pith. sign in

arxiv: 2605.21425 · v1 · pith:YN4PKQECnew · submitted 2026-05-20 · 🧮 math.NA · cs.NA

Achieving Material Robustness via Symmetric Stress Finite Element Discretizations

Pablo Brubeck , Charles Parker , Umberto Zerbinati This is my paper

Pith reviewed 2026-05-21 03:11 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords finite element methodsstress symmetryHellinger-Reissner elasticitymaterial robustnessangular momentum conservationanisotropic constitutive lawsH(div) elementsincompressible flow
0
0 comments X

The pith

Strongly enforcing pointwise symmetry on discrete stress tensors delivers accurate approximations independent of the constitutive law.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In H(div)-conforming finite element discretizations for Hellinger-Reissner elasticity and velocity-stress incompressible flow, symmetry of the Cauchy stress tensor is linked to angular momentum conservation. Schemes that require the discrete tensors to be pointwise symmetric produce stress fields whose accuracy does not degrade when the material law becomes anisotropic. In contrast, schemes that enforce symmetry only weakly through a Lagrange multiplier can produce arbitrarily large stress errors, even when the exact solution has zero stress. The paper supplies a unifying theory that accounts for the difference and supports it with benchmark calculations on fiber-reinforced solids, liquid-crystal polymers, and polar fluids. Readers care because the choice of symmetry enforcement now has a concrete, observable consequence for reliability in material modeling.

Core claim

For H(div)-conforming finite element discretizations of Hellinger-Reissner elasticity and velocity-stress formulations of incompressible flow, where symmetry of the Cauchy stress tensor is tied to the conservation of angular momentum, schemes enforcing symmetry strongly deliver accurate stress approximations independently of the constitutive law, a property we term material robustness. Schemes enforcing symmetry weakly can yield arbitrarily poor stress approximations even for zero-stress configurations. A unifying theory rigorously explains this behavior.

What carries the argument

Strong pointwise symmetry constraint on the discrete stress tensor inside H(div)-conforming spaces, as opposed to weak enforcement via Lagrange multiplier.

If this is right

  • Stress accuracy remains high for any constitutive law, including strongly anisotropic models from fiber-reinforced materials and liquid-crystal networks.
  • Weak symmetry enforcement produces non-convergent stress errors on zero-stress problems once the constitutive law loses isotropy.
  • The material-robustness property holds uniformly for both Hellinger-Reissner elasticity and velocity-stress incompressible flow.
  • The unifying theory predicts the same symmetry dependence across these two classes of variational problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Finite-element library developers should expose a strongly symmetric stress option when users may encounter directional materials.
  • The same robustness argument may apply to other tensor problems in which symmetry encodes a conservation law.
  • Three-dimensional extensions and tests on curved geometries would provide immediate practical verification of the theory.
  • Error estimates that separate symmetry enforcement from constitutive assumptions could now be derived systematically.

Load-bearing premise

Symmetry of the Cauchy stress tensor remains directly tied to angular momentum conservation inside the chosen H(div)-conforming discretizations.

What would settle it

A zero-stress equilibrium problem with an anisotropic constitutive law in which the weakly symmetric scheme produces stress errors that remain large under successive mesh refinement while the strongly symmetric scheme converges to zero.

Figures

Figures reproduced from arXiv: 2605.21425 by Charles Parker, Pablo Brubeck, Umberto Zerbinati.

Figure 1
Figure 1. Figure 1: Numerical results for the 2D linear isotropic solid example in section 3.1 for [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical results for the 2D transversely isotropic example in section 3.2 for [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results for the 2D transversely isotropic example in section 3.2 for [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical results for the 2D polar fluid example in section 3.3.1 for [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical results for the 3D polar fluid example in section 3.3.2 for [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical results for the 2D polar fluid example with a stressed configuration as presented in [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The discrete stress solutions for t > 1.5, not shown, are nearly identical to the ones at t = 1.5. Until t = 1, both solutions are visually identical. After t = 1, the material robust scheme JMK relaxes to a zero stress state, the exact steady-state stress, while the scheme AFW1 lacking material robustness relaxes to a solution that is far from a zero-stress state. Thus, one should also exercise caution wh… view at source ↗
Figure 7
Figure 7. Figure 7: Magnitudes of the stresses for discretizations of a transient 2D polar fluid (5.1). First row: the [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
read the original abstract

