In the discretisation of incompressible flow, it has long been understood that schemes which are only discretely divergence-free can produce velocity errors that scale with the size of the pressure, while pointwise divergence-free schemes are pressure robust: their velocity error is insensitive to the irrotational part of the data.
In this talk, based on joint work with Pablo Brubeck and Charles Parker, I will argue that an entirely analogous phenomenon governs mixed discretisations whose underlying constitutive law involves a symmetric stress tensor — including Hellinger–Reissner elasticity, polar fluids, and viscous flow — and that it is controlled by the choice of strong versus weak stress symmetry.
We introduce the notion of a material-robust discretisation: a scheme is material robust if, upon scaling the component of the solution that lives in the kernel of the constitutive law by a parameter, the discrete stress error remains bounded as such parameter blow up. Material robustness is the natural counterpart of pressure robustness on the constitutive side, and it is intimately tied to angular momentum conservation: in the absence of body torques and couple stresses, pointwise symmetry of the discrete stress is equivalent to discrete conservation of angular momentum, and is precisely what is needed to rule out the loss of robustness.
For isotropic constitutive laws, we show that weakly symmetric schemes (Arnold–Falk–Winther and relatives) are already material robust, provided the auxiliary rotation space contains rigid-body motions — explaining why the issue has gone largely unnoticed. For anisotropic laws, motivated by liquid-crystal polymer networks and polar fluids, weakly symmetric schemes can produce arbitrarily large stress errors, even for stress-free configurations, while strongly symmetric schemes (Hu–Zhang, Johnson–Mercier–Křížek) remain robust. We unify these observations through an abstract saddle-point theory in which material robustness is encoded in the kernel inclusion discrete kernel in the continuous one — the same structure-preservation principle that underlies pressure robustness in FEEC discretisations.