Linearised elasticity via AFW symmetric-tensor mixed elements

Intuition Beginner

Elasticity asks how a solid deforms under load. The displacement tells where each material point moves. The stress tells how internal forces pass through small surfaces inside the body.

For computation, stress is not just another vector field. It is a tensor field, it must balance forces, and in ordinary elasticity it has a symmetry condition tied to conservation of angular momentum.

Mixed finite element methods approximate stress and displacement as separate unknowns. This is useful because force balance is then built directly into the numerical system.

The hard part is choosing stress elements that keep balance and symmetry stable on a mesh. The Arnold-Falk-Winther construction is a central solution to that problem.

Visual Beginner

The method treats stress and displacement as different mesh quantities linked by force balance.

Worked example Beginner

Imagine a loaded beam. A displacement-only method approximates the shape change directly. A mixed method also solves for the internal stress that carries the load.

If the stress space is poorly chosen, the computed stress may fail to balance forces or may mishandle the symmetry condition. The displacement can then look plausible while the internal force field is unreliable.

AFW-type mixed elements are designed so the stress space fits the balance law and the symmetry constraint.

Check your understanding Beginner

Formal definition Intermediate+

Let $\Omega \subset {\mathbb{R}}^n$ be an elastic body. In the stress-displacement mixed formulation, the stress is a symmetric tensor field $$ \sigma\in H(\operatorname{div};\Omega;\mathbb S), $$ where $\mathbb{S}$ denotes symmetric matrices, and the displacement is $$ u\in L^2(\Omega;\mathbb R^n). $$ The compliance tensor is denoted by $A$. A common Hellinger-Reissner weak form seeks $(\sigma ,u)$ such that $$ (A\sigma,\tau)+(\operatorname{div}\tau,u)=0 \quad\text{for all }\tau\in H(\operatorname{div};\Omega;\mathbb S), $$ and $$ (\operatorname{div}\sigma,v)=(f,v) \quad\text{for all }v\in L^2(\Omega;\mathbb R^n). $$ Sign conventions vary; the structural point is that stress is controlled in an $H(\mathrm{div})$ tensor space and displacement is controlled in $L^2$.

The elasticity complex replaces the de Rham complex as the organizing structure. In three-dimensional schematic form, it has the shape $$ 0\to \mathrm{RM}\to V\xrightarrow{\epsilon}\mathbb S \xrightarrow{J}\mathbb S\xrightarrow{\operatorname{div}}V\to 0, $$ where $\mathrm{RM}$ denotes rigid motions, $ϵ$ is the symmetric gradient, $J$ is the compatibility operator, and $\mathrm{div}$ is the row-wise divergence of stress.

AFW-type elements construct stress and displacement spaces that fit this complex and satisfy the mixed inf-sup requirements. Some formulations enforce stress symmetry directly; the Arnold-Falk-Winther 2007 construction uses weakly imposed symmetry with an additional multiplier to obtain stable families in two and three dimensions.

Counterexamples to common slips

• A stable Stokes or Poisson mixed element does not automatically give a stable elasticity element.
• The stress symmetry condition cannot be ignored without changing the physical and variational problem.
• The elasticity complex is not just the de Rham complex with new labels; it uses the symmetric gradient and tensor-valued operators.

Key theorem with proof Intermediate+

Theorem (AFW mixed elasticity stability pattern). For the linear elasticity mixed problem, finite element spaces built to fit the elasticity complex and equipped with a suitable Fortin/cochain projection satisfy the Babuška-Brezzi stability conditions. Consequently, the mixed approximation of stress and displacement is stable and converges with the approximation order of the chosen AFW-type spaces.

Proof. The continuous Hellinger-Reissner formulation is a saddle-point problem. Its stability requires coercivity of the compliance form on the divergence-free stress kernel and an inf-sup condition linking stress divergence to displacement.

The elasticity complex identifies the exact differential relations behind those two requirements. Rigid motions form the initial kernel, the symmetric gradient produces compatible strains, and stress divergence gives the force-balance operator.

The discrete spaces are chosen so that the same operator relations hold at mesh level. A Fortin or cochain projection transfers the continuous inf-sup test functions into the discrete stress space while preserving the relevant divergence relation.

With discrete inf-sup stability in place, the Babuška-Brezzi theorem from 24.01.04 gives well-posedness. Galerkin orthogonality and polynomial approximation then give the standard mixed-method error estimate for stress and displacement. $\square$

Bridge. The Maxwell unit 24.04.02 showed how FEEC treats a vector curl problem through the de Rham complex. Elasticity is harder: the unknown stress is symmetric-tensor-valued, and the organizing complex is the elasticity complex rather than the ordinary de Rham complex.

Exercises Intermediate+

Advanced results Master

The Hellinger-Reissner formulation is attractive because stress is computed as a primary variable. For engineering and mechanics, this is more than a numerical preference: stress is the quantity that enters yield criteria, traction balance, and force transmission.

The finite element challenge is that stress must lie in an $H(\mathrm{div})$-conforming tensor space while satisfying symmetry. Direct symmetric stress elements are possible but intricate. The Arnold-Winther two-dimensional element is a landmark construction; the Arnold-Falk-Winther 2007 weak-symmetry framework gives a systematic stable route in both two and three dimensions.