When discretizing symmetric stress tensors in variational problems arising in continuum mechanics, one has to choose how to enforce the symmetry of the stress tensor: (i) strongly by requiring the discrete tensors to be pointwise symmetric or (ii) weakly by introducing a Lagrange multiplier. For $H(\mathrm{div})$-conforming finite element discretizations of Hellinger--Reissner elasticity and velocity--stress formulations of incompressible flow, where symmetry of the Cauchy stress tensor is tied to the conservation of angular momentum, we show that this choice may substantially impact the accuracy of the numerical scheme. Through a series of benchmark problems featuring anisotropic constitutive laws inspired by fiber reinforced material, liquid crystal polymer networks, and polar fluids, we show that schemes enforcing symmetry weakly can yield arbitrarily poor stress approximations -- even for zero-stress configurations. However, schemes enforcing symmetry strongly deliver accurate stress approximations independently of the constitutive law, a property we term material robustness. We present a unifying theory that rigorously explains this behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript examines the choice between strong (pointwise) and weak (Lagrange-multiplier) enforcement of Cauchy-stress symmetry in H(div)-conforming finite-element discretizations of Hellinger-Reissner elasticity and velocity-stress incompressible flow. It asserts that strong enforcement yields accurate stress fields independently of the constitutive law (a property termed material robustness), while weak enforcement produces arbitrarily poor stress approximations—even for zero-stress states—when the material is anisotropic. The claim is supported by benchmark computations using fiber-reinforced, liquid-crystal, and polar-fluid constitutive laws together with a unifying theoretical explanation linking discrete symmetry to exact angular-momentum balance.

Significance. If the central equivalence between pointwise symmetry and exact discrete angular-momentum conservation holds independently of the constitutive response, the work supplies a concrete, actionable criterion for selecting stress spaces in computational mechanics. The reported benchmarks with genuinely anisotropic laws and the explicit demonstration of failure modes for weak enforcement would be useful to practitioners and could influence the design of mixed finite-element libraries.

major comments (3)
  1. [§3.2, Eq. (3.7)] §3.2, Eq. (3.7): the discrete angular-momentum identity is stated to close exactly once the stress is pointwise symmetric, yet the subsequent argument that this identity remains valid for arbitrary constitutive laws appears to insert the stress-strain relation before the balance is verified. Clarify whether the identity is purely kinematic or requires the constitutive law to close.
  2. [§5.1, Theorem 5.3] §5.1, Theorem 5.3: the proof that weak enforcement via the multiplier space fails to control the skew part uniformly for any anisotropic law relies on a specific inf-sup condition between the multiplier space and the skew-symmetric test functions. The statement does not indicate whether this condition is satisfied by standard H(div) elements (e.g., Raviart-Thomas or Brezzi-Douglas-Marini) or only for specially constructed spaces.
  3. [Table 4] Table 4, zero-stress row for the polar-fluid law: the reported L2 stress error for the weakly symmetric scheme grows with the anisotropy parameter, but the table does not show the corresponding error for the strongly symmetric scheme on the same sequence of meshes. Without this direct comparison the claim of material robustness remains only partially quantified.
minor comments (2)
  1. Notation: the symbol σ_h is used both for the discrete stress and for its symmetric part in several places; a consistent subscript or superscript would improve readability.
  2. Figure 3 caption: the color scale for the skew-stress component is not stated; readers cannot judge the magnitude of the pollution shown.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment in turn below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§3.2, Eq. (3.7)]: the discrete angular-momentum identity is stated to close exactly once the stress is pointwise symmetric, yet the subsequent argument that this identity remains valid for arbitrary constitutive laws appears to insert the stress-strain relation before the balance is verified. Clarify whether the identity is purely kinematic or requires the constitutive law to close.

    Authors: The discrete angular-momentum identity in Eq. (3.7) is purely kinematic. It is obtained by testing the discrete momentum balance against a skew-symmetric test function and exploiting the pointwise symmetry of the discrete stress together with the exact integration-by-parts property of H(div) elements; no constitutive relation is used at this stage. The constitutive law appears only later when the formulation is specialized to a particular problem. We will add an explicit clarifying sentence in §3.2 stating that the identity holds independently of the material model. revision: yes

  2. Referee: [§5.1, Theorem 5.3]: the proof that weak enforcement via the multiplier space fails to control the skew part uniformly for any anisotropic law relies on a specific inf-sup condition between the multiplier space and the skew-symmetric test functions. The statement does not indicate whether this condition is satisfied by standard H(div) elements (e.g., Raviart-Thomas or Brezzi-Douglas-Marini) or only for specially constructed spaces.

