Abstract
We design a hybrid high-order (HHO) method to approximate the Stokes problem on curved domains using unfitted meshes. We prove inf-sup stability and a priori estimates with optimal convergence rates. Moreover, we provide numerical simulations that corroborate the theoretical convergence rates. A cell-agglomeration procedure is used to prevent the appearance of small cut cells.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
References
Badia, S., Verdugo, F., Martín, A.F.: The aggregated unfitted finite element method for elliptic problems. Comput. Methods Appl. Mech. Engrg. 336, 533–553 (2018)
Botti, L., Di Pietro, D. A. Droniou, J.: A hybrid high-order method for the incompressible Navier-Stokes equations based on Temam’s device. J. Comput. Phys. 376, 786–816 (2019)
Burman, E.: Ghost penalty. C. R. Math. Acad. Sci. Paris. 348(21–22), 1217–1220 (2010)
Burman, E., Cicuttin, M., Delay, G., Ern, A.: An unfitted Hybrid High-Order method with cell agglomeration for elliptic interface problems. submitted. https://hal.archives-ouvertes.fr/hal-02280426/
Burman, E., Claus, S., Hansbo, P., Larson, M. G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Internat. J. Numer. Methods Engrg. 104(7), 472–501 (2015)
Burman, E., Delay, G., Ern, A.: An unfitted hybrid high-order method for the Stokes interface problem, to appear in IMA J. Numer. Anal. https://hal.archives-ouvertes.fr/hal-02519896/
Burman, E., Ern, A.: An unfitted hybrid high-order method for elliptic interface problems. SIAM J. Numer. Anal. 56(3), 1525–1546 (2018)
Burman, E., Hansbo, P.: Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM Math. Model. Anal. 48(3), 859–874 (2014)
Cicuttin, M., Di Pietro, D. A., Ern, A.: Implementation of discontinuous skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming. J. Comput. Appl. Math. 344, 852–874 (2018)
Cockburn, B., Di Pietro, D. A., Ern, A.: Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin methods. ESAIM: Math. Model Numer. Anal. (M2AN) 50(3), 635–650 (2016)
de Prenter, F., Lehrenfeld, C., Massing, A.: A note on the stability parameter in Nitsche’s method for unfitted boundary value problems. Comput. Math. Appl. 75(12), 4322–4336 (2018)
Di Pietro, D. A., Ern, A.: A Hybrid High-Order locking-free method for linear elasticity on general meshes. Comput. Meth. Appl. Mech. Engrg. 283, 1–21 (2015)
Di Pietro, D. A., Ern, A., Lemaire, S.: An arbitrary-order and compact- stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Meth. Appl. Math. 14(4), 461–472 (2014)
Di Pietro, D. A., Ern, A., Linke, A., Schieweck, F.: A discontinuous skeletal method for the viscosity-dependent Stokes problem. Comput. Methods Appl. Mech. Engrg. 306, 175–195 (2016)
Fournié, M., Lozinski, A.: Stability and optimal convergence of unfitted extended finite element methods with Lagrange multipliers for the Stokes equations. In: Lect. Notes Comput. Sci. Eng., pp. 143–182, Springer, Cham (2017)
Guzmán, J., Olshanskii, M.: Inf-sup stability of geometrically unfitted Stokes finite elements. Math. Comp. 87(313), 2091–2112 (2018)
Johansson, A., Larson, M. G.: A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123(4), 607–628 (2013)
Lehrenfeld, C.: Removing the stabilization parameter in fitted and unfitted symmetric Nitsche formulations. ar**v:1603.00617 (2016)
Lehrenfeld, C., Reusken, A.: Analysis of a high-order unfitted finite element method for elliptic interface problems. IMA J. Numer. Anal. 38, 1351–1387 (2018)
Massing, A., Larson, M. G., Logg, A., Rognes, M. E.: A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comput. 61(3), 604–628 (2014)
Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971)
Solano, M., Vargas, F.: A high order HDG method for Stokes flow in curved domains. J. Sci. Comput.79(3), 1505–1533 (2019)
Sollie, W. E. H., Bokhove, O., van der Vegt, J. J. W.: Space-time discontinuous Galerkin finite element method for two-fluid flows. J. Comput. Phys. 230(3), 789–817 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Burman, E., Delay, G., Ern, A. (2021). The Unfitted HHO Method for the Stokes Problem on Curved Domains. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_38
Download citation
DOI: https://doi.org/10.1007/978-3-030-55874-1_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-55873-4
Online ISBN: 978-3-030-55874-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)