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Weakly Symmetric and Traceless Tangential-Normal Tensor Finite Elements: Application to the Brinkman Equations

Xuehai Huang, Xinyue Zhao

arXiv:2609.10331v2math.NAcs.NA

TL;DR

Existing conforming symmetric-traceless tensor elements can require high polynomial degrees and continuity on lower-dimensional subsimplices. This paper develops weakly symmetric, pointwise traceless tangential-normal elements and applies them to a Brinkman mixed method that is uniformly stable, exactly divergence-free, and pressure-robust.

  • Problem

    Available conforming symmetric-traceless tensor constructions can require high polynomial degrees and continuity constraints on lower-dimensional subsimplices.

    Source: §1 · p. 1

  • Method

    The paper constructs weakly symmetric, pointwise traceless tangential-normal tensor elements and couples them with BDM velocities and discontinuous pressures for a distributional Brinkman method.

    Source: §1 · pp. 1, 3

  • Results

    The Brinkman method is stabilization-free, uniformly well posed in viscosity-dependent norms, exactly divergence-free, pressure-robust, and supports optimal-order error estimates.

    Source: §1 · p. 3, §4.2 · p. 15, §5.1 · p. 17

  • Takeaways & Limitations

    An H(div)-conforming velocity space is compatible with the Darcy-limit normal boundary condition, and pairing it with a compatible pressure space gives exact mass conservation.

    Source: §1 · p. 2

    The global tensor space does not ensure continuity of the full normal trace across element boundaries.

    Source: §2.5 · p. 9

Abstract

from arXiv · show

We develop a family of weakly symmetric and pointwise traceless tangential-normal tensor finite elements in arbitrary space dimension and for all polynomial orders. Symmetry is imposed through local cell moments, while the only globally coupled stress degrees of freedom are tangential-normal facet moments; no vertex degrees of freedom are required. In dimensions three and higher, a lowest-order linear enrichment restores the rigid-motion facet control required for discrete Korn stability. As a principal application, we construct a distributional mixed method for the incompressible Brinkman equations using the physical viscous stress. Coupled with divergence-conforming BDM velocities and discontinuous pressures, the method is stabilization-free, uniformly stable with respect to the viscosity parameter, exactly divergence-free, and pressure-robust. We establish optimal-order error estimates in the natural norms. Under suitable parameter-explicit regularity assumptions, we also obtain a parameter-uniform boundary-layer estimate with optimal Darcy approximation order. Relaxing tangential-normal continuity yields an algebraically equivalent stress-hybridized formulation and a stabilization-free virtual element realization.

1. Introduction

The paper develops pointwise-traceless tangential-normal tensor elements with weak symmetry and facet-only coupling, then applies them to the incompressible Brinkman equations. Its distributional method is stabilization-free, uniformly stable in the viscosity parameter, exactly divergence-free, and pressure-robust.

1. Introduction

  • The new elements preserve pointwise tracelessness, impose symmetry through cell moments, and couple neighboring elements only through tangential-normal facet moments.They require no vertex degrees of freedom.Source: §1 · pp. 1–2
  • Earlier conforming symmetric-traceless constructions used high polynomial degrees and continuity constraints on lower-dimensional subsimplices, motivating relaxed interelement continuity.Examples in three dimensions include degree seven with vertex derivatives through order three, or degree four with tensor values at vertices.Source: §1 · pp. 1–2
  • For lowest order in dimensions three and higher, linear stress enrichment restores the rigid-motion facet control needed for discrete Korn stability.No such enrichment is needed in two dimensions.Source: §1 · p. 2
  • The distributional Brinkman method is stabilization-free, uniformly stable in viscosity-fitted norms, exactly divergence-free, and pressure-robust.It uses the physical viscous stress, BDM velocities, and discontinuous pressures; the paper also establishes optimal-order natural-norm errors and a parameter-uniform boundary-layer estimate by comparison with Darcy.Source: §1 · p. 3
  • Relaxing tangential-normal continuity gives an algebraically equivalent stress-hybridized formulation and a stabilization-free virtual element realization on simplices.Omitting the drag term gives the corresponding Stokes method.Source: §1 · p. 3

2. Weakly Symmetric and Traceless Tangential-Normal Tensor Elements

The tensor elements are pointwise traceless, weakly symmetric, and built from facet and cell degrees of freedom. Their construction combines tangential-normal traces with cell-moment constraints, adds lowest-order rotational control where needed, and yields a global space coupled only through facet moments.