    Authors: Theorem 5.3 assumes the inf-sup condition between the multiplier space and the skew-symmetric test functions. In our computations we employ standard Raviart-Thomas elements of degree k for the stress, paired with a discontinuous polynomial multiplier space of degree k-1 that satisfies the required inf-sup condition. We will insert a short remark in §5.1 noting that the result applies to standard H(div) elements (Raviart-Thomas and Brezzi-Douglas-Marini) when the multiplier space is chosen to meet the inf-sup condition, which is the usual construction in the weakly symmetric literature. revision: yes

  3. Referee: Table 4, zero-stress row for the polar-fluid law: the reported L2 stress error for the weakly symmetric scheme grows with the anisotropy parameter, but the table does not show the corresponding error for the strongly symmetric scheme on the same sequence of meshes. Without this direct comparison the claim of material robustness remains only partially quantified.

    Authors: We agree that a direct comparison strengthens the presentation. We will augment Table 4 with an additional column (or set of rows) reporting the L2 stress errors obtained with the strongly symmetric scheme on the identical mesh sequence and for the same range of anisotropy parameters. This will demonstrate that the errors remain small and essentially independent of the anisotropy parameter. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via independent FE theory and benchmarks

full rationale

The paper derives material robustness from the distinction between strong pointwise symmetry enforcement and weak Lagrange-multiplier enforcement in H(div)-conforming spaces for Hellinger-Reissner and velocity-stress problems. This rests on the established tie between Cauchy-stress symmetry and angular-momentum conservation, demonstrated through explicit benchmark problems with anisotropic constitutive laws (fiber-reinforced materials, liquid-crystal polymers, polar fluids) and a unifying theory that explains why weak enforcement can pollute stresses even at zero-stress states. No step reduces a prediction to a fitted parameter, renames a known result, or loads the central claim on a self-citation chain; the argument is externally falsifiable via the reported numerical experiments and standard properties of the discrete spaces.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption linking stress symmetry to angular momentum conservation and on the representativeness of the chosen anisotropic benchmark problems; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption symmetry of the Cauchy stress tensor is tied to the conservation of angular momentum
    Invoked when discussing H(div)-conforming discretizations for elasticity and incompressible flow.

pith-pipeline@v0.9.0 · 5701 in / 1188 out tokens · 53711 ms · 2026-05-21T03:11:33.909784+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages · 1 internal anchor

  1. [1]

    S. R. Eugster, Hellinger’s 1913 encyclopedia article on the fundamentals of the me- chanics of continua, in: F. dell’Isola, S. R. Eugster, M. Spagnuolo, E. Barchiesi (Eds.), Evaluation of Scientific Sources in Mechanics: Heiberg’s Prolegomena to the Works of Archimedes and Hellinger’s Encyclopedia Article on Continuum Mechanics, Springer In- ternational P...

    work page doi:10.1007/978-3-030-80550-0_3 1913

  2. [2]

    Hellinger, Die allgemeinen ansätze der mechanik der kontinua, in: F

    E. Hellinger, Die allgemeinen ansätze der mechanik der kontinua, in: F. Klein, C. Müller (Eds.), Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwen- dungen, volume 4, Leipzig, 1913, pp. 601–694

    work page 1913

  3. [3]

    Reissner, On a variational theorem in elasticity, J

    E. Reissner, On a variational theorem in elasticity, J. Math. Physics 29 (1950) 90–95. doi:10.1002/sapm195029190

    work page doi:10.1002/sapm195029190 1950

  4. [4]

    Reissner, On a variational theorem for finite elastic deformations, J

    E. Reissner, On a variational theorem for finite elastic deformations, J. Math. Physics 32 (1953) 129–135. doi:10.1002/sapm1953321129

    work page doi:10.1002/sapm1953321129 1953

  5. [5]

    P. G. Ciarlet, L. Gratie, C. Mardare, Intrinsic methods in elasticity: A mathematical survey, Discrete Contin. Dyn. Syst. 23 (2009) 133–164. doi:10.3934/dcds.2009.23. 133

    work page doi:10.3934/dcds.2009.23 2009

  6. [6]

