An improved understanding of the divergence-free constraint for the incompressible Navier–Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of pressure-robustness allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order
DOI : 10.5802/smai-jcm.44
Mots-clés : incompressible Navier–Stokes, pressure-robust methods, Helmholtz–Hodge projector, Discontinuous Galerkin method, divergence-free
@article{SMAI-JCM_2019__5__89_0, author = {Gauger, Nicolas R. and Linke, Alexander and Schroeder, Philipp W.}, title = {On high-order pressure-robust space discretisations, their advantages for incompressible high {Reynolds} number generalised {Beltrami} flows and beyond}, journal = {The SMAI Journal of computational mathematics}, pages = {89--129}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {5}, year = {2019}, doi = {10.5802/smai-jcm.44}, zbl = {07090176}, language = {en}, url = {http://www.numdam.org/articles/10.5802/smai-jcm.44/} }
TY - JOUR AU - Gauger, Nicolas R. AU - Linke, Alexander AU - Schroeder, Philipp W. TI - On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond JO - The SMAI Journal of computational mathematics PY - 2019 SP - 89 EP - 129 VL - 5 PB - Société de Mathématiques Appliquées et Industrielles UR - http://www.numdam.org/articles/10.5802/smai-jcm.44/ DO - 10.5802/smai-jcm.44 LA - en ID - SMAI-JCM_2019__5__89_0 ER -
%0 Journal Article %A Gauger, Nicolas R. %A Linke, Alexander %A Schroeder, Philipp W. %T On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond %J The SMAI Journal of computational mathematics %D 2019 %P 89-129 %V 5 %I Société de Mathématiques Appliquées et Industrielles %U http://www.numdam.org/articles/10.5802/smai-jcm.44/ %R 10.5802/smai-jcm.44 %G en %F SMAI-JCM_2019__5__89_0
Gauger, Nicolas R.; Linke, Alexander; Schroeder, Philipp W. On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond. The SMAI Journal of computational mathematics, Tome 5 (2019), pp. 89-129. doi : 10.5802/smai-jcm.44. http://www.numdam.org/articles/10.5802/smai-jcm.44/
[1] On really locking-free mixed finite element methods for the transient incompressible Stokes equations, SIAM J. Numer. Anal., Volume 56 (2018) no. 1, pp. 185-209 | MR | Zbl
[2] Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations, Comput. Methods Appl. Math., Volume 18 (2018) no. 3, pp. 353-372 | MR | Zbl
[3] The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes, Comput. Methods Appl. Mech. Engrg., Volume 341 (2018), pp. 917-938 | DOI | MR
[4] Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem, Numer. Methods Partial Differential Equations, Volume 31 (2015) no. 4, pp. 1224-1250 | MR | Zbl
[5] A stable finite element for the Stokes equations, Calcolo, Volume 21 (1984) no. 4, p. 337-344 (1985) | DOI | MR | Zbl
[6] Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., Volume 25 (1997) no. 2–3, pp. 151-167 | DOI | MR | Zbl
[7] Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., Volume 32 (1995) no. 3, pp. 797-823 | DOI | MR | Zbl
[8] Mathematics and turbulence: where do we stand?, J. Turbul., Volume 14 (2013) no. 3, pp. 42-76 | MR
[9] On stabilized finite element methods for the Stokes problem in the small time step limit, Internat. J. Numer. Methods Fluids, Volume 53 (2007) no. 4, pp. 573-597 | DOI | MR | Zbl
[10] Mixed Finite Element Methods and Applications, Springer-Verlag Berlin Heidelberg, 2013 | Zbl
[11] Well balanced finite volume methods for nearly hydrostatic flows, J. Comput. Phys., Volume 196 (2004) no. 2, pp. 539-565 | MR | Zbl
[12] New connections between finite element formulations of the Navier–Stokes equations, J. Comput. Phys., Volume 229 (2010) no. 24, pp. 9020-9025 | DOI | MR | Zbl
[13] Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Springer-Verlag New York, 2013 | Zbl
[14] Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 32 (1982) no. 1-3, pp. 199-259 FENOMECH ’81, Part I (Stuttgart, 1981) | DOI | MR | Zbl
