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A Hybrid High-Order method for the incompressible Navier--Stokes equations based on Temam's device

Lorenzo Botti, Daniele Di Pietro, Jérôme Droniou

arXiv:1807.07345v2math.NA

TL;DR

The paper addresses stable and flexible discretisation of incompressible Navier–Stokes equations, particularly the discrete preservation of convective non-dissipation. It proposes an HHO method using Temam’s device, proves h^(k+1) energy error estimates under small data, and validates convergence and physical test cases numerically.

  • Problem

    Discretising incompressible Navier–Stokes equations requires preserving convective non-dissipation despite discrete velocities that may not be divergence free or vanish on the boundary.

  • Method

    The paper develops an HHO method with a Temam-device convective trilinear form, optional convective stabilisation, flexible mesh support, and weak or strong velocity boundary enforcement.

  • Results

    An energy error estimate in h^(k+1) is proved under a data smallness assumption, and numerical tests assess convergence, stabilisation, boundary enforcement, and convection-dominated robustness.

  • Takeaways & Limitations

    The method combines discrete convective non-dissipation with inf-sup stability, local conservation, general meshes, and efficient elimination of locally coupled unknowns.

  • Takeaways & Limitations

    The proved error estimate is derived under a smallness assumption on the data.

Abstract

from arXiv · show

In this work we propose a novel Hybrid High-Order method for the incompressible Navier--Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in $h^{k+1}$ (with $h$ denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known two- and three-dimensional lid-driven cavity problems).

1. Introduction

The paper introduces an HHO method for incompressible Navier–Stokes equations that combines Temam’s device with flexible meshes, stable pressure–velocity coupling, and efficient implementation. It establishes theoretical error estimates and validates the method numerically.

  • Method and novelty: The proposed HHO method incorporates Temam’s device into the convective term to support stability and discrete non-dissipation.The convective trilinear form is designed to mimic the continuous non-dissipation property.
  • Method and novelty: HHO uses broken polynomial unknowns on mesh elements and their skeleton, with local reconstructions replacing explicit reference-element basis functions.These reconstructions support consistent Galerkin terms, while stabilisation terms provide stability.
  • Method capabilities: The method supports arbitrary approximation orders on polyhedral meshes and non-matching junctions, while remaining locally conservative and amenable to efficient implementations.Only face velocities and elementwise mean pressure values remain globally coupled at each nonlinear iteration.
  • Method capabilities: It satisfies a uniform inf-sup condition, supports weak or strong velocity boundary enforcement, and is designed to remain robust for large Reynolds numbers.The weak enforcement extends Nitsche’s technique and can improve boundary-layer resolution while simplifying implementation.
  • Analysis and validation: An energy error estimate in h^(k+1) is proved under a data smallness assumption, alongside numerical tests of convergence, stabilisation, boundary enforcement, and convection-dominated robustness.Kovasznay’s solution is used to assess convergence rates with or without convective stabilisation and with either boundary-condition treatment.

2. Continuous setting and a key remark

The continuous analysis identifies non-dissipation of the convective term as a stability requirement and motivates Temam’s modified trilinear form. This form remains non-dissipative without requiring divergence-free velocity or homogeneous boundary values.

  • Stability requirements: Navier–Stokes well-posedness for small data relies on viscous coercivity, pressure–velocity inf-sup stability, and convective non-dissipation.These properties, together with consistency, guide the design of the discrete HHO method.
  • Continuous non-dissipation: The standard convective term does not contribute to the kinetic-energy balance when the velocity is divergence free and vanishes on the boundary.This follows from integration by parts and the incompressibility and wall-boundary conditions.
  • Motivation for Temam’s device: At the discrete level, non-dissipation is difficult because the discrete velocity may not be sufficiently divergence free or zero on the boundary.Temam’s device is introduced to overcome these two sources of nonzero contributions.
  • Temam’s device: The modified trilinear form satisfies ˜t(w, v, v) = 0 for all w and v in H1(Ω)^d.Thus it is non-dissipative even when w is not divergence free and v does not vanish on ∂Ω.

3. Discrete setting

The discrete setting combines polyhedral meshes, broken polynomial unknowns, and local reconstructions that provide the building blocks for the HHO discretisation. Discrete divergence and directional-derivative operators are designed for pressure coupling, convection, and stability analysis.

