An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

Hanot, Marien-Lorenzo

doi:10.1090/mcom/3837

An arbitrary-order fully discrete Stokes complex on general polyhedral meshes
HTML articles powered by AMS MathViewer

by Marien-Lorenzo Hanot HTML | PDF

Math. Comp. 92 (2023), 1977-2023 Request permission

Abstract:

In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enrich the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into the complex. We show a complete set of results on the novelties of this complex: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. The Stokes complex is especially well suited for problem involving Jacobian, divergence and curl, like the Stokes problem or magnetohydrodynamic systems. The framework developed here eases the design and analysis of schemes for such problems. Schemes built that way are nonconforming and benefit from the exactness of the complex. We illustrate with the design and study of a scheme solving the Stokes equations and validate the convergence rates with various numerical tests.

References

Douglas N. Arnold, Finite element exterior calculus, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 93, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018. MR 3908678, DOI 10.1137/1.9781611975543.ch1
Douglas N. Arnold, Richard S. Falk, and Jay Gopalakrishnan, Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions, Math. Models Methods Appl. Sci. 22 (2012), no. 9, 1250024, 26. MR 2974162, DOI 10.1142/S0218202512500248
Franck Boyer and Pierre Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013. MR 2986590, DOI 10.1007/978-1-4614-5975-0
Erik Burman, Snorre H. Christiansen, and Peter Hansbo, Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics, Comput. Methods Appl. Mech. Engrg. 369 (2020), 113224, 20. MR 4118823, DOI 10.1016/j.cma.2020.113224
Snorre H. Christiansen and Kaibo Hu, Generalized finite element systems for smooth differential forms and Stokes’ problem, Numer. Math. 140 (2018), no. 2, 327–371. MR 3851060, DOI 10.1007/s00211-018-0970-6
L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo, A family of three-dimensional virtual elements with applications to magnetostatics, SIAM J. Numer. Anal. 56 (2018), no. 5, 2940–2962. MR 3860122, DOI 10.1137/18M1169886
L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo, Lowest order virtual element approximation of magnetostatic problems, Comput. Methods Appl. Mech. Engrg. 332 (2018), 343–362. MR 3764431, DOI 10.1016/j.cma.2017.12.028
L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, $H(\text {div})$ and $H(\mathbf {curl})$-conforming virtual element methods, Numer. Math. 133 (2016), no. 2, 303–332. MR 3489088, DOI 10.1007/s00211-015-0746-1
Lourenço Beirão da Veiga, Franco Dassi, Daniele A. Di Pietro, and Jérôme Droniou, Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes, Comput. Methods Appl. Mech. Engrg. 397 (2022), Paper No. 115061, 31. MR 4436363, DOI 10.1016/j.cma.2022.115061
L. Beirão da Veiga, F. Dassi, and G. Vacca, The Stokes complex for virtual elements in three dimensions, Math. Models Methods Appl. Sci. 30 (2020), no. 3, 477–512. MR 4082227, DOI 10.1142/S0218202520500128
Lourenco Beirão da Veiga, Carlo Lovadina, and Giuseppe Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 2, 509–535. MR 3626409, DOI 10.1051/m2an/2016032
Daniele A. Di Pietro and Jérôme Droniou, A third Strang lemma and an Aubin-Nitsche trick for schemes in fully discrete formulation, Calcolo 55 (2018), no. 3, Paper No. 40, 39. MR 3850320, DOI 10.1007/s10092-018-0282-3
Daniele Antonio Di Pietro and Jérôme Droniou, The hybrid high-order method for polytopal meshes, MS&A. Modeling, Simulation and Applications, vol. 19, Springer, Cham, [2020] ©2020. Design, analysis, and applications. MR 4230986, DOI 10.1007/978-3-030-37203-3
Daniele A. Di Pietro, Jérôme Droniou, and Francesca Rapetti, Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, Math. Models Methods Appl. Sci. 30 (2020), no. 9, 1809–1855. MR 4151796, DOI 10.1142/S0218202520500372
Richard S. Falk and Michael Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal. 51 (2013), no. 2, 1308–1326. MR 3045658, DOI 10.1137/120888132
Johnny Guzmán and Michael Neilan, A family of nonconforming elements for the Brinkman problem, IMA J. Numer. Anal. 32 (2012), no. 4, 1484–1508. MR 2991835, DOI 10.1093/imanum/drr040
Johnny Guzmán and Michael Neilan, Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comp. 83 (2014), no. 285, 15–36. MR 3120580, DOI 10.1090/S0025-5718-2013-02753-6
Kaibo Hu, Qian Zhang, and Zhimin Zhang, A family of finite element Stokes complexes in three dimensions, SIAM J. Numer. Anal. 60 (2022), no. 1, 222–243. MR 4369574, DOI 10.1137/20M1358700
X. Huang, Nonconforming finite element Stokes complexes in three dimensions, Sci. China Math. (2023), DOI 10.1007/s10092-018-0282-3.
Volker John, Alexander Linke, Christian Merdon, Michael Neilan, and Leo G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev. 59 (2017), no. 3, 492–544. MR 3683678, DOI 10.1137/15M1047696
Giovanni Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, RI, 2009. MR 2527916, DOI 10.1090/gsm/105
Kent Andre Mardal, Xue-Cheng Tai, and Ragnar Winther, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal. 40 (2002), no. 5, 1605–1631. MR 1950614, DOI 10.1137/S0036142901383910
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
Michael Neilan, Discrete and conforming smooth de Rham complexes in three dimensions, Math. Comp. 84 (2015), no. 295, 2059–2081. MR 3356019, DOI 10.1090/S0025-5718-2015-02958-5
Daniele A. Di Pietro and Jérôme Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes: exactness, Poincaré inequalities, and consistency, Found. Comput. Math. 23 (2023), no. 1, 85–164. MR 4546145, DOI 10.1007/s10208-021-09542-8
Xue-Cheng Tai and Ragnar Winther, A discrete de Rham complex with enhanced smoothness, Calcolo 43 (2006), no. 4, 287–306. MR 2283095, DOI 10.1007/s10092-006-0124-6
Shangyou Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp. 74 (2005), no. 250, 543–554. MR 2114637, DOI 10.1090/S0025-5718-04-01711-9
Shangyou Zhang, Divergence-free finite elements on tetrahedral grids for $k\geq 6$, Math. Comp. 80 (2011), no. 274, 669–695. MR 2772092, DOI 10.1090/S0025-5718-2010-02412-3
Shuo Zhang, Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids, Numer. Math. 133 (2016), no. 2, 371–408. MR 3489090, DOI 10.1007/s00211-015-0749-y