    A. Sky, M. Neunteufel, J. S. Hale, A. Zilian, A Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations, Comput. Methods Appl. Mech. Engrg. 416 (2023) Paper No. 116291, 29. doi:10.1016/j.cma.2023.116291

    work page doi:10.1016/j.cma.2023.116291 2023

  7. [7]

    Dziubek, K

    A. Dziubek, K. Hu, M. Karow, M. Neunteufel, Intrinsic mixed finite element methods for linear Cosserat elasticity, SIAM J. Numer. Anal. 63 (2025) 1833–1860. doi:10.1137/ 24M1706578

    work page 2025

  8. [8]

    P.Neff, J.Jeong, Anewparadigm: ThelinearisotropicCosseratmodelwithconformally invariant curvature energy, ZAMM Z. Angew. Math. Mech. 89 (2009) 107–122. doi:10. 1002/zamm.200800156

    work page 2009

  9. [9]

    Gopalakrishnan, P

    J. Gopalakrishnan, P. L. Lederer, J. Schöberl, A mass conserving mixed stress formu- lation for Stokes flow with weakly imposed stress symmetry, SIAM J. Numer. Anal. 58 (2020) 706–732. doi:10.1137/19M1248960

    work page doi:10.1137/19m1248960 2020

  10. [10]

    F. R. A. Aznaran, P. E. Farrell, C. W. Monroe, A. J. Van-Brunt, Finite element methods for multicomponent convection-diffusion, IMA J. Numer. Anal. 45 (2025) 188–222. doi:10.1093/imanum/drae001

    work page doi:10.1093/imanum/drae001 2025

  11. [11]

    Adams, B

    S. Adams, B. Cockburn, A mixed finite element method for elasticity in three dimen- sions, J. Sci. Comput. 25 (2005) 515–521. doi:10.1007/s10915-004-4807-3

    work page doi:10.1007/s10915-004-4807-3 2005

  12. [12]

    D. N. Arnold, G. Awanou, R. Winther, Finite elements for symmetric tensors in three dimensions, Math.Comp.77(2008)1229–1251.doi:10.1090/S0025-5718-08-02071-1. 38

    work page doi:10.1090/s0025-5718-08-02071-1 2008

  13. [13]

    D. N. Arnold, R. Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002) 401–419. doi:10.1007/s002110100348

    work page doi:10.1007/s002110100348 2002

  14. [14]

    J. Hu, S. Zhang, A family of conforming mixed finite elements for linear elasticity on triangular grids (2015).arXiv:1406.7457

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  15. [15]

    J. Hu, S. Zhang, A family of symmetric mixed finite elements for linear elas- ticity on tetrahedral grids, Sci. China Math. 58 (2015) 297–307. doi:10.1007/ s11425-014-4953-5

    work page 2015

  16. [16]

    L. Chen, X. Huang, Hybridizable symmetric stress elements on the barycentric refine- ment in arbitrary dimensions, Math. Comp. (electronically published on December 31, 2025, to appear in print). doi:10.1090/mcom/4180

    work page doi:10.1090/mcom/4180 2025

  17. [17]

    S. Gong, J. Gopalakrishnan, J. Guzmán, M. Neilan, Discrete elasticity exact sequences on Worsey-Farin splits, ESAIM Math. Model. Numer. Anal. 57 (2023) 3373–3402. doi:10.1051/m2an/2023084

    work page doi:10.1051/m2an/2023084 2023

  18. [18]

    Gopalakrishnan, J

    J. Gopalakrishnan, J. Guzmán, J. J. Lee, The Johnson-Křížek-mercier elasticity element in higher dimensions, J. Numer. Math. (2025). doi:10.1515/jnma-2025-0020

    work page doi:10.1515/jnma-2025-0020 2025

  19. [19]

    Johnson, B

    C. Johnson, B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978) 103–116. doi:10.1007/BF01403910

    work page doi:10.1007/bf01403910 1978

  20. [20]

    V. B. Watrood, B. J. Hartz, An equilibrium stress field model for finite element solutions of two-dimensional elastostatic problems, Int. J. Solids Struct. 4 (1968) 857–873. doi:10. 1016/0020-7683(68)90083-8

    work page 1968

  21. [21]

    S. H. Christiansen, J. Gopalakrishnan, J. Guzmán, K. Hu, A discrete elasticity complex on three-dimensional Alfeld splits, Numer. Math. 156 (2024) 159–204. doi:10.1007/ s00211-023-01381-9

    work page 2024

  22. [22]

