[Paper Review] The Stokes complex for Virtual Elements with application to Navier--Stokes flows

L. Beirão da Veiga, David Mora+1 more|arXiv (Cornell University)|Jul 27, 2018

Advanced Numerical Methods in Computational Mathematics10 references3 citations

TL;DR

This paper establishes the Stokes complex structure for a virtual element method (VEM) applied to 2D incompressible Navier--Stokes flows by introducing a $ C^1 $-conforming virtual element space $ \Phi_h \subset H^2(\Omega) $, proving that the triad $ \{\Phi_h, \mathbf{V}_h, Q_h\} $ forms an exact Stokes complex. The method enables a discrete $ \mathrm{curl} $ formulation of the Navier--Stokes problem with divergence-free velocity and computable operators via degrees of freedom, validated by numerical tests showing optimal convergence and robustness.

ABSTRACT

In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier--Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space $Φ_h \subset H^2(Ω)$ and prove that the triad $\{Φ_h, V_h, Q_h\}$ (with $V_h$ and $Q_h$ denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.

Motivation & Objective

• To uncover the underlying Stokes complex structure in the virtual element method for Stokes and Navier--Stokes problems.
• To construct a $ C^1 $-conforming virtual element space $ \Phi_h \subset H^2(\Omega) $ that completes the discrete Stokes complex with velocity and pressure spaces.
• To prove that the triad $ \{\Phi_h, \mathbf{V}_h, Q_h\} $ forms an exact Stokes complex, ensuring discrete divergence-free velocity and stable solution existence.
• To develop a computable discrete $ \mathrm{curl} $ formulation of the Navier--Stokes problem using the $ \Phi_h $ space and degrees of freedom.
• To compare the complex-preserving VEM with a direct $ C^1 $ discretization of the stream function formulation, analyzing condition numbers and system size.

Proposed method

• Introduces a virtual element space $ \Phi_h $ consisting of $ H^2 $-conforming functions on polygonal meshes, with degrees of freedom including point values, moments of gradients, and moments of Laplacians up to order $ k-1 $.
• Defines the discrete bilinear form $ \widetilde{a}_h(\cdot,\cdot) $ using projections of the Hessian and a stabilizing term $ \widetilde{\mathcal{S}}^E $, ensuring computability via degrees of freedom.
• Constructs a computable trilinear form $ \widetilde{c}_h(\cdot;\cdot,\cdot) $ via projections of $ \Delta\psi_h $, $ \mathrm{curl}\,\psi_h $, and $ \nabla\psi_h $, all expressible from degrees of freedom.
• Derives a discrete $ \mathrm{curl} $ formulation of the Navier--Stokes problem in $ \Phi_h $, yielding the same velocity as the original method but requiring global pressure recovery.
• Proposes and analyzes a third discretization of the convective term, extending convergence results to this variant.
• Compares the Stokes-complex-based VEM with a direct $ C^1 $-VEM discretization of the stream function formulation, evaluating condition numbers and system size.

Experimental results

Research questions

• RQ1 Can the virtual element method for Stokes and Navier--Stokes problems be framed within a discrete Stokes complex structure?
• RQ2 Is it possible to construct a $ C^1 $-conforming virtual element space $ \Phi_h \subset H^2(\Omega) $ such that $ \{\Phi_h, \mathbf{V}_h, Q_h\} $ forms an exact Stokes complex?
• RQ3 Are the differential operators (e.g., Laplacian, curl) in the complex computable from the degrees of freedom of the virtual element space?
• RQ4 Does the resulting discrete $ \mathrm{curl} $ formulation preserve the velocity solution while enabling pressure recovery via a global system?
• RQ5 How does the performance of the Stokes-complex-preserving VEM compare to a direct $ C^1 $ stream function discretization in terms of condition number and system size?

Key findings

• The triad $ \{\Phi_h, \mathbf{V}_h, Q_h\} $ forms an exact Stokes complex, ensuring discrete divergence-free velocity and existence of a unique solution.
• The discrete bilinear and trilinear forms are computable solely from the degrees of freedom, with the stabilizing term $ \widetilde{\mathcal{S}}^E $ satisfying stability bounds independent of mesh size.
• The discrete $ \mathrm{curl} $ formulation of the Navier--Stokes problem yields the same velocity as the original VEM, with pressure reconstructed via a global rectangular system.
• Numerical tests confirm optimal convergence rates for velocity and pressure, with condition numbers and system sizes comparable to or better than standard FEM.
• The proposed alternative discretization of the convective term maintains convergence properties and offers a viable alternative for high-order accuracy.
• The Stokes-complex-based VEM outperforms the direct $ C^1 $ stream function discretization in terms of condition number and system size, especially for higher-order elements.