  • Mesh: The method uses regular meshes of polyhedral elements and planar faces, with element diameters and face sizes uniformly comparable.The number of faces per element is also uniformly bounded independently of the meshsize.
  • Discrete spaces: For degree k, velocity unknowns are vector-valued polynomials of degree at most k on elements and faces, while pressure unknowns are discontinuous polynomials of degree at most k.The global velocity space is assembled from element and face degrees of freedom, with interpolation from smooth functions into these discrete variables.
  • Local reconstructions: HHO local reconstructions mimic integration by parts by combining element unknowns in volume terms with face unknowns in boundary terms.The gradient reconstruction contains the gradient of the element unknown and a boundary correction based on element-face differences.
  • Local reconstructions: The velocity reconstruction has degree k+1, its gradient is the L2-orthogonal projection of the reconstructed gradient, and its element mean matches the element unknown.This reconstruction is central to the approximation of viscous terms.
  • Discrete operators: The discrete divergence is used with degree k for pressure-velocity coupling and degree 2k for incorporating Temam’s device into the convection term.A discrete directional derivative likewise represents the advective velocity through element and face unknowns and has stated approximation properties.

4. The Hybrid High-Order method

The HHO method combines locally reconstructed polynomial unknowns with Temam-stabilized convection, pressure coupling, optional convective stabilization, and weak or strong velocity boundary enforcement.

  • Viscous and pressure terms: Local reconstructions and stabilization yield consistent viscous terms, local norm equivalence, and stable bilinear forms.The stabilization penalizes differences between reconstructed velocities and discrete unknowns.
  • Convective term: Temam’s device supplies discrete skew-symmetry and non-dissipation even when discrete velocities are not sufficiently divergence-free or boundary-zero.Its second and third terms are crucial for these properties.
  • Convective stabilization: A generic Lipschitz function ρ unifies upwind, locally upwinded θ, and Scharfetter–Gummel convective stabilizations.The choice ρ(s)=1/2|s| gives the upwind scheme, while other choices recover the locally upwinded θ and Scharfetter–Gummel variants.
  • Boundary conditions: The formulation supports both weak Nitsche enforcement and strong enforcement of velocity boundary conditions.Weak enforcement can improve boundary-layer resolution and simplify implementation.
  • Conservation: The discrete incompressibility constraint implies elementwise vanishing reconstructed divergence and enables local mass balance through elementwise pressure tests.Continuity of mass fluxes follows from single-valued face unknowns.
  • Analysis and relation to prior methods: An energy error estimate is proved under a data smallness assumption, and the method is positioned as a higher-order extension of an HMM Navier–Stokes scheme.The convergence analysis focuses on the weak-boundary-condition version but carries over to strong enforcement.

5. Numerical tests

Numerical tests assess convergence and physical cavity flows in two and three dimensions. They confirm the predicted rates, while high-order cavity computations are accurate on coarse meshes but can become under-dissipative at high Reynolds numbers.

  • Test suite: The numerical study combines analytical-solution convergence tests with two- and three-dimensional lid-driven cavity benchmarks.The experiments monitor velocity and pressure errors, convergence rates, computational times, and condensed-system properties.
  • Kovasznay flow: Pressure errors approach order k+1, while velocity errors approach orders k+1 in energy norm and k+2 in L2 norm.Weakly enforced boundary conditions provide rates close to optimal, whereas strong enforcement is slightly sub-optimal for velocity.
  • Kovasznay flow: Convective stabilization explains the slightly sub-optimal strong-boundary-condition rates, while degree-k=0 results are strongly affected by upwind stabilization.The low-degree effect is associated with larger jumps between element and face unknowns in under-resolved computations.
  • Two-dimensional cavity flow: At Re = 20,000, k=1 cavity computations agree better with Erturk et al. reference solutions than k=7 computations.The reported behavior suggests k=1 calculations are over-dissipative at high Reynolds numbers, despite high-order accuracy on coarse meshes at low Reynolds numbers.
  • Three-dimensional cavity flow: At Re = 1,000, k=4 and k=8 three-dimensional cavity computations agree very well with reference solutions on 8^3 and 8^3/16^3 grids.Lower-order k=1 and k=2 computations show an over-dissipative mismatch near the negative peak of the u1 distribution.
  • Computational cost: Doubling mesh step size together with polynomial degree improves accuracy and computational cost in three-dimensional cavity calculations.The paper links the resulting condensed global systems to potential efficiency gains for implicit time discretizations.

6. Flux formulation

The discrete problem is reformulated using conservative numerical fluxes, yielding local momentum and mass balances and conservative interface fluxes.

  • Flux reformulation: The strongly enforced discrete problem admits an equivalent formulation in terms of conservative numerical fluxes.This reformulation is established for the discrete problem (52).
  • Interface conservation: Numerical normal traces of the global momentum and mass fluxes are conservative across internal interfaces.The interface balance follows from the discrete equations and the single-valuedness of face unknowns.
  • Local balances: The formulation provides finite-volume-like local momentum balances on each mesh element.Testing with elementwise constant velocity functions yields the local balance.
  • Local balances: The formulation also provides finite-volume-like local mass balances on each mesh element.The local mass balance follows by testing with an elementwise constant pressure function.
  • Consequences: Local conservation of momentum and mass is relevant for engineering applications and can support flux-equilibration a posteriori error estimators.The paper explicitly connects the conservation relations to both engineering preservation and mathematical error estimation.