[15] IsoGeometric Analysis: Stable elements for the 2D Stokes equation, Int. J. Numer. Meth. Fluids, Volume 65 (2011) no. 11–12, pp. 1407-1422 | DOI | MR | Zbl
[16] Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence, Numer. Math., Volume 107 (2007) no. 1, pp. 39-77 | MR | Zbl
[17] A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier–Stokes equations, SIAM J. Numer. Anal., Volume 49 (2011) no. 4, pp. 1461-1481 | DOI | MR | Zbl
[18] A mathematical introduction to fluid mechanics, Texts in Applied Mathematics, 4, Springer-Verlag, New York, 1993, xii+169 pages | DOI | MR | Zbl
[19] A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations, J. Sci. Comput., Volume 31 (2007) no. 1–2, pp. 61-73 | DOI | MR | Zbl
[20] Mixed finite elements for numerical weather prediction, J. Comput. Phys., Volume 231 (2012) no. 21, pp. 7076-7091 | DOI | MR | Zbl
[21] A finite element exterior calculus framework for the rotating shallow-water equations, J. Comput. Phys., Volume 257 (2014) no. part B, pp. 1506-1526 | DOI | MR | Zbl
[22] Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math., Volume 44 (2018) no. 1, pp. 195-225 | DOI | MR | Zbl
[23] A discontinuous skeletal method for the viscosity-dependent Stokes problem, Comput. Methods Appl. Mech. Engrg., Volume 306 (2016), pp. 175-195 | DOI | MR
[24] Mathematical Aspects of Discontinuous Galerkin Methods, Springer-Verlag Berlin, 2012 | Zbl
[25] Aspects of finite element discretizations for solving the Boussinesq approximation of the Navier–Stokes Equations, Notes on Numerical Fluid Mechanics: Numerical Methods for the Navier–Stokes Equations., Volume 47 (1994), pp. 50-61 | Zbl
[26] The Navier–Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge University Press, 2006 | Zbl
[27] Theory and Practice of Finite Elements, Springer New York, 2004 | Zbl
[28] Exact fully 3D Navier–Stokes solutions for benchmarking, Int. J. Numer. Meth. Fluids, Volume 19 (1994) no. 5, pp. 369-375 | DOI | Zbl
[29] IsoGeometric divergence-conforming B-splines for the steady Navier–Stokes equations, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 8, pp. 1421-1478 | DOI | MR | Zbl
[30] Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations, J. Comput. Phys., Volume 241 (2013), pp. 141-167 | DOI | MR | Zbl
[31] Newer and newer elements for incompressible flows, Finite Elements in Fluids, Volume 6 (1985), pp. 171-187 | Zbl
[32] Two classes of mixed finite element methods, Comput. Methods Appl. Mech. Engrg., Volume 69 (1988) no. 1, pp. 89-129 | MR | Zbl
[33] Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Engrg., Volume 237/240 (2012), pp. 166-176 | DOI | MR | Zbl
[34] Pressure separation — a technique for improving the velocity error in finite element discretisations of the Navier–Stokes equations, Appl. Math. Comp., Volume 165 (2005) no. 2, pp. 275-290 | MR | Zbl
[35] Spurious velocities in the steady flow of an incompressible fluid subjected to external forces, Int. J. Numer. Meth. Fluids, Volume 25 (1997) no. 6, pp. 679-695 | DOI | MR | Zbl
[36] Max-norm estimates for Stokes and Navier–Stokes approximations in convex polyhedra, Numer. Math., Volume 131 (2015) no. 4, pp. 771-822 | DOI | MR | Zbl
[37] On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: Implementation, Int. J. Numer. Meth. Fluids, Volume 11 (1990) no. 5, pp. 621-659 | DOI | Zbl
[38] A new finite element for incompressible or Boussinesq fluids, Proceedings of the Third International Conference on Finite Elements in Flow Problems, Wiley (1980) | Zbl
[39] Development of the Discontinuous Galerkin Method for High-Resolution, Large Scale CFD and Acoustics in Industrial Geometries, Ph.D. thesis, Université catholique de Louvain, 2013