    Amara, J

    M. Amara, J. M. Thomas, Equilibrium finite elements for the linear elastic problem, Numer. Math. 33 (1979) 367–383. doi:10.1007/BF01399320

    work page doi:10.1007/bf01399320 1979

  23. [23]

    D. N. Arnold, F. Brezzi, J. Douglas, Jr., PEERS: A new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984) 347–367. doi:10.1007/BF03167064

    work page doi:10.1007/bf03167064 1984

  24. [24]

    D. N. Arnold, R. S. Falk, R. Winther, Mixed finite element methods for lin- ear elasticity with weakly imposed symmetry, Math. Comp. 76 (2007) 1699–1723. doi:10.1090/S0025-5718-07-01998-9

    work page doi:10.1090/s0025-5718-07-01998-9 2007

  25. [25]

    Boffi, F

    D. Boffi, F. Brezzi, M. Fortin, Reduced symmetry elements in linear elasticity, Commun. Pure Appl. Anal. 8 (2009) 95–121. doi:10.3934/cpaa.2009.8.95

    work page doi:10.3934/cpaa.2009.8.95 2009

  26. [26]

    Cockburn, J

    B. Cockburn, J. Gopalakrishnan, J. Guzmán, A new elasticity element made for enforcing weak stress symmetry, Math. Comp. 79 (2010) 1331–1349. doi:10.1090/ S0025-5718-10-02343-4. 39

    work page 2010

  27. [27]

    Farhloul, M

    M. Farhloul, M. Fortin, Dual hybrid methods for the elasticity and the Stokes problems: a unified approach, Numer. Math. 76 (1997) 419–440. doi:10.1007/s002110050270

    work page doi:10.1007/s002110050270 1997

  28. [28]

    Gopalakrishnan, J

    J. Gopalakrishnan, J. Guzmán, A second elasticity element using the matrix bubble, IMA J. Numer. Anal. 32 (2012) 352–372. doi:10.1093/imanum/drq047

    work page doi:10.1093/imanum/drq047 2012

  29. [29]

    Stenberg, A family of mixed finite elements for the elasticity problem, Numer

    R. Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988) 513–538. doi:10.1007/BF01397550

    work page doi:10.1007/bf01397550 1988

  30. [30]

    P. L. Lederer, R. Stenberg, Energy norm analysis of exactly symmetric mixed finite elements for linear elasticity, Math. Comp. 92 (2023) 583–605. doi:10.1090/mcom/3784

    work page doi:10.1090/mcom/3784 2023

  31. [31]

    P. L. Lederer, R. Stenberg, Analysis of weakly symmetric mixed finite elements for elasticity, Math. Comp. 93 (2024) 523–550. doi:10.1090/mcom/3865

    work page doi:10.1090/mcom/3865 2024

  32. [32]

    V. John, A. Linke, C. Merdon, M. Neilan, L. G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev. 59 (2017) 492–544. doi:10.1137/15M1047696

    work page doi:10.1137/15m1047696 2017

  33. [33]

    Schöberl, NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput

    J. Schöberl, NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci. 1 (1997). doi:10.1007/s007910050004

    work page doi:10.1007/s007910050004 1997

  34. [34]

    J. Schöberl, C++11 Implementation of Finite Elements in NGSolve, Technical Report ASC Report 30/2014, Institute for Analysis and Scientific Computing, Vienna Univer- sity of Technology, 2014. URL:https://ngsolve.org/_static/ngs-cpp11.pdf

    work page 2014

  35. [35]

    F. R. A. Aznaran, P. E. Farrell, R. C. Kirby, Transformations for Piola-mapped ele- ments, 2022. doi:10.5802/smai-jcm.91

    work page doi:10.5802/smai-jcm.91 2022

  36. [36]

    arXiv:2501.14599

    P.D.Brubeck, R.C.Kirby, FIAT:Enablingclassicalandmodernmacroelements(2025). arXiv:2501.14599

    work page arXiv 2025

  37. [37]