2. Weakly Symmetric and Traceless Tangential-Normal Tensor Elements

  • The element family uses only facet and cell degrees of freedom and is valid in all dimensions d ≥ 2.The weak-symmetry conditions are imposed through cell moments against skew-symmetric tensors.Source: §2 · p. 3

2.2. An auxiliary traceless tangential-normal element.

  • The cited passage recalls a local tangential-normal bubble space for traceless tensors and states that its k = 0 instance is the zero space.Source: §2.2 · p. 5

2.3. Weak symmetry through cell moments for k ≥1. τ ∈Σ+

  • For k ≥ 1, weak symmetry annihilates skew-symmetric cell moments through degree k, coupling facet and bubble modes.Symmetry is imposed by cell-moment conditions rather than pointwise symmetry.Source: §2.3 · p. 6

2.4. Lowest-order element and RM(F) facet moments.

  • At lowest order in dimensions d ≥ 3, linear enrichment supplies the rotational facet control required by the projected discrete Korn estimate.The additional rotational facet moments are trivial in two dimensions.Source: §2.4 · pp. 7–8

2.5. Global tensor space.

  • The global tensor space is pointwise traceless and weakly symmetric, with tangential-normal facet moments as its only interelement coupling and no vertex degrees of freedom.Its full normal trace need not be continuous, so the space is generally not H(div)-conforming.Source: §2.5 · p. 9

3. Continuous Distributional Formulation

The continuous distributional mixed formulation uses a stable decomposition of the dual of H0(div) to obtain viscosity-uniform well-posedness, and is equivalent to the primal Brinkman formulation.

3. Continuous Distributional Formulation

  • A stable decomposition writes each dual-space datum as div τ + ∇q, with a norm bound for the symmetric-traceless stress and scalar pressure components.This decomposition supplies the continuous inf–sup ingredient for parameter-uniform well-posedness.Source: §3 · pp. 9–10
  • The formulation retains standard norms on H0(div) and L2 to establish stability uniformly with respect to ν.Its bilinear form couples the stress-pressure variables to velocity through divergence and the distributional stress divergence.Source: §3 · pp. 9–10
  • The distributional mixed problem has a unique solution for f in (H0(div, Ω))′, with a viscosity-uniform solution bound.The theorem applies for 0 < ν ≤ 1.Source: §3 · pp. 10–11
  • The distributional formulation is equivalent to the primal Brinkman weak formulation, with σ = νε(u).The equivalence is stated in both directions for solutions in the specified spaces.Source: §3 · p. 11

4. The Distributional Mixed Element Method

The discrete method pairs BDM velocities and discontinuous pressures with tangential-normal stress elements, using commuting interpolation and projected Korn control to establish viscosity-uniform stability and exact incompressibility.

4.1.1. Velocity–pressure spaces.

  • For k ≥ 0, velocities use the H(div)-conforming Brezzi–Douglas–Marini (BDM) element of order k + 1, while pressure is approximated by discontinuous piecewise polynomials of degree k.Source: §4.1.1 · pp. 11–12

4.1.2. Interpolation and projections.

  • The BDM velocity interpolant has a commuting property, and a stable discrete divergence lifting provides wh with div wh = qh and |wh|1,h ≲ ∥qh∥0.The lifting bound supports the discrete stability argument.Source: §4.1.2 · p. 12
  • Stress interpolation preserves single-valued tangential-normal facet moments, while the projected discrete Korn inequality controls purely tangential velocity jumps.The Korn argument uses moments against rigid-motion traces on facets.Source: §4.1.2 · p. 12, §4.2 · p. 13

