Stabilized schemes for the hydrostatic Stokes equations.

*(English)*Zbl 1328.35168This article deals with some numerical schemes for the finite element approximation of the hydrostatic Stokes equations that are used in oceanography. Considering appropriate functional spaces for the velocity and the pressure, the problem is reformulated by adding a residual term to the vertical momentum equation, a term that stabilizes the vertical velocity. This technique allows to recover partially the \(H^1\)-coercivity, which disappears in hydrostatic problems, to prove stability for Stokes-stable finite element combinations and to provide some error estimates.

In the next section another residual term is added to the continuity equation in order to obtain a better approximation of the vertical derivative of the pressure. The well-posedness of the discrete scheme is proved and some error estimates are established, including the \(L^2\)-norm of the vertical derivative of the pressure.

Finally, some numerical simulations are presented, in order to illustrate the advantages of the schemes previously described.

In the next section another residual term is added to the continuity equation in order to obtain a better approximation of the vertical derivative of the pressure. The well-posedness of the discrete scheme is proved and some error estimates are established, including the \(L^2\)-norm of the vertical derivative of the pressure.

Finally, some numerical simulations are presented, in order to illustrate the advantages of the schemes previously described.

Reviewer: Ruxandra Stavre (Bucureşti)

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

86A05 | Hydrology, hydrography, oceanography |

76D07 | Stokes and related (Oseen, etc.) flows |

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\textit{F. Guillén González} and \textit{J. R. Rodríguez Galván}, SIAM J. Numer. Anal. 53, No. 4, 1876--1896 (2015; Zbl 1328.35168)

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##### References:

[1] | P. Azérad, Analyse et approximation du problème de Stokes dans un bassin peu profond, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), pp. 53–58. |

[2] | P. Azérad, Analyse des Équations de Navier–Stokes en Bassin peu Profond et de l’Équation de Transport, Ph.D. thesis, Neuchatel, 1996. |

[3] | P. Azérad and F. Guillén, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal., 33 (2001), pp. 847–859. |

[4] | O. Besson and M.R. Laydi, Some estimates for the anisotropic Navier–Stokes equations and for the hydrostatic approximation, Math. Model. Numer. Anal, 26 (1992), pp. 855–865. · Zbl 0765.76017 |

[5] | S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Appl. Math. Texts, 3rd ed., Springer-Verlag, New York, 2008. · Zbl 1135.65042 |

[6] | F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New-York, 1991. · Zbl 0788.73002 |

[7] | C. Cao and E.S. Titi, Global well–posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), pp. 245–267. · Zbl 1151.35074 |

[8] | T. Chacón-Rebollo and F. Guillén-González, An intrinsic analysis of the hydrostatic approximation of Navier–Stokes equations, C. R. Acad. Sci. Paris, Sér. I Math, 330 (2000), pp. 841–846. · Zbl 0959.35134 |

[9] | T. Chacón-Rebollo and D. Rodríguez-Gómez, A numerical solver for the primitive equations of the ocean using term-by-term stabilization, Appl. Numer. Math., 55 (2005), pp. 1–31. |

[10] | B. Cushman-Roisin and J.M. Beckers, Introduction to Geophysical Fluid Dynamics—Physical and Numerical Aspects, Academic Press, New York, 2009. · Zbl 1319.86001 |

[11] | A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. · Zbl 1059.65103 |

[12] | V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, New York, 1986. · Zbl 0413.65081 |

[13] | F. Guillén-González, N. Masmoudi, and M.A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations, 14 (2001), pp. 1381–1408. · Zbl 1161.76454 |

[14] | F. Guillén-González and J.R. Rodríguez-Galván, On the stability of approximations for the Stokes problem using different finite element spaces for each component of the velocity, Appl. Numer. Math., to appear. · Zbl 1329.76172 |

[15] | F. Guillén-González and J.R. Rodríguez-Galván, Analysis of the hydrostatic Stokes problem and element approximation in unstructured meshes, Numer. Math., 130 (2015), pp. 225–256. · Zbl 1330.35324 |

[16] | F. Guillén-González and D. Rodríguez-Gómez, Bubble finite elements for the primitive equations of the ocean, Numer. Math., 101 (2005), pp. 689–728. · Zbl 1095.76031 |

[17] | Y. He, First order decoupled method of the primitive equations of the ocean I: Time discretization, J. Math. Anal. Appl., 412 (2014), pp. 895–921. · Zbl 1308.65143 |

[18] | Y. He, Second order decoupled implicit/explicit method of the primitive equations of the ocean I: Time discretization., Int. J. Numer. Anal. Model., 12 (2015), pp, 1–30. · Zbl 1334.86002 |

[19] | Y. He and J. Wu, Global H\(2\)-regularity results of the \(3\)D primitive equations of the ocean, Int. J. Numer. Anal. Model., 11 (2014), pp. 452–477. |

[20] | F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), pp. 251–265. · Zbl 1266.68090 |

[21] | M. Kimmritz and M. Braack, Equal-order finite elements for the hydrostatic Stokes problem, Comput. Methods Appl. Math., 12 (2012), pp. 306–329. · Zbl 1284.65151 |

[22] | J.-L. Lions, R. Temam, and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, 5 (1992), pp. 237–288. · Zbl 0746.76019 |

[23] | J.-L. Lions, R. Temam, and S. Wang, On the equations of large scale ocean, Nonlinearity, 5 (1992), pp. 1007–1053. · Zbl 0766.35039 |

[24] | U.S.A. National Geophysical Data Center NOAA, 2-Minute Gridded Global Relief Data (etopo2v2), www.ngdc.noaa.gov/mgg/global/etopo2.html. |

[25] | F. Ortegón Gallego, Regularization by monotone perturbations of the hydrostatic approximation of Navier–Stokes equations, Math. Models Methods Appl. Sci., 14 (2004), pp. 1819–1848. · Zbl 1096.35105 |

[26] | J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. · Zbl 0713.76005 |

[27] | R. Temam, Some mathematical aspects of geophysical fluid dynamic equations, Milan J. Math., 71 (2003), pp. 175–198. · Zbl 1048.86005 |

[28] | R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of Mathematical Fluid Dynamics, Vol. 3, S. Friedlander and D. Serre, eds., Elsevier, New York, 2004, pp. 535–658. · Zbl 1222.35145 |

[29] | M. Ziane, Regularity results for Stokes type systems, Applicable Analysis, 58 (1995), pp. 263–292. · Zbl 0837.35030 |

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