilization effects on coarser meshes while diminishing them under refinement.
  • 4.4. Smagorinsky term: The section establishes the first VEM analysis of the Smagorinsky term, including discrete existence and uniqueness and a priori error estimates.The analysis recovers the classical finite-element estimate and improved convergence under additional regularity.
  • 4.4. Smagorinsky term: The three-dimensional extension is not treated specifically, although the main results are stated to hold up to constants with a priori scaling h1/2.This remark relies on coercivity, continuity, and inf-sup properties.

5. Theoretical analysis: well-posedness of the discrete formulation

The discrete formulation is shown to be well posed through a fixed-point argument for existence and a small-data bound for uniqueness.

  • 5. Theoretical analysis: well-posedness of the discrete formulation: The analysis addresses well-posedness of the discrete weak problem including the Smagorinsky term.The proof framework studies the discrete velocity and pressure formulation together.
  • Existence: A continuous operator and the Riesz–Fréchet theorem provide a unique auxiliary solution for each input velocity.This operator construction precedes the fixed-point argument.
  • Existence: A fixed-point theorem yields existence of a discrete divergence-free velocity solution.The resulting velocity is then used to establish pressure existence.
  • Existence: The weak Navier–Stokes–Smagorinsky problem admits at least one velocity–pressure solution.Pressure existence follows after the velocity result and stability estimates.
  • Uniqueness: Under the stated bound involving α, ζconv, and Lsmag, the discrete problem has a unique solution.The bound is used to control the nonlinear convective and Smagorinsky contributions.
  • Uniqueness: When 0 < κ < 1, the difference between two velocity solutions vanishes, and pressure uniqueness follows from the second equation.The argument uses coercivity, nonnegativity, continuity, and the small-data condition.

6. Theoretical analysis: a priori error estimates

The error analysis derives velocity and pressure estimates from approximation, consistency, continuity, and stability bounds, with improved velocity orders under divergence-free discretization.

  • 6. Theoretical analysis: a priori error estimates: The section derives a priori convergence rates for VEM velocity and pressure approximations of the Navier–Stokes–Smagorinsky problem.The results are organized around approximation and consistency estimates for the discrete forms.
  • Preliminaries: The analysis uses approximation properties, forcing estimates, and continuity bounds for the viscous, convective, and Smagorinsky terms.These estimates control the component errors before they are combined globally.
  • 6.2. Error analysis: The improved estimate relies on nonnegativity of the Smagorinsky term and bounds the turbulent contribution through projector continuity and discrete inverse estimates.The proof separates interpolation and discrete errors before estimating the nonlinear terms.
  • 6.2. Error analysis: The pressure error estimate is established under additional regularity for both velocity and pressure, together with the discrete stability bounds.The estimate separates high-order and low-order coefficients.

7. Numerical results

Numerical tests on sinusoidal and polynomial problems assess velocity and pressure errors across mesh types and Reynolds numbers. The observed convergence agrees with theory, while larger Reynolds numbers require finer meshes to reach the asymptotic regime.

  • Experimental setup: The experiments evaluate H1 velocity errors, L2 pressure errors, and convergence rates on Cartesian, hexagonal, and triangular meshes.Tests use degree k ∈ {2, 3, 4} and include sinusoidal and polynomial cases.
  • Convergence behavior: As Reynolds number increases, the preasymptotic region grows because stronger convection produces larger constants in the error estimates.Consequently, finer meshes are needed before asymptotic convergence becomes visible.
  • Convergence behavior: The reported convergence rates agree with the theoretical findings, including the pressure-rate behavior.This agreement is stated for both analytical test cases.
  • Sinusoidal test: The sinusoidal test reports velocity and pressure errors for Re = 100 and Re = 1000 across all three mesh families.Tables 1–4 organize the corresponding velocity and pressure errors and convergence rates.
  • Polynomial test: The polynomial test likewise reports velocity and pressure errors for Re = 100 and Re = 1000 on Cartesian, hexagonal, and triangular meshes.Tables 5–8 provide the corresponding velocity and pressure results.

8. Conclusions

The paper establishes a theoretical and numerical foundation for divergence-free VEM applied to the two-dimensional Navier–Stokes–Smagorinsky model. It proves classical and sharper convergence results under small-data assumptions and validates them numerically.

  • Scope and contribution: The study extends incompressible-flow analysis in the virtual element framework to the classical Smagorinsky turbulence model.The authors identify this model as a first step toward more comprehensive turbulence models.
  • Theoretical results: Under the usual small-data assumption, the analysis provides the classical h convergence rates for a priori error estimates.The result concerns the Navier–Stokes–Smagorinsky equations discretized with divergence-free VEM.
  • Theoretical results: Sharper convergence is obtained with weaker regularity assumptions than those used in finite element analyses for the Smagorinsky model.Numerical tests support the theoretical estimates.
  • Future directions: The framework opens future investigation of turbulence models incorporating anisotropy or backscattering.Backscattering is described as energy transfer from small to large scales.
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