FEEC clarifies why these constructions are hard. The scalar Poisson and Maxwell cases are governed by the de Rham complex. Linear elasticity is governed by a different complex whose operators include the symmetric gradient, compatibility, and divergence. The finite element spaces must approximate this tensor-valued complex, not only individual fields.

In the weak-symmetry formulation, the stress may be allowed to range over a larger tensor space while a skew-symmetric multiplier enforces symmetry weakly. This trades a difficult pointwise constraint for a saddle-point constraint that can be stabilized by compatible spaces.

The AFW result is therefore a second major FEEC application beside the mixed Hodge Laplacian theorem. It shows that homological finite element design extends beyond differential forms into tensor-valued complexes that arise from continuum mechanics.

Synthesis. Linearised elasticity exposes the full strength of the FEEC philosophy. The numerical method is stable when its finite element spaces reproduce the operator complex of the PDE. For elasticity, that means the symmetric-tensor stress complex, not only scalar approximation quality.

Full proof set Master

Proposition 1 (stress symmetry encodes angular-momentum balance). In ordinary linear elasticity without body couples, the Cauchy stress tensor is symmetric.

Proof. Balance of angular momentum says that internal tractions cannot produce a net infinitesimal torque in the absence of couple stresses. Localizing that balance to an infinitesimal material volume gives equality of the off-diagonal stress components. Hence the stress tensor lies in $\mathbb{S}$. $\square$

Proposition 2 (the mixed elasticity system is a saddle-point problem). The Hellinger-Reissner equations have the Babuška-Brezzi structure.

Proof. The stress bilinear form $(A\sigma ,\tau )$ is coercive on the kernel of the divergence constraint under the usual material assumptions. The coupling term $(\mathrm{div}\tau ,u)$ links stress to displacement as a constraint equation. Thus the unknown pair $(\sigma ,u)$ fits the mixed saddle-point pattern of a coercive form on a constrained kernel plus an inf-sup condition for the constraint operator. $\square$

Proposition 3 (the elasticity complex supplies the Fortin mechanism). A projection that commutes with the elasticity-complex operators yields the discrete stability transfer needed by the AFW method.

Proof. The continuous inf-sup proof chooses stress test fields whose divergence controls a given displacement test. A commuting projection maps those stress fields into the discrete stress space while preserving the divergence relation up to the chosen discrete displacement projection. Boundedness of the projection keeps the test-field norm under control. This is exactly the Fortin criterion for the discrete inf-sup condition. $\square$

Connections Master

• Babuška-Brezzi condition 24.01.04. Mixed elasticity is a saddle-point problem whose stability is measured by an inf-sup condition.

• FEEC convergence theorem 24.03.06. The proof pattern mirrors FEEC: compatible complexes plus bounded projections give stability and approximation.

• Mixed FEM for the Hodge Laplacian 24.04.01. The Hodge Laplacian is the model Hilbert-complex problem; elasticity is the symmetric-tensor analogue.

• Maxwell FEEC edge elements 24.04.02. Maxwell uses the de Rham complex, while elasticity uses the elasticity complex.

• Continuum mechanics 09.07.01. The mechanics unit supplies stress, strain, and displacement as physical variables.

Historical & philosophical context Master

Mixed elasticity elements were difficult because the natural physical stress variable is both tensor-valued and symmetric. Standard scalar finite element intuition did not supply the right spaces.

Arnold and Winther's 2002 element gave a stable mixed finite element for elasticity in two dimensions with symmetric stress ^{[Arnold-Winther]}. Arnold, Falk, and Winther's 2007 work developed stable mixed methods with weakly imposed symmetry, extending the construction and making the complex-based design more systematic ^{[Arnold-Falk-Winther]}.

The philosophical shift is the same one seen throughout FEEC: stability comes from respecting the PDE's operator complex. Elasticity shows that this principle is not limited to exterior derivatives and ordinary differential forms. It also governs tensor-valued complexes built from mechanics.

Bibliography Master

@article{ArnoldWinther2002Elasticity,
  author = {Arnold, Douglas N. and Winther, Ragnar},
  title = {Mixed finite elements for elasticity},
  journal = {Numerische Mathematik},
  volume = {92},
  pages = {401--419},
  year = {2002}
}

@article{ArnoldFalkWinther2007Elasticity,
  author = {Arnold, Douglas N. and Falk, Richard S. and Winther, Ragnar},
  title = {Mixed finite element methods for linear elasticity with weakly imposed symmetry},
  journal = {Mathematics of Computation},
  volume = {76},
  pages = {1699--1723},
  year = {2007}
}

@article{ArnoldFalkWinther2006ElasticityContext,
  author = {Arnold, Douglas N. and Falk, Richard S. and Winther, Ragnar},
  title = {Finite element exterior calculus, homological techniques, and applications},
  journal = {Acta Numerica},
  volume = {15},
  pages = {1--155},
  year = {2006}
}

@book{Arnold2018ElasticityFEEC,
  author = {Arnold, Douglas N.},
  title = {Finite Element Exterior Calculus},
  series = {CBMS-NSF Regional Conference Series in Applied Mathematics},
  volume = {93},
  publisher = {SIAM},
  year = {2018}
}

@book{BoffiBrezziFortin2013Elasticity,
  author = {Boffi, Daniele and Brezzi, Franco and Fortin, Michel},
  title = {Mixed Finite Element Methods and Applications},
  series = {Springer Series in Computational Mathematics},
  volume = {44},
  publisher = {Springer},
  year = {2013}
}