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Virtual Elements for the Navier-Stokes problem on polygonal meshes

L. Beirão da Veiga, C. Lovadina, G. Vacca

arXiv:1703.00437v2math.NA

TL;DR

The paper addresses how to construct conforming VEM discretizations for 2D Navier–Stokes equations on general polygonal meshes while retaining point-wise divergence-free discrete velocities. It proposes a family using enhanced velocity spaces and computable discrete forms, and reports rigorous stability and optimal-convergence analysis with numerical confirmation. The methods also support pressure-partially-decoupled velocity errors and applications such as Stokes–Darcy coupling, while remaining non-pressure-robust and subject to stated mesh-regularity assumptions.

  • Problem

    Existing pressure-robust Galerkin schemes for Stokes and Navier–Stokes equations work on standard simplicial or hexahedral meshes, leaving conforming divergence-free methods for arbitrary polygonal meshes as a distinct need.

  • Method

    The paper develops a family of 2D Navier–Stokes VEMs with enhanced velocity spaces, discontinuous polynomial pressures, projector-based convective discretizations, and computable discrete forms.

  • Results

    The methods provide point-wise divergence-free discrete velocities, rigorous stability and optimal error estimates, and numerical tests that assess their performance.

  • Takeaways & Limitations

    The divergence-free construction partly decouples velocity errors from discrete pressures, appears more robust for small viscosity, and permits direct reuse for mixed-form diffusion and easier Stokes–Darcy coupling.

  • Takeaways & Limitations

    The methods are not pressure-robust, and the analysis assumes mesh shape-regularity constants and polynomial consistency degree conditions.

Abstract

from arXiv · show

A family of Virtual Element Methods for the 2D Navier-Stokes equations is proposed and analysed. The schemes provide a discrete velocity field which is point-wise divergence-free. A rigorous error analysis is developed, showing that the methods are stable and optimally convergent. Several numerical tests are presented, confirming the theoretical predictions. A comparison with some mixed finite elements is also performed.

1 Introduction

The paper extends VEM to 2D Navier–Stokes equations on general polygonal meshes, constructing point-wise divergence-free velocity approximations and analysing their performance. It develops rigorous error estimates and numerical tests covering flexibility, stability-related behaviour, and coupling advantages.

  • Background: VEM approximates PDEs on highly general polygonal meshes, including meshes with hanging vertices and non-convex elements, using computable information rather than pointwise-known functions.The available degrees of freedom suffice to construct the stiffness matrix and right-hand side.
  • Proposed methods: The proposed family applies an enhanced VEM velocity space and polynomially discontinuous pressure space to 2D Navier–Stokes equations.Suitable projectors address the nonlinear convective term, and the family contains one method for each consistency order k ≥2.
  • Analysis: The paper combines established Navier–Stokes Galerkin analysis with VEM-specific techniques, including a new interpolant construction.The authors identify these techniques as potentially useful for other VEM spaces with different regularity requirements.
  • Numerical assessment: The tested k = 2 scheme provides point-wise divergence-free discrete velocities, partly decoupling velocity errors from discrete pressures and appearing more robust at small viscosity than standard mixed finite elements.The pressure influence remains indirect through loading and convection-term approximation.
  • Applications: The same divergence-free virtual-space pair can approximate mixed-form diffusion directly, simplifying Stokes–Darcy coupling and joining different polygonal meshes across subdomains.Polygons that permit hanging nodes make gluing different meshes easier.
  • Validation: The paper presents numerical tests and comparisons with mixed finite element schemes after deriving optimal error estimates in its theoretical analysis.The outline identifies the tests as assessing actual performance and the analysis as leading to optimal error estimates.

2 The continuous Navier-Stokes equation

The paper formulates the steady 2D Navier–Stokes problem variationally on a polygonal domain with homogeneous Dirichlet conditions. It introduces velocity and pressure spaces, the inf-sup structure, and an equivalent divergence-free kernel formulation with a skew-symmetric convective form.

  • Problem setting: The continuous problem is the steady Navier–Stokes equation on a polygonal domain with homogeneous Dirichlet boundary conditions.The viscosity satisfies ν > 0, and u and p denote velocity and pressure fields.
  • Variational formulation: Its variational formulation combines the viscous bilinear form, convective trilinear form, pressure-velocity bilinear form, and external-force functional.The equation is required for all test velocities v in V.
  • Functional setting: The spaces V and Q and the bilinear form b satisfy the inf-sup condition, while the relevant forms are continuous.These properties support the stated well-posedness framework.
  • Well-posedness: The formulation has a unique solution (u, p) in V × Q under the stated assumptions.The paper explicitly states uniqueness for Problem (7).
  • Kernel formulation: Defining Z as the kernel of b removes the pressure equation and yields an equivalent formulation tested only with divergence-free functions.On Z, the equation is ν a(u, v) + c(u; u, v) = (f, v).
  • Convection: For fixed divergence-free u, the convective bilinear form is skew-symmetric, motivating the skew-symmetrized trilinear form used in the analysis.The modified form is introduced for arbitrary w, u, and v in V.

3 Virtual formulation of the problem

The paper constructs a computable VEM discretization for 2D Navier–Stokes equations on general polygonal meshes, using enhanced velocity spaces, polynomial projections, stabilization, and locally polynomial pressures. The resulting scheme is stable, point-wise divergence-free, and can also accommodate Darcy flow.