[40] On the parameter choice in grad-div stabilization for the Stokes equations, Adv. Comput. Math., Volume 40 (2014) no. 2, pp. 491-516 | DOI | MR | Zbl
[41] Finite Element Methods for Incompressible Flow Problems, Springer International Publishing, 2016 | Zbl
[42] On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., Volume 59 (2017) no. 3, pp. 492-544 | DOI | MR | Zbl
[43] Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, Oxford, 2005 | DOI | Zbl
[44] Arbitrary Order Finite Volume Well-Balanced Schemes for the Euler Equations with Gravity, SIAM J. Sci. Comput., Volume 41 (2019) no. 2, p. A695-A721 | DOI | MR | Zbl
[45] Hybrid Discontinuous Galerkin methods for solving incompressible flow problems, Rheinisch-Westfälische Technische Hochschule Aachen (2010) (Masters thesis)
[46] A divergence-free velocity reconstruction for incompressible flows, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 17-18, pp. 837-840 | DOI | MR | Zbl
[47] On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Engrg., Volume 268 (2014), pp. 782-800 | DOI | MR | Zbl
[48] Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 311 (2016), pp. 304-326 | DOI | MR | Zbl
[49] Towards pressure-robust mixed methods for the incompressible Navier–Stokes equations, Comput. Methods Appl. Math., Volume 18 (2018) no. 3, pp. 353-372 | MR | Zbl
[50] Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-Problem, Math. Comp., Volume 87 (2018) no. 312, pp. 1543-1566 | DOI | MR | Zbl
[51] Pressure-induced locking in mixed methods for time-dependent (Navier–)Stokes equations, J. Comput. Phys., Volume 388 (2019), pp. 350-356 | DOI | MR
[52] Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors, ESAIM: M2AN, Volume 50 (2016) no. 1, pp. 289-309 | DOI | MR | Zbl
[53] Stable finite-element calculation of incompressible flows using the rotation form of convection, IMA J. Numer. Anal., Volume 22 (2002) no. 3, pp. 437-461 | MR | Zbl
[54] Grad-div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 198 (2009) no. 49-52, pp. 3975-3988 | DOI | MR | Zbl
[55] Are FEM solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!), Internat. J. Numer. Methods Fluids, Volume 9 (1989) no. 1, pp. 99-112 | DOI | MR | Zbl
[56] Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, SIAM, 2008 | DOI | Zbl
[57] Parallele Lösung der Navier–Stokes-Gleichungen, University of Magdeburg (1997) (Habilitation) | Zbl
[58] C++11 Implementation of Finite Elements in NGSolve (2014) no. 30/2014 https://www.asc.tuwien.ac.at/~schoeberl/wiki/publications/ngs-cpp11.pdf (ASC Report)
[59] Robustness of High-Order Divergence-Free Finite Element Methods for Incompressible Computational Fluid Dynamics, Georg-August-Universität Göttingen (2019) https://hdl.handle.net/11858/00-1735-0000-002E-E5BC-8 (Ph. D. Thesis)
[60] Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations, SeMA J., Volume 75 (2018) no. 4, pp. 629-653 | DOI | MR | Zbl
[61] Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier–Stokes flows, J. Numer. Math., Volume 25 (2017) no. 4, pp. 249-276 | DOI | MR | Zbl
[62] A locally mass conserving quadratic velocity, linear pressure element, 1987 (Numerical Analysis Report No, 147, Manchester University/UMIST)
[63] Spectral/hp element methods: Recent developments, applications, and perspectives, J. Hydrodyn., Volume 30 (2018), pp. 1-22 | DOI
[64] On the Rotation and Skew-symmetric Forms for Incompressible Flow Simulations, Appl. Numer. Math., Volume 7 (1991) no. 1, pp. 27-40 | Zbl
[65] A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp., Volume 74 (2005) no. 250, pp. 543-554 | DOI | MR
- A Reynolds-semi-robust and pressure-robust Hybrid High-Order method for the time dependent incompressible Navier–Stokes equations on general meshes, Computer Methods in Applied Mechanics and Engineering, Volume 436 (2025), p. 117660 | DOI:10.1016/j.cma.2024.117660
