Convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations
Tóm tắt
We prove the convergence of certain second-order numerical methods to weak solutions of the Navier–Stokes equations satisfying, in addition, the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli–Kohn–Nirenberg. More precisely, we treat the space-periodic case in three space dimensions and consider a full discretization in which the classical Crank–Nicolson method (θ-method with
$\theta =1/2$
) is used to discretize the time variable. In contrast, in the space variables, we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions toward weak solutions (satisfying a precise local energy balance) without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes, providing alternate proofs of the existence of “physically relevant” solutions in three space dimensions.
Tài liệu tham khảo
Albritton, D., Brué, E., Colombo, M.: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations. Ann. Math. (2) 196(1), 415–455 (2022)
Baker, G.A.: Projection Methods for Boundary-Value Problems for Equations of Elliptic and Parabolic Type with Discontinuous Coefficients. ProQuest LLC, Ann Arbor (1973). Thesis (Ph.D.)–Cornell University
Berselli, L.C.: Weak solutions constructed by semi-discretization are suitable: the case of slip boundary conditions. Int. J. Numer. Anal. Model. 15, 479–491 (2018)
Berselli, L.C.: Three-Dimensional Navier-Stokes Equations for Turbulence. Mathematics in Science and Engineering. Academic Press, London (2021)
Berselli, L.C., Fagioli, S., Spirito, S.: Suitable weak solutions of the Navier-Stokes equations constructed by a space-time numerical discretization. J. Math. Pures Appl. 9(125), 189–208 (2019)
Berselli, L.C., Spirito, S.: On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains. Commun. Math. Phys. 316, 171–198 (2012)
Berselli, L.C., Spirito, S.: An elementary approach to the inviscid limits for the 3D Navier-Stokes equations with slip boundary conditions and applications to the 3D Boussinesq equations. NoDEA Nonlinear Differ. Equ. Appl. 21, 149–166 (2014)
Berselli, L.C., Spirito, S.: Weak solutions to the Navier-Stokes equations constructed by semi-discretization are suitable. In: Recent Advances in Partial Differential Equations and Applications. Contemp. Math., vol. 666, pp. 85–97. Am. Math. Soc., Providence (2016)
Bramble, J.H., Xu, J.: Some estimates for a weighted \(L^{2}\) projection. Math. Comput. 56, 463–476 (1991)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer, New York (2008)
Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)
Carstensen, C.: Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for \(H^{1}\)-stability of the \(L^{2}\)-projection onto finite element spaces. Math. Comput. 71, 157–163 (2002)
Demlow, A., Guzmán, J., Schatz, A.H.: Local energy estimates for the finite element method on sharply varying grids. Math. Comput. 80, 1–9 (2011)
Diening, L., Storn, J., Tscherpel, T.: On the Sobolev and \(L^{p}\)-stability of the \(L^{2}\)-projection. SIAM J. Numer. Anal. 59, 2571–2607 (2021)
Douglas, J. Jr., Dupont, T., Wahlbin, L.: The stability in \(L^{q}\) of the \(L^{2}\)-projection into finite element function spaces. Numer. Math. 23, 193–197 (1974/75)
Guermond, J.-L.: Finite-element-based Faedo-Galerkin weak solutions to the Navier-Stokes equations in the three-dimensional torus are suitable. J. Math. Pures Appl. 9(85), 451–464 (2006)
Guermond, J.-L.: Faedo-Galerkin weak solutions of the Navier-Stokes equations with Dirichlet boundary conditions are suitable. J. Math. Pures Appl. (9) 88, 87–106 (2007)
Guermond, J.-L.: On the use of the notion of suitable weak solutions in CFD. Int. J. Numer. Methods Fluids 57, 1153–1170 (2008)
Guermond, J.-L., Oden, J.T., Prudhomme, S.: Mathematical perspectives on large Eddy simulation models for turbulent flows. J. Math. Fluid Mech. 6, 194–248 (2004)
He, Y., Li, K.: Nonlinear Galerkin method and two-step method for the Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 12, 283–305 (1996)
He, Y., Sun, W.: Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 45, 837–869 (2007)
Heywood, J.G., Rannacher, R.: Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)
Horiuti, K.: Comparison of conservative and rotational forms in large Eddy simulation of turbulent channel flow. J. Comput. Phys. 71, 343–370 (1987)
Ingram, R.: A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations. Math. Comput. 82, 1953–1973 (2013)
Ingram, R.: Unconditional convergence of high-order extrapolations of the Crank-Nicolson, finite element method for the Navier-Stokes equations. Int. J. Numer. Anal. Model. 10, 257–297 (2013)
Layton, W., Manica, C.C., Neda, M., Olshanskii, M., Rebholz, L.G.: On the accuracy of the rotation form in simulations of the Navier-Stokes equations. J. Comput. Phys. 228, 3433–3447 (2009)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 2. Oxford Lecture Series in Mathematics and Its Applications, vol. 10. Clarendon Press, New York (1998). Compressible models, Oxford Science Publications
Marion, M., Temam, R.: Navier-Stokes equations: theory and approximation. In: Handbook of Numerical Analysis, Vol. VI, Handb. Numer. Anal., vol. VI, pp. 503–688. North-Holland, Amsterdam (1998)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)
Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55, 97–112 (1977)
Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland, Amsterdam (1977)
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin (1997)
Tone, F.: Error analysis for a second order scheme for the Navier-Stokes equations. Appl. Numer. Math. 50, 93–119 (2004)
Vasseur, A.F.: A new proof of partial regularity of solutions to Navier-Stokes equations. NoDEA Nonlinear Differ. Equ. Appl. 14, 753–785 (2007)
Zang, T.: On the rotation and skew-symmetric forms for incompressible flow simulations. Appl. Numer. Math. 7, 27–40 (1991)