  • 3.1 Virtual element space and polynomial projections: The velocity space is defined on general polygonal meshes under star-shapedness and minimum vertex-separation assumptions.The mesh hypotheses require each element to be star-shaped with respect to a ball of radius proportional to its diameter and vertex distances bounded below proportionally to the diameter.
  • 3.1 Virtual element space and polynomial projections: Velocity degrees of freedom comprise vertex values, edge-point values, velocity moments, and divergence moments.These four subsets are denoted DV1–DV4 and are illustrated for polynomial orders k = 2 and k = 3.
  • 3.1 Virtual element space and polynomial projections: The enhanced virtual space makes the required polynomial projections and moments exactly computable from the degrees of freedom.The construction supports exact computation of the H1 projection and higher-order projections, while integration by parts provides additional moments.
  • 3.2 Discrete bilinear forms and load term approximation: Discrete bilinear and trilinear forms are built from computable projections, local contributions, and a stabilizing bilinear form spectrally equivalent to the continuous energy form.The stabilization satisfies bounds independent of the element and mesh size, ensuring stability of the discrete bilinear form.
  • 3.2 Discrete bilinear forms and load term approximation: The pressure–velocity coupling is left unapproximated and is computable from selected velocity degrees of freedom because the pressure is polynomial on each element.The discrete convection form is uniformly continuous with respect to the mesh size.
  • 3.3 The discrete problem: The discrete spaces satisfy an inf-sup condition and div Vh = Qh, yielding a point-wise divergence-free discrete velocity and a uniquely solvable discrete problem.Coercivity, skew-symmetry of the discrete convection form, and the inf-sup condition provide well-posedness.
  • 3.3 The discrete problem: The same virtual velocity–pressure spaces can accommodate both Stokes/Navier–Stokes and Darcy problems, simplifying coupled-flow implementations and interface conditions.This follows from the stability property of the proposed velocity–pressure pair.

4 Theoretical analysis

The theoretical analysis establishes interpolation estimates, stability, and error bounds for the enhanced virtual element method. Under regularity, mesh, and stability assumptions, velocity and pressure converge with the predicted optimal orders, while the velocity estimate is independent of the continuous pressure.

  • 4 Theoretical analysis: An interpolant in the enhanced velocity space is constructed for functions in Hs+1(Ω) ∩ V with 0 < s ≤ k.The construction is developed through several steps using auxiliary approximants, degree-of-freedom interpolation, and local arguments.
  • 4 Theoretical analysis: The interpolation analysis relies on polynomial approximation, scaled norms, mixed-problem well-posedness, and mesh-shape regularity constants.The constants depend on the polynomial degree and shape-regularity parameters, while the relevant estimates use Sobolev embeddings and inverse inequalities.
  • 4 Theoretical analysis: The divergence-free subspace Zh is approximated with the same accuracy order as the full discrete velocity space Vh.For v ∈ Hs+1(Ω) ∩ Z, the discrete-kernel approximation follows from the inf-sup condition and the interpolation theorem.
  • 4 Theoretical analysis: Theorem 4.2 gives the velocity estimate ∥u − uh∥V ≤ hs F(u; ν, γ, γh) + hs+2 H(f; ν, γh) for 0 < s ≤ k.The functions F and H are independent of h; the estimate combines velocity approximation, load approximation, and convection terms.
  • 4 Theoretical analysis: The velocity error estimate does not depend on the continuous pressure because the proposed discrete velocity is divergence-free.Pressure affects the velocity estimate only indirectly through the approximation of loading and convection terms.
  • 4 Theoretical analysis: The pressure estimate includes approximation, load, and velocity-dependent contributions, as expressed in bound (94).The stated estimate is ∥p − ph∥Q ≤ C hs |p|s + C hs+2 |f|s+1 + hs K(u; ν, γ, γh).

5 Numerical Tests

The numerical tests assess second-order VEM against mixed finite elements across hydrostatic, low-viscosity, distorted-mesh, and coupled-flow settings. They confirm pressure-decoupled velocity behavior, optimal convergence, and robustness on general polygonal and glued non-conforming meshes.

  • Experimental setup: The experiments use second-order VEM and compare it with Q2-P1 or P2-P1 mixed finite elements across multiple polygonal mesh families.The tested meshes include quadrilateral, triangular, Voronoi, WEB-like, distorted, and glued non-conforming grids.
  • Tests 5.1–5.2: In hydrostatic tests with polynomial pressure, VEM produces an exact velocity while Q2-P1 retains non-negligible velocity errors from pressure approximation.Both pressures lie outside the discrete pressure space, but the VEM velocity is exact because the load is polynomial of degree two.
  • Test 5.1: For trigonometric pressure, VEM attains fourth-order H1-velocity convergence and quadratic L2-pressure convergence, compared with quadratic convergence for Q2-P1.The VEM still exhibits slight velocity pollution from higher-order load approximation, but less than the finite element method.
  • Test 5.2: For the first Navier–Stokes benchmark, VEMnon-skew is exact whereas VEMskew and P2-P1 are not; the latter is affected by pressure approximation.The second benchmark similarly favors VEMnon-skew over VEMskew, although it does not reproduce the exact velocity.
  • Test 5.3: For ν = 10^-4 and 10^-5, P2-P1 does not converge, while VEM remains accurate except at very small viscosities.The reported robustness is attributed to velocity errors that do not depend directly on pressure.
  • Tests 5.5–5.6: On strongly distorted meshes, VEM maintains optimal second-order accuracy while Q2-P1 becomes sub-optimal; glued polygonal meshes also show appropriate convergence.The glued meshes demonstrate handling of independent non-conforming grids and coupling between Darcy and Stokes flow problems.
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