×

Time-discretization of stochastic 2-D Navier-Stokes equations with a penalty-projection method. (English) Zbl 1423.76093

Summary: A time-discretization of the stochastic incompressible Navier-Stokes problem by penalty method is analyzed. Some error estimates are derived, combined, and eventually arrive at a speed of convergence in probability of order 1/4 of the main algorithm for the pair of variables velocity and pressure. Also, using the law of total probability, we obtain the strong convergence of the scheme for both variables.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65J15 Numerical solutions to equations with nonlinear operators
65M15 Error bounds 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
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20(3), 179-192 (1973) · Zbl 0258.65108 · doi:10.1007/BF01436561
[2] Baňas, Ľ., Brzeźniak, Z., Neklyudov, M., Prohl, A.: A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation. IMA J. Numer. Anal. 34(2), 502 (2014) · Zbl 1298.65012 · doi:10.1093/imanum/drt020
[3] Bernard, P., Wallace, J.: Turbulent Flow: Analysis, Measurement, and Prediction. Wiley, New York (2002)
[4] Bessaih, H., Brzeźniak, Z., Millet, A.: Splitting up method for the 2D stochastic Navier-Stokes equations. Stoch. Partial Differ. Equ. Anal. Comput. 2(4), 433-470 (2014) · Zbl 1312.60080
[5] Pinsky, M., Birnir, B.: Turbulence of a unidirectional flow. In: Probability, Geometry and Integrable Systems for Henry McKean’s Seventy-Fifth Birthday, pp. 29-52. Cambridge University Press, Cambridge (2008) · Zbl 1171.35094
[6] Birnir, B.: The Kolmogorov-Obukhov statistical theory of turbulence. J. Nonlinear Sci. 23(4), 657-688 (2013) · Zbl 1282.76112 · doi:10.1007/s00332-012-9164-z
[7] Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183. Springer, Berlin (2012) · Zbl 1286.76005
[8] Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. ESAIM Math. Model. Numer. Anal. Model. Math. Anal. Numer. 8(R2), 129-151 (1974) · Zbl 0338.90047
[9] Brzeźniak, Z., Carelli, E., Prohl, A.: Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing. IMA J. Numer. Anal. 33(3), 771-824 (2013) · Zbl 1426.76227 · doi:10.1093/imanum/drs032
[10] Brzeźniak, Z., Ferrario, B.: A note on stochastic Navier-Stokes equations with a generalized noise. Stoch. Partial Differ. Equ. Anal. Comput