4.2. Distributional mixed finite element method and stability.

  • The discrete method is uniformly well posed for 0 < ν ≤ 1 and produces exactly divergence-free velocities.The stability constant is independent of mesh size h and ν, with dependence on dimension, polynomial degree, domain, and mesh shape regularity.Source: §4.2 · p. 15
  • Gradient perturbations leave the discrete velocity and stress unchanged, while the lowest-order RM(F)-enriched stress space supplies rotational trace control without additional stabilization.The comparison states that the minimal-facet MCS method instead uses consistent vorticity-divergence stabilization.Source: §4.2 · pp. 15–16

5. Error Analysis

The error analysis establishes optimal-order estimates in natural norms and a parameter-uniform boundary-layer estimate by comparison with the Darcy limit, under parameter-explicit regularity assumptions.

5.1. Optimal-order error estimates.

  • The method has optimal-order error estimates in the natural norms, with a stated pressure-error result when p ∈ Hk+1(Ω).The section presents these estimates before treating the viscosity-uniform boundary-layer bound.Source: §5 · p. 16, §5.1 · p. 17

5.2. Parameter-uniform boundary-layer estimate.

  • Comparison with the Darcy limit yields a parameter-uniform boundary-layer estimate under parameter-explicit regularity assumptions.The subsection further assumes a convex domain and f ∈ H(curl, Ω).Source: §5.2 · p. 17

6. Equivalent formulations

Section 6 presents two algebraically equivalent alternatives to the distributional mixed method: stress hybridization and a stabilization-free virtual element realization. The hybridized method relaxes tangential–normal stress continuity, while the virtual element formulation is equivalent to it.

6.1. Stress-hybridized formulation.

  • Relaxing tangential–normal stress continuity yields a hybridized formulation with a tangential facet multiplier.The formulation is derived from the distributional mixed method.Source: §6.1 · p. 19, §6 · p. 19
  • The hybridized method is well posed and algebraically equivalent to the distributional method, with stress recovered as ν dev εw(uh, λh).The recovered stress and the velocity-pressure pair solve the original formulation.Source: §6.1 · p. 19

6.2. Stabilization-free virtual element method.

  • Adapting the local Stokes–Neumann construction of yields a stabilization-free virtual element realization of the hybridized method.Source: §6.2 · p. 20
  • The virtual element degrees of freedom are unisolvent because the traction space contains rigid-motion traces and the local Stokes–Neumann argument applies.Source: §6.2 · p. 20
  • The virtual element degrees of freedom determine a BDM reconstruction with a commuting divergence property.The construction uses facet and cell degrees of freedom and the degree-k Koszul decomposition.Source: §6.2 · p. 20
  • Its bilinear form uses computable facet and volume terms, with weak symmetry supporting the required identity.Source: §6.2 · p. 21
  • The virtual element method is well posed and algebraically equivalent to the hybridized method, with a reconstructed stress solving the distributional method.Source: §6.2 · p. 21

7. Numerical Examples

The numerical tests show order k + 1 convergence, uniformly bounded normalized boundary-layer errors over the tested mesh sizes and viscosities, and obstacle-flow behavior consistent with Brinkman screening.

7. Numerical Examples

  • The examples test convergence, Darcy-limit robustness, and obstacle through-flow with slip boundary conditions on uniform simplicial meshes.Source: §7 · p. 21

7.1. Error Estimates for a Smooth Exact Solution.

  • Order k + 1 convergence holds for weighted errors across both polynomial degrees and all tested viscosities.The observed convergence confirms Corollary 5.3.Source: §7.1 · p. 22

7.2. Darcy Limit and Boundary Layer.

7.3. Obstacle Through-Flow.

  • The velocity disturbance localizes near the obstacle, with essentially parallel downstream flow consistent with Brinkman screening.The nonzero wall-parallel velocity reflects the slip condition.Source: Figure 2 · p. 23, §7.3 · p. 23
  • The divergence diagnostic remains at roundoff, and the pressure difference stabilizes under refinement faster than obstacle drag.The drag is more sensitive to local mesh resolution near the obstacle because it is a traction functional.Source: §7.3 · p. 23, Table 2 · p. 24
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