2000 character limit reached
A pressure-robust Discrete de Rham scheme for the Navier-Stokes equations (2401.04456v1)
Published 9 Jan 2024 in math.NA and cs.NA
Abstract: In this work we design and analyse a Discrete de Rham (DDR) method for the incompressible Navier-Stokes equations. Our focus is, more specifically, on the SDDR variant, where a reduction in the number of unknowns is obtained using serendipity techniques. The main features of the DDR approach are the support of general meshes and arbitrary approximation orders. The method we develop is based on the curl-curl formulation of the momentum equation and, through compatibility with the Helmholtz-Hodge decomposition, delivers pressure-robust error estimates for the velocity. It also enables non-standard boundary conditions, such as imposing the value of the pressure on the boundary. In-depth numerical validation on a complete panel of tests including general polyhedral meshes is provided. The paper also contains an appendix where bounds on DDR potential reconstructions and differential operators are proved in the more general framework of Polytopal Exterior Calculus.
- “A Recursive Algebraic Coloring Technique for Hardware-Efficient Symmetric Sparse Matrix-Vector Multiplication” In ACM Trans. Parallel Comput. 7.3 New York, NY, USA: Association for Computing Machinery, 2020 DOI: 10.1145/3399732
- “Vector potentials in three-dimensional non-smooth domains” In Math. Methods Appl. Sci. 21.9, 1998, pp. 823–864 DOI: 10.1002/(SICI)1099-1476(199806)21:9¡823::AID-MMA976¿3.0.CO;2-B
- “An arbitrary order scheme on generic meshes for miscible displacements in porous media” In SIAM J. Sci. Comput. 40.4, 2018, pp. B1020–B1054 DOI: 10.1137/17M1138807
- Franck Assous, Patrick Ciarlet and Simon Labrunie “Mathematical foundations of computational electromagnetism” 198, Applied Mathematical Sciences Springer, Cham, 2018, pp. ix+458 DOI: 10.1007/978-3-319-70842-3
- “Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes” Accepted for publication In Comput. Meth. Appl. Mech. Engrg., 2022 arXiv:2112.09750 [math.NA]
- “Boundary conditions involving pressure for the Stokes problem and applications in computational hemodynamics” In Comput. Methods Appl. Mech. Engrg. 322, 2017, pp. 58–80 DOI: https://doi.org/10.1016/j.cma.2017.04.024
- “An exterior calculus framework for polytopal methods”, 2023, pp. 41p URL: https://arxiv.org/abs/2303.11093
- D. Castanon Quiroz and D.A. Di Pietro “A Hybrid High-Order method for the incompressible Navier–Stokes problem robust for large irrotational body forces” In Comput. Math. Appl. 79.8, 2020, pp. 2655–2677 DOI: 10.1016/j.camwa.2019.12.005
- D. Castañón Quiroz and D.A. Di Pietro “A pressure-robust HHO method for the solution of the incompressible Navier–Stokes equations on general meshes” Published online In IMA J. Numer. Anal., 2023 DOI: 10.1093/imanum/drad007
- D.A. Di Pietro and J. Droniou “The Hybrid High-Order method for polytopal meshes”, Modeling, Simulation and Application 19 Springer International Publishing, 2020 DOI: 10.1007/978-3-030-37203-3
- D.A. Di Pietro, J. Droniou and F. Rapetti “Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra” In Math. Models Methods Appl. Sci. 30.9, 2020, pp. 1809–1855 DOI: 10.1142/S0218202520500372
- “A discontinuous skeletal method for the viscosity-dependent Stokes problem” In Comput. Meth. Appl. Mech. Engrg. 306, 2016, pp. 175–195 DOI: 10.1016/j.cma.2016.03.033
- Daniele A. Di Pietro and Jérôme Droniou “An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency” In Found. Comput. Math. 23, 2023, pp. 85–164 DOI: 10.1007/s10208-021-09542-8
- Daniele A. Di Pietro and Jérôme Droniou “Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method” In M2AN Math. Model. Numer. Anal. 57, 2023, pp. 191–225 DOI: 10.1051/m2an/2022067
- “The gradient discretisation method” 82, Mathematics & Applications Springer, 2018, pp. 511p DOI: 10.1007/978-3-319-79042-8
- Richard S. Falk and Michael Neilan “Stokes complexes and the construction of stable finite elements with pointwise mass conservation” In SIAM J. Numer. Anal. 51.2, 2013, pp. 1308–1326 DOI: 10.1137/120888132
- “Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem” Published online In IMA J. Numer. Anal., 2020 DOI: 10.1093/imanum/draa073
- V. Girault “Curl-conforming finite element methods for Navier-Stokes equations with nonstandard boundary conditions in 𝐑3superscript𝐑3{\bf R}^{3}bold_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT” In The Navier-Stokes equations (Oberwolfach, 1988) 1431, Lecture Notes in Math. Springer, Berlin, 1990, pp. 201–218 DOI: 10.1007/BFb0086071
- “Finite element methods for Navier-Stokes equations” Theory and algorithms 5, Springer Series in Computational Mathematics Berlin: Springer-Verlag, 1986, pp. x+374
- V. Gol’dshtein, I. Mitrea and M. Mitrea “Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds” Problems in mathematical analysis. No. 52 In J. Math. Sci. (N.Y.) 172.3, 2011, pp. 347–400 DOI: 10.1007/s10958-010-0200-y
- Martin W. Licht “Smoothed projections and mixed boundary conditions” In Math. Comp. 88.316, 2019, pp. 607–635 DOI: 10.1090/mcom/3330
- “Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations” In Comput. Methods Appl. Mech. Engrg. 311, 2016, pp. 304–326 DOI: 10.1016/j.cma.2016.08.018
- Alexander Linke “On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime” In Comput. Methods Appl. Mech. Engrg. 268, 2014, pp. 782–800 DOI: 10.1016/j.cma.2013.10.011
- “Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls” In Arch. Ration. Mech. Anal. 207.3, 2013, pp. 919–968 DOI: 10.1007/s00205-012-0585-5
- Shuo Zhang “Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids” In Numer. Math. 133.2, 2016, pp. 371–408 DOI: 10.1007/s00211-015-0749-y