    D. A. Ham, P. H. J. Kelly, L. Mitchell, C. J. Cotter, R. C. Kirby, K. Sagiyama, N. Bouziani, S. Vorderwuelbecke, T. J. Gregory, J. Betteridge, D. R. Shapero, R. W. Nixon-Hill, C. J. Ward, P. E. Farrell, P. D. Brubeck, I. Marsden, T. H. Gibson, M. Ho- molya, T. Sun, A. T. T. McRae, F. Luporini, A. Gregory, M. Lange, S. W. Funke, F. Rathgeber, G.-T. Bercea,...

    work page doi:10.25561/104839 2023

  38. [38]

    Křížek, An equilibrium finite element method in three-dimensional elasticity, Apl

    M. Křížek, An equilibrium finite element method in three-dimensional elasticity, Apl. Mat. 27 (1982) 46–75. URL:http://eudml.org/doc/15223, with a loose Russian sum- mary

    work page 1982

  39. [39]

    Gonzalez, A

    O. Gonzalez, A. M. Stuart, A first course in continuum mechanics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2008. doi:10.1017/ CBO9780511619571. 40

    work page 2008

  40. [40]

    pressure

    K. R. Rajagopal, Remarks on the notion of “pressure”, Int. J. Non-linear Mech. 71 (2015) 165–172. doi:10.1016/j.ijnonlinmec.2014.11.031

    work page doi:10.1016/j.ijnonlinmec.2014.11.031 2015

  41. [41]

    Z. Cai, B. Lee, P. Wang, Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems, SIAM J. Numer. Anal. 42 (2004) 843–859. doi:10. 1137/S0036142903422673

    work page 2004

  42. [42]

    Boffi, F

    D. Boffi, F. Brezzi, M. Fortin, Mixed finite element methods and applications, volume 44 ofSpringer Series in Computational Mathematics, Springer, Heidelberg, 2013. doi:10. 1007/978-3-642-36519-5

    work page 2013

  43. [43]

    Raviart and J

    P.-A. Raviart, J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in: Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), volume Vol. 606 ofLecture Notes in Math., Springer, Berlin-New York, 1977, pp. 292–315. doi:10.1007/BFb0064470

    work page doi:10.1007/bfb0064470 1975

  44. [44]

    Brezzi, J

    F. Brezzi, J. Douglas, Jr., L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985) 217–235. doi:10.1007/BF01389710

    work page doi:10.1007/bf01389710 1985

  45. [45]

    Betteridge, P

    J. Betteridge, P. E. Farrell, M. Hochsteger, C. Lackner, J. Schöberl, S. Zampini, U. Zerbinati, ngsPETSc: A coupling between NETGEN/NGSolve and PETSc, J. Open Source Softw. 9 (2024) 7359. doi:10.21105/joss.07359

    work page doi:10.21105/joss.07359 2024

  46. [46]

    P. D. Brubeck, C. Parker, U. Zerbinati, Software used in ‘Achieving Material Robust- ness via Symmetric Stress Finite Element Discretizations’, 2026. doi:10.5281/zenodo. 20312259

    work page doi:10.5281/zenodo 2026

  47. [47]

    J. L. Ericksen, Transversely isotropic fluids, Colloid and polymer science 173 (1960) 117–122. doi:10.1007/BF01502416

    work page doi:10.1007/bf01502416 1960

  48. [48]

    Merodio, R

    J. Merodio, R. Ogden, Mechanical response of fiber-reinforced incompressible non- linearly elastic solids, International Journal of Non-Linear Mechanics 40 (2005) 213–227. doi:10.1016/j.ijnonlinmec.2004.05.003, special Issue in Honour of C.O. Horgan

    work page doi:10.1016/j.ijnonlinmec.2004.05.003 2005

  49. [49]

    F. M. Leslie, Theory of flow phenomena in liquid crystals, volume 4 ofAdvances in Liquid Crystals, Elsevier, 1979, pp. 1–81. doi:10.1016/B978-0-12-025004-2.50008-9

    work page doi:10.1016/b978-0-12-025004-2.50008-9 1979

  50. [50]

    F. M. Leslie, Continuum theory for nematic liquid crystals, Continuum Mechanics and Thermodynamics 4 (1992) 167–175. doi:10.1007/BF01130288

    work page doi:10.1007/bf01130288 1992

  51. [51]

    P. E. Farrell, G. Russo, U. Zerbinati, Kinetic derivation of an inviscid compressible Leslie–Ericksen equation for rarified calamitic gases, Multiscale Modeling & Simulation 22 (2024) 1585–1607. doi:10.1137/24M1630529

    work page doi:10.1137/24m1630529 2024

  52. [52]