Appendix A. Proofs of intermediate results

The appendix collects proofs of the intermediate results used in the method's analysis.

  • The appendix gathers proofs of the intermediate analytical results required for the convergence analysis.

Appendix A.1. Discrete Sobolev embeddings

The appendix establishes discrete Sobolev-embedding tools using jump estimates, boundary-face estimates, and mesh-norm definitions.

  • Discrete jumps across internal faces are bounded by the element-to-face differences on the two neighboring elements.The estimate is obtained by inserting and subtracting the single-valued face unknown.
  • Boundary traces are bounded by element-to-face differences together with the face unknown itself.This estimate is combined with the discrete trace inequality.
  • The resulting bounds are inserted into the mesh-norm definitions to obtain the stated discrete embedding estimate.

Appendix A.2. Approximation properties of the discrete directional derivative

The appendix derives approximation estimates for the discrete directional derivative using projector properties, Hölder inequalities, and discrete trace bounds.

  • The proof introduces the reconstructed quantity ˆv_T and uses the defining directional-derivative relation with matching reconstructed arguments.
  • Hölder inequalities with exponents (4, 2, 4) and (2, 4, 4) control separate terms in the approximation estimate.The argument combines projector orthogonality and L4-boundedness with approximation properties.
  • Projector approximation properties and gradient-reconstruction estimates bound the factors appearing in the first term.The proof uses l = 0, p = 4 for the projector and ℓ = 2k for the gradient reconstruction.
  • The second term is bounded through optimal L2-projector approximation, while the third uses a discrete trace inequality.The stated parameter choices include s = k + 1 for the projector and α = 4 for the trace estimate.
  • Substitution of the intermediate bounds yields the claimed approximation result.

Appendix A.3. Viscous term

The viscous-term analysis establishes stability and boundedness through approximation, consistency, seminorm equivalence, and meshwise inequalities.

  • The resulting estimates are obtained after taking suprema over admissible discrete functions.
  • The proof uses discrete norm equivalences and mesh-size relations to control element and face contributions.These include the definition of the discrete norm and h_F ≤ h_T.
  • Elementwise integration by parts and single-valued face unknowns organize the viscous consistency argument.Boundary face unknowns vanish where required, cancelling the remaining boundary contributions.
  • Stability follows by combining approximation estimates with viscous-stabilisation consistency and local seminorm equivalence.The argument then uses Cauchy–Schwarz inequalities over elements and faces.

Appendix A.4. Pressure-velocity coupling

The pressure–velocity coupling analysis proves consistency and inf-sup stability using discrete divergence properties and a continuous right-inverse of the divergence.

  • Consistency uses the commuting property of the discrete divergence and removes projections because the reconstructed divergence lies in the relevant polynomial space.
  • Inf-sup stability is obtained from the surjectivity of the continuous divergence operator and a velocity field with prescribed negative divergence.The constructed field satisfies ∥v_qh∥_H1(Ω)^d ≲ ∥q_h∥.
  • The discrete interpolator transfers the continuous divergence lifting into the discrete stability estimate.
  • Elementwise integration by parts and single-valued internal face unknowns control the pressure–velocity consistency terms.
  • Cauchy–Schwarz, Hölder inequalities, and projector approximation properties complete the consistency bounds.

Appendix A.5. Convective term

The convective-term analysis establishes skew-symmetry, non-dissipation, and boundedness through discrete integration by parts, embeddings, and approximation estimates.

  • The discrete convective trilinear form is skew-symmetric and non-dissipative when the test functions coincide.The non-dissipation property follows immediately by setting z_h = v_h.
  • Boundedness is proved by decomposing the convective expression into multiple contributions and estimating each with discrete inequalities.
  • Discrete Sobolev and trace embeddings control element and face terms, with q = 4 used throughout the key estimates.
  • Approximation properties of directional derivatives, divergence reconstructions, and L2 projectors control consistency contributions.
  • The argument assumes piecewise W^{k+1,4} regularity with compatible traces, ensuring global W^{1,4} regularity.
  • Collecting the componentwise estimates yields the final convective bound under the stated regularity and boundary conditions.The proof also uses that the continuous velocity vanishes on ∂Ω.

Appendix A.6. Convective stabilisation

The convective-stabilisation analysis proves continuity by bounding the stabilisation contribution with the same estimates used for the convective term.

  • Lipschitz continuity of ρ with ρ(0) = 0 gives the linear bound needed for the stabilisation residual.
  • The face residual is controlled by separating face and element contributions, then applying trace, Hölder, and Sobolev inequalities.
  • The boundary penalisation vanishes when the interpolated velocity satisfies the homogeneous boundary condition.
  • The continuity estimate follows by inserting the residual bound into the preceding inequality and summing with Cauchy–Schwarz.
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