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Two-grid mixed finite-element approximations to the Navier-Stokes equations based on a Newton-type step. (English) Zbl 1404.65166

Summary: A two-grid scheme to approximate the evolutionary Navier-Stokes equations is introduced and analyzed. A standard mixed finite element approximation is first obtained over a coarse mesh of size \(H\) at any positive time \(T>0\). Then, the approximation is postprocessed by means of solving a steady problem based on one step of a Newton iteration over a finer mesh of size \(h<H\). The method increases the rate of convergence of the standard Galerkin method in one unit in terms of \(H\) and equals the rate of convergence of the standard Galerkin method over the fine mesh \(h\). However, the computational cost is essentially the cost of approaching the Navier-Stokes equations with the plain Galerkin method over the coarse mesh of size \(H\) since the cost of solving one single steady problem is negligible compared with the cost of computing the Galerkin approximation over the full time interval \((0,T]\). For the analysis we take into account the loss of regularity at initial time of the solution of the Navier- Stokes equations in the absence of nonlocal compatibility conditions. Some numerical experiments are shown.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
65H10 Numerical computation of solutions to systems of equations
76D07 Stokes and related (Oseen, etc.) flows
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[1] Layton, W., Tobiska, L.: A two-level method with backtracking for the Navier-Stokes equations. SIAM J. Numer. Anal. 35, 2035-2054 (1998) · Zbl 0913.76050 · doi:10.1137/S003614299630230X
[2] Layton, W., Lenferink, W.: Two-level picard and modified picard methods for the Navier-Stokes equations. Appl. Math. Comput. 80, 1-12 (1995) · Zbl 0828.76017
[3] García-Archilla, B., Novo, J., Titi, E.S.: Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds. SIAM J. Numer. Anal. 35, 941-972 (1998) · Zbl 0914.65105 · doi:10.1137/S0036142995296096
[4] García-Archilla, B., Novo, J., Titi, E.S.: An approximate inertial manifold approach to postprocessing Galerkin methods for the Navier-Stokes equations. Math. Comp. 68, 893-911 (1999) · Zbl 0930.76063 · doi:10.1090/S0025-5718-99-01057-1
[5] de Frutos, J., Novo, J.: A spectral element method for the Navier-Stokes equations with improved accuracy. SIAM J. Numer. Anal. 38, 799-819 (2000) · Zbl 0982.76070 · doi:10.1137/S0036142999351984
[6] Margolin, L.G., Titi, E.S., Wynne, S.: The postprocessing Galerkin and nonlinear Galerkin methods—a truncation analysis point of view. SIAM J. Numer. Anal. 41, 695-714 (2003) · Zbl 1130.65314 · doi:10.1137/S0036142901390500
[7] Ayuso, B., de Frutos, J., Novo, J.: Improving the accuracy of the mini-element approximation to Navier-Stokes equations. IMA J. Numer. Anal. 27, 198-218 (2007) · Zbl 1104.76058 · doi:10.1093/imanum/drl010
[8] Ayuso, B., Garca-Archilla, B., Novo, J.: The postprocessed mixed finite element method for the Navier-Stokes equations. SIAM J. Numer. Anal. 43, 1091-1111 (2005) · Zbl 1094.76037 · doi:10.1137/040602821
[9] de Frutos, J., García-Archilla, B., Novo, J.: The postprocessed mixed finite-element method for the Navier-Stokes equations: refined error bounds. SIAM J. Numer. Anal. 46, 201-230 (2007) · Zbl 1166.65381 · doi:10.1137/06064458
[10] de Frutos, J., García-Archilla, B., Novo, J.: Postprocessing finite-element methods for the Navier-Stokes equations: the fully discrete case. SIAM J. Numer. Anal. 47, 596-621 (2008) · Zbl 1391.76379 · doi:10.1137/070707580
[11] de Frutos, J., García-Archilla, B., Novo, J.: Static two-grid mixed finite element approximations to the Navier-Stokes equations. J. Sci. Comput. 52, 619-637 (2012) · Zbl 1264.76065 · doi:10.1007/s10915-011-9562-7
[12] de Frutos, J., García-Archilla, B., Novo, J.: Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations. Appl. Math. Comput. 218, 7034-7051 (2012) · Zbl 1426.76244
[13] Goswami, D., Damázio, P.D.: A two-grid finite element method for time-dependent incompressible Navier-Stokes equations with non-smooth initial data. Numer. Math. Theory Method Appl. 8, 549-581 (2015) · Zbl 1363.76036 · doi:10.4208/nmtma.2015.m1414
[14] Liu, Q., Hou, Y., Wang, W., Zhao, J.: Two-level consitent splitting methods based on three corrections for the time-dependent Navier-Stokes equations. Int. J. Numer. Method Fluid 80(7), 429-450 (2015). doi:10.1002/fld.4087 · doi:10.1002/fld.4087
[15] Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275-311 (1982) · Zbl 0487.76035 · doi:10.1137/0719018
[16] Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. III: Smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25, 489-512 (1988) · Zbl 0646.76036 · doi:10.1137/0725032
[17] Girault, V., Raviart, P.A.: Finite Element Methods for the Navier-Stokes Equations: Theory and Algorithms, vol. 5. Springer-Verlag, Berlin (1986) · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5
[18] Brezzi, F., Falk, R.S.: Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28, 581-590 (1991) · Zbl 0731.76042 · doi:10.1137/0728032
[19] Hood, P., Taylor, C.: A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluid 1, 73-100 (1973) · Zbl 0328.76020 · doi:10.1016/0045-7930(73)90027-3
[20] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991) · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1
[21] Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numer. Math. 107(1), 39-77 (2007) · Zbl 1117.76032 · doi:10.1007/s00211-007-0070-5
[22] Arndt, D., Dallmann, H., Lube, G.: Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem. Numer. Method Partial Differ. Equ. 31(4), 1224-1250 (2015) · Zbl 1446.76126 · doi:10.1002/num.21944
[23] Dallmann, H., Arndt, D.: Stabilized finite element methods for the Oberbeck-Boussinesq model. J. Sci. Comput. 69, 244273 (2016) · Zbl 1457.65074 · doi:10.1007/s10915-016-0191-z
[24] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001) · Zbl 1042.35002
[25] de Frutos, J., García-Archilla, B., John, V., Novo, J.: Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math. arXiv:1610.05017 · Zbl 1404.65188
[26] de Frutos, J., García-Archilla, B., Novo, J.: A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations. J. Comput. Appl. Math. 6, 1103-1122 (2011) · Zbl 1238.76031 · doi:10.1016/j.cam.2011.07.033
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