- Energy conservation for 3D Euler and Navier-Stokes equations in a bounded domain: applications to Beltrami flows, Journal of Nonlinear Science, Volume 35 (2025) no. 1, p. 30 (Id/No 10) | DOI:10.1007/s00332-024-10102-x | Zbl:7948925
- A low-cost, penalty parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations, Computers Mathematics with Applications, Volume 166 (2024), pp. 51-64 | DOI:10.1016/j.camwa.2024.04.023 | Zbl:7859659
- Pressure-robust enriched Galerkin methods for the Stokes equations, Journal of Computational and Applied Mathematics, Volume 436 (2024), p. 19 (Id/No 115449) | DOI:10.1016/j.cam.2023.115449 | Zbl:1522.65213
- Gradient-robust hybrid DG discretizations for the compressible Stokes equations, Journal of Scientific Computing, Volume 100 (2024) no. 2, p. 26 (Id/No 54) | DOI:10.1007/s10915-024-02605-2 | Zbl:7902980
- Inf-sup stabilized Scott-Vogelius pairs on general shape-regular simplicial grids by Raviart-Thomas enrichment, M
AS. Mathematical Models Methods in Applied Sciences, Volume 34 (2024) no. 5, pp. 919-949 | DOI:10.1142/s0218202524500180 | Zbl:1540.65488 - Chaotic dynamics of two-dimensional flows around a cylinder, Physics of Fluids, Volume 36 (2024) no. 2 | DOI:10.1063/5.0186496
- Velocity-vorticity geometric constraints for the energy conservation of 3D ideal incompressible fluids, The Journal of Geometric Analysis, Volume 34 (2024) no. 8, p. 16 (Id/No 259) | DOI:10.1007/s12220-024-01704-8 | Zbl:1543.35140
- The virtual element method, Acta Numerica, Volume 32 (2023), pp. 123-202 | DOI:10.1017/s0962492922000095 | Zbl:7736655
- An EMA-conserving, pressure-robust and re-semi-robust method with a robust reconstruction method for Navier-Stokes, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 57 (2023) no. 2, pp. 467-490 | DOI:10.1051/m2an/2022093 | Zbl:1529.65076
- Virtual element method for the Navier–Stokes equation coupled with the heat equation, IMA Journal of Numerical Analysis, Volume 43 (2023) no. 6, p. 3396 | DOI:10.1093/imanum/drac072
- Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow, Journal of Computational Physics, Volume 478 (2023), p. 18 (Id/No 111961) | DOI:10.1016/j.jcp.2023.111961 | Zbl:7660332
- Pressure-robustness in the context of optimal control, SIAM Journal on Control and Optimization, Volume 61 (2023) no. 1, pp. 340-358 | DOI:10.1137/22m1482603 | Zbl:1510.49039
- An arbitrary order and pointwise divergence-free finite element scheme for the incompressible 3D Navier-Stokes equations, SIAM Journal on Numerical Analysis, Volume 61 (2023) no. 2, pp. 784-811 | DOI:10.1137/21m1443686 | Zbl:1512.35439
- Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization, SIAM Journal on Scientific Computing, Volume 45 (2023) no. 6, p. b853-b883 | DOI:10.1137/22m1478598 | Zbl:1533.65046
- Analysis and computation of a pressure-robust method for the rotation form of the incompressible Navier-Stokes equations with high-order finite elements, Computers Mathematics with Applications, Volume 112 (2022), pp. 1-22 | DOI:10.1016/j.camwa.2022.02.017 | Zbl:1524.76144
- Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem, IMA Journal of Numerical Analysis, Volume 42 (2022) no. 1, p. 597 | DOI:10.1093/imanum/draa073
- A low-order divergence-free H(div)-conforming finite element method for Stokes flows, IMA Journal of Numerical Analysis, Volume 42 (2022) no. 4, p. 3711 | DOI:10.1093/imanum/drab080
- Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations, Journal of Numerical Mathematics, Volume 30 (2022) no. 4, pp. 267-294 | DOI:10.1515/jnma-2021-0078 | Zbl:1527.76033
- HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations, Journal of Scientific Computing, Volume 92 (2022) no. 1, p. 38 (Id/No 28) | DOI:10.1007/s10915-022-01864-1 | Zbl:1490.65189
