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The long-time \(L^2\) and \(H^1\) stability of linearly extrapolated second-order time-stepping schemes for the 2D incompressible Navier-Stokes equations. (English) Zbl 1428.76097

Summary: Herein, we present a study on the long-time stability of finite element discretizations of a generalized class of semi-implicit second-order time-stepping schemes for the 2D incompressible Navier-Stokes equations. These remarkably efficient schemes require only a single linear solve per time-step through the use of a linearly-extrapolated advective term. Our result develops a class of sufficient conditions such that if external forcing is uniformly bounded in time, velocity solutions are uniformly bounded in time in both the \(L^2\) and \(H^1\) norms. We provide numerical verification of these results. We also demonstrate that divergence-free finite elements are critical for long-time \(H^1\) stability.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations

Software:

Gmsh; FEniCS; RODAS; UFL
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Full Text: DOI

References:

[1] Heywood, J.; Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem, Part II: stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23, 4, 750-777 (1986) · Zbl 0611.76036
[2] Heywood, J.; Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem. Part IV: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27, 2, 353-384 (1990) · Zbl 0694.76014
[3] Simo, J.; Armero, F., Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. Methods Appl. Mech. Eng., 111, 1-2, 111-154 (1994) · Zbl 0846.76075
[4] Tone, F.; Wirosoetisno, D., On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 44, 1, 29-40 (2006) · Zbl 1108.76050
[5] Tone, F., On the long-time stability of the Crank-Nicolson scheme for the 2D Navier-Stokes equations, Numer. Methods Part. Diff. Eq., 23, 1, 1235-1248 (2007) · Zbl 1127.76042
[6] Badia, S.; Codina, R.; Gutierrez-Santacreu, J., Long-term stability estimates and existence of a global attractor in a finite element approximation of the Navier-Stokes equations with numerical subgrid scale modeling, SIAM J. Numer. Anal., 48, 3, 1013-1037 (2010) · Zbl 1305.76051
[7] Wang, X., An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations, Numer. Math., 121, 4, 753-779 (2012) · Zbl 1435.76054
[8] Gottlieb, S.; Tone, F.; Wang, C.; Wang, X.; Wirosoetisno, D., Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 50, 1, 126-150 (2012) · Zbl 1237.76033
[9] Akbas, M.; Rebholz, L.; Tone, F., A note on the importance of mass conservation in long-time stability of Navier-Stokes simulations using finite elements, Appl. Math. Lett., 45, 98-102 (2015) · Zbl 1311.76044
[10] Heister, T.; Olshanksii, M.; Rebholz, L., Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135, 143-167 (2017) · Zbl 1387.76052
[11] Akbas, M.; Kaya, S.; Rebholz, L., On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems, Numer. Methods Part. Diff. Eq., 33, 4, 999-1017 (2017) · Zbl 1439.76043
[12] Jiang, N.; Mohebujjaman, M.; Rebholz, L.; Trenchea, C., An optimally accurate discrete regularization for second order timestepping methods for Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 310, 388-405 (2016) · Zbl 1439.76076
[13] Taylor, C.; Hood, P., A numerical solution of the NSE using the finite element technique, Int. J. Comput. Fluids, 1, 73-100 (1973) · Zbl 0328.76020
[14] Zhang, S., On the P1 Powell-Sabin divergence-free finite element for the Stokes equations, J. Comput. Math., 456-470 (2008) · Zbl 1174.65039
[15] Qin, J., On the convergence of some low order mixed finite elements for incompressible fluids (1994), The Pennsylvania State University, Ph.D. thesis
[16] Guzman, J.; Scott, R., The Scott-Vogelius finite elements revisited, Math. Comput. (2018) · Zbl 1405.65150
[17] Zhang, S., Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids, Calcolo, 48, 211-244 (2011) · Zbl 1232.65151
[18] Zhang, S., A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comput., 74, 250, 543-554 (2005) · Zbl 1085.76042
[19] Zhang, S., Divergence-free finite elements on tetrahedral grids for k ≥ 6, Math. Comput., 80, 274, 669-695 (2011) · Zbl 1410.76204
[20] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (2002), Springer-Verlag
[21] Shen, J., Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38, 4, 201-229 (1990) · Zbl 0684.65095
[22] Alnæs, M.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M.; Wells, G., The FEniCS project version 1.5, Arch. Numer. Softw., 3, 100 (2015)
[23] Alnæs, M.; Logg, A.; Ølgaard, K.; Rognes, M.; Wells, G., Unified form language: a domain-specific language for weak formulations of partial differential equations, ACM Trans. Math. Softw., 40, 2 (2014) · Zbl 1308.65175
[24] Geuzaine, C.; Remacle, J., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities., Int. J. Numer. Methods Eng., 79, 11, 1309-1331 (2009) · Zbl 1176.74181
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