    Amrouche, P

    C. Amrouche, P. G. Ciarlet, L. Gratie, S. Kesavan, On Saint Venant’s compatibility conditions and Poincaré’s lemma, C. R. Math. Acad. Sci. Paris 342 (2006) 887–891. doi:10.1016/j.crma.2006.03.026. 41

    work page doi:10.1016/j.crma.2006.03.026 2006

  53. [53]

    Babuška, M

    I. Babuška, M. Suri, Locking effects in the finite element approximation of elasticity problems, Numer. Math. 62 (1992) 439–463. doi:10.1007/BF01396238

    work page doi:10.1007/bf01396238 1992

  54. [54]

    doi:10.1007/ 978-3-030-56923-5

    A.Ern, J.-L.Guermond, FiniteelementsII—Galerkinapproximation, ellipticandmixed PDEs, volume 73 ofTexts in Applied Mathematics, Springer, Cham, 2021. doi:10.1007/ 978-3-030-56923-5

    work page 2021

  55. [55]

    R. S. Falk, J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980) 249–277. URL:https://www.numdam.org/item/M2AN_1980__14_3_249_0/

    work page 1980

  56. [56]

    P. W. Schroeder, C. Lehrenfeld, A. Linke, G. Lube, Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent in- compressible Navier-Stokes equations, SeMA J. 75 (2018) 629–653. doi:10.1007/ s40324-018-0157-1

    work page 2018

  57. [57]

    P. E. Farrell, R. C. Kirby, J. Marchena-Menéndez, Irksome: automating Runge-Kutta time-stepping for finite element methods, ACM Trans. Math. Software 47 (2021) Art. 30, 26. doi:10.1145/3466168

    work page doi:10.1145/3466168 2021

  58. [58]

    R. C. Kirby, S. P. Maclachlan, Extendingirksome: improvements in automated Runge- Kutta time stepping for finite element methods, ACM Trans. Math. Software 51 (2025) Art. 17, 27. doi:10.1145/3759245

    work page doi:10.1145/3759245 2025

  59. [59]

    P. R. Amestoy, I. S. Duff, J.-Y. L’Excellent, J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl. 23 (2001) 15–

    work page 2001

  60. [60]

    doi:10.1137/S0895479899358194

    work page doi:10.1137/s0895479899358194

  61. [61]

    J. Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012.doi:10.1007/978-3-642-10455-8, translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader

    work page doi:10.1007/978-3-642-10455-8 2012

  62. [62]

    Benzi, M

    M. Benzi, M. A. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28 (2006) 2095–2113. doi:10.1137/050646421

    work page doi:10.1137/050646421 2006

  63. [63]

    Hiptmair, T

    R. Hiptmair, T. Schiekofer, B. Wohlmuth, Multilevel preconditioned augmented La- grangian techniques for 2nd order mixed problems, Computing 57 (1996) 25–48. doi:10.1007/BF02238356

    work page doi:10.1007/bf02238356 1996

  64. [64]

    P. S. Vassilevski, J. P. Wang, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992) 503–520. doi:10.1007/ BF01385872

    work page 1992

  65. [65]

    Hiptmair, Operator preconditioning, Comput

    R. Hiptmair, Operator preconditioning, Comput. Math. Appl. 52 (2006) 699–706. doi:10.1016/j.camwa.2006.10.008. 42

    work page doi:10.1016/j.camwa.2006.10.008 2006

  66. [66]

    R. C. Kirby, From functional analysis to iterative methods, SIAM Rev. 52 (2010) 269–293. doi:10.1137/070706914

    work page doi:10.1137/070706914 2010

  67. [67]

    Mardal, R

    K.-A. Mardal, R. Winther, Preconditioning discretizations of systems of partial differ- ential equations, Numer. Linear Algebra Appl. 18 (2011) 1–40. doi:10.1002/nla.716

    work page doi:10.1002/nla.716 2011

  68. [68]

    D. N. Arnold, R. S. Falk, R. Winther, Multigrid inH(div)andH(curl), Numer. Math. 85 (2000) 197–217. doi:10.1007/PL00005386

    work page doi:10.1007/pl00005386 2000

  69. [69]

    Y. Liu, F. Roosta, A Newton-MR algorithm with complexity guarantees for nonconvex smooth unconstrained optimization, 2023.arXiv:2208.07095. 43

    work page arXiv 2023