- A lowest-degree conservative finite element scheme for incompressible Stokes problems on general triangulations, Journal of Scientific Computing, Volume 93 (2022) no. 1, p. 31 (Id/No 28) | DOI:10.1007/s10915-022-01974-w | Zbl:1497.65229
- Numerical analysis of a discontinuous Galerkin method for the Borrvall-Petersson topology optimization problem, SIAM Journal on Numerical Analysis, Volume 60 (2022) no. 5, pp. 2538-2564 | DOI:10.1137/21m1438943 | Zbl:1503.35180
- An augmented Lagrangian preconditioner for the magnetohydrodynamics equations at high Reynolds and coupling numbers, SIAM Journal on Scientific Computing, Volume 44 (2022) no. 4, p. b1018-b1044 | DOI:10.1137/21m1416539 | Zbl:1505.65305
- A low-degree strictly conservative finite element method for incompressible flows on general triangulations, SMAI Journal of Computational Mathematics, Volume 8 (2022), pp. 225-248 | DOI:10.5802/smai-jcm.85 | Zbl:1501.65135
- An introduction to second order divergence-free VEM for fluidodynamics, The virtual element method and its applications, Cham: Birkhäuser, 2022, pp. 185-225 | DOI:10.1007/978-3-030-95319-5_5 | Zbl:1503.65286
- Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading, Calcolo, Volume 58 (2021) no. 2, p. 20 (Id/No 15) | DOI:10.1007/s10092-021-00402-z | Zbl:1477.65187
- A pressure-robust virtual element method for the Stokes problem, Computer Methods in Applied Mechanics and Engineering, Volume 382 (2021), p. 19 (Id/No 113879) | DOI:10.1016/j.cma.2021.113879 | Zbl:1506.76102
- Locking-free and gradient-robust
-conforming HDG methods for linear elasticity, Journal of Scientific Computing, Volume 86 (2021) no. 3, p. 31 (Id/No 39) | DOI:10.1007/s10915-020-01396-6 | Zbl:1460.65146 - A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation, SIAM Journal on Numerical Analysis, Volume 59 (2021) no. 5, pp. 2746-2774 | DOI:10.1137/20m1351230 | Zbl:1477.65185
- A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations, SMAI Journal of Computational Mathematics, Volume 7 (2021), pp. 75-96 | DOI:10.5802/smai-jcm.72 | Zbl:1472.65155
- Implicit LES with High-Order H(div)-Conforming FEM for Incompressible Navier-Stokes Flows, Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2018, Volume 135 (2020), p. 157 | DOI:10.1007/978-3-030-41800-7_10
- A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem, Computer Methods in Applied Mechanics and Engineering, Volume 367 (2020), p. 24 (Id/No 113069) | DOI:10.1016/j.cma.2020.113069 | Zbl:1442.76059
- Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem, ETNA. Electronic Transactions on Numerical Analysis, Volume 52 (2020), pp. 281-294 | DOI:10.1553/etna_vol52s281 | Zbl:1446.65146
- On the significance of pressure-robustness for the space discretization of incompressible high Reynolds number flows, Finite volumes for complex applications IX – methods, theoretical aspects, examples. FVCA 9, Bergen, Norway, June 15–19, 2020. In 2 volumes, Cham: Springer, 2020, pp. 103-112 | DOI:10.1007/978-3-030-43651-3_7 | Zbl:1454.65169
- Divergence-free tangential finite element methods for incompressible flows on surfaces, International Journal for Numerical Methods in Engineering, Volume 121 (2020) no. 11, pp. 2503-2533 | DOI:10.1002/nme.6317 | Zbl:7841929
- Pressure robust weak Galerkin finite element methods for Stokes problems, SIAM Journal on Scientific Computing, Volume 42 (2020) no. 3, p. b608-b629 | DOI:10.1137/19m1266320 | Zbl:1440.65224
- Helmholtz's decomposition for compressible flows and its application to computational aeroacoustics, SN Partial Differential Equations and Applications, Volume 1 (2020) no. 6, p. 19 (Id/No 46) | DOI:10.1007/s42985-020-00044-w | Zbl:1457.76016
- The Stokes complex for virtual elements with application to Navier-Stokes flows, Journal of Scientific Computing, Volume 81 (2019) no. 2, pp. 990-1018 | DOI:10.1007/s10915-019-01049-3 | Zbl:1446.65185
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