Convergence to weak solutions of a space-time hybridized discontinuous Galerkin method for the incompressible Navier–Stokes equations

Kirk, Keegan; Çeşmeli̇oğlu, Ayçıl; Rhebergen, Sander

doi:10.1090/mcom/3780

Convergence to weak solutions of a space-time hybridized discontinuous Galerkin method for the incompressible Navier–Stokes equations
HTML articles powered by AMS MathViewer

by Keegan L. A. Kirk, Ayçıl Çeşmeli̇oğlu and Sander Rhebergen HTML | PDF

Math. Comp. 92 (2023), 147-174 Request permission

Abstract:

We prove that a space-time hybridized discontinuous Galerkin method for the evolutionary Navier–Stokes equations converges to a weak solution as the time step and mesh size tend to zero. Moreover, we show that this weak solution satisfies the energy inequality. To perform our analysis, we make use of discrete functional analysis tools and a discrete version of the Aubin–Lions–Simon theorem.

References

Naveed Ahmed and Gunar Matthies, Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier-Stokes equations, IMA J. Numer. Anal. 41 (2021), no. 4, 3113–3144. MR 4328410, DOI 10.1093/imanum/draa053
S. Badia, R. Codina, T. Gudi, and J. Guzmán, Error analysis of discontinuous Galerkin methods for the Stokes problem under minimal regularity, IMA J. Numer. Anal. 34 (2014), no. 2, 800–819. MR 3194809, DOI 10.1093/imanum/drt022
Bergh, J. and J. Löfström. 2012. Interpolation Spaces: An Introduction, Grundlehren der mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin, Heidelberg.
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
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829, DOI 10.1007/978-0-387-70914-7
Annalisa Buffa and Christoph Ortner, Compact embeddings of broken Sobolev spaces and applications, IMA J. Numer. Anal. 29 (2009), no. 4, 827–855. MR 2557047, DOI 10.1093/imanum/drn038
Aycil Cesmelioglu, Bernardo Cockburn, and Weifeng Qiu, Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations, Math. Comp. 86 (2017), no. 306, 1643–1670. MR 3626531, DOI 10.1090/mcom/3195
Konstantinos Chrysafinos and Noel J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations, Math. Comp. 79 (2010), no. 272, 2135–2167. MR 2684359, DOI 10.1090/S0025-5718-10-02348-3
Matteo Cicuttin, Alexandre Ern, and Nicolas Pignet, Hybrid high-order methods—a primer with applications to solid mechanics, SpringerBriefs in Mathematics, Springer, Cham, [2021] ©2021. MR 4387067, DOI 10.1007/978-3-030-81477-9
Bernardo Cockburn, Guido Kanschat, and Dominik Schotzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp. 74 (2005), no. 251, 1067–1095. MR 2136994, DOI 10.1090/S0025-5718-04-01718-1
Bernardo Cockburn, Daniele A. Di Pietro, and Alexandre Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 635–650. MR 3507267, DOI 10.1051/m2an/2015051
Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259, DOI 10.7208/chicago/9780226764320.001.0001
Monique Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74–97. MR 977489, DOI 10.1137/0520006
Daniele A. Di Pietro and Jérôme Droniou, A hybrid high-order method for Leray-Lions elliptic equations on general meshes, Math. Comp. 86 (2017), no. 307, 2159–2191. MR 3647954, DOI 10.1090/mcom/3180
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 and Alexandre Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations, Math. Comp. 79 (2010), no. 271, 1303–1330. MR 2629994, DOI 10.1090/S0025-5718-10-02333-1
Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148, DOI 10.1007/978-3-642-22980-0
Daniele A. Di Pietro and Alexandre Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21. MR 3283758, DOI 10.1016/j.cma.2014.09.009
Daniele A. Di Pietro and Stella Krell, A hybrid high-order method for the steady incompressible Navier-Stokes problem, J. Sci. Comput. 74 (2018), no. 3, 1677–1705. MR 3767823, DOI 10.1007/s10915-017-0512-x
Vít Dolejší and Miloslav Feistauer, Discontinuous Galerkin method, Springer Series in Computational Mathematics, vol. 48, Springer, Cham, 2015. Analysis and applications to compressible flow. MR 3363720, DOI 10.1007/978-3-319-19267-3
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, The stability in $L^{q}$ of the $L^{2}$-projection into finite element function spaces, Numer. Math. 23 (1974/75), 193–197. MR 383789, DOI 10.1007/BF01400302
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, and Raphaèle Herbin, The gradient discretisation method, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 82, Springer, Cham, 2018. MR 3898702, DOI 10.1007/978-3-319-79042-8
Shukai Du and Francisco-Javier Sayas, An invitation to the theory of the hybridizable discontinuous Galerkin method, SpringerBriefs in Mathematics, Springer, Cham, 2019. Projections, estimates, tools. MR 3970243, DOI 10.1007/978-3-030-27230-2
Alexandre Ern and Jean-Luc Guermond, Finite elements I—Approximation and interpolation, Texts in Applied Mathematics, vol. 72, Springer, Cham, [2021] ©2021. MR 4242224, DOI 10.1007/978-3-030-56341-7
Ern, A. and J.-L. Guermond. 2021. Finite Elements III, Texts in Applied Mathematics, no. 74, Springer International Publishing.
Tamás L. Horváth and Sander Rhebergen, A locally conservative and energy-stable finite-element method for the Navier-Stokes problem on time-dependent domains, Internat. J. Numer. Methods Fluids 89 (2019), no. 12, 519–532. MR 3935520, DOI 10.1002/fld.4707
Tamás L. Horváth and Sander Rhebergen, An exactly mass conserving space-time embedded-hybridized discontinuous Galerkin method for the Navier-Stokes equations on moving domains, J. Comput. Phys. 417 (2020), 109577, 17. MR 4107052, DOI 10.1016/j.jcp.2020.109577
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
Fumio Kikuchi, Rellich-type discrete compactness for some discontinuous Galerkin FEM, Jpn. J. Ind. Appl. Math. 29 (2012), no. 2, 269–288. MR 2931412, DOI 10.1007/s13160-012-0057-1
Kirk, K. L. A., T. L. Horváth, and S. Rhebergen. 2021. Analysis of an exactly mass conserving space-time hybridized discontinuous Galerkin method for the time-dependent Navier–Stokes equations, Preprint, arXiv:2103.13492.
Keegan L. A. Kirk and Sander Rhebergen, Analysis of a pressure-robust hybridized discontinuous Galerkin method for the stationary Navier-Stokes equations, J. Sci. Comput. 81 (2019), no. 2, 881–897. MR 4019980, DOI 10.1007/s10915-019-01040-y
Jizhou Li, Beatrice Riviere, and Noel Walkington, Convergence of a high order method in time and space for the miscible displacement equations, ESAIM Math. Model. Numer. Anal. 49 (2015), no. 4, 953–976. MR 3371899, DOI 10.1051/m2an/2014059
Charalambos Makridakis and Ricardo H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems, Numer. Math. 104 (2006), no. 4, 489–514. MR 2249675, DOI 10.1007/s00211-006-0013-6
Arif Masud and Thomas J. R. Hughes, A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems, Comput. Methods Appl. Mech. Engrg. 146 (1997), no. 1-2, 91–126. MR 1453437, DOI 10.1016/S0045-7825(96)01222-4
Issei Oikawa, Hybridized discontinuous Galerkin method with lifting operator, JSIAM Lett. 2 (2010), 99–102. MR 3009390, DOI 10.14495/jsiaml.2.99
Sander Rhebergen and Bernardo Cockburn, A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys. 231 (2012), no. 11, 4185–4204. MR 2911790, DOI 10.1016/j.jcp.2012.02.011
Sander Rhebergen, Bernardo Cockburn, and Jaap J. W. van der Vegt, A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys. 233 (2013), 339–358. MR 3000935, DOI 10.1016/j.jcp.2012.08.052
Sander Rhebergen and Garth N. Wells, Analysis of a hybridized/interface stabilized finite element method for the Stokes equations, SIAM J. Numer. Anal. 55 (2017), no. 4, 1982–2003. MR 3686803, DOI 10.1137/16M1083839
Sander Rhebergen and Garth N. Wells, A hybridizable discontinuous Galerkin method for the Navier-Stokes equations with pointwise divergence-free velocity field, J. Sci. Comput. 76 (2018), no. 3, 1484–1501. MR 3833698, DOI 10.1007/s10915-018-0671-4
Tomáš Roubíček, Nonlinear partial differential equations with applications, 2nd ed., International Series of Numerical Mathematics, vol. 153, Birkhäuser/Springer Basel AG, Basel, 2013. MR 3014456, DOI 10.1007/978-3-0348-0513-1
Dominik Schötzau and Thomas P. Wihler, A posteriori error estimation for $hp$-version time-stepping methods for parabolic partial differential equations, Numer. Math. 115 (2010), no. 3, 475–509. MR 2640055, DOI 10.1007/s00211-009-0285-8
Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI 10.1007/BF01762360
Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024
Noel J. Walkington, Convergence of the discontinuous Galerkin method for discontinuous solutions, SIAM J. Numer. Anal. 42 (2005), no. 5, 1801–1817. MR 2139223, DOI 10.1137/S0036142902412233
Noel J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations, SIAM J. Numer. Anal. 47 (2010), no. 6, 4680–4710. MR 2595054, DOI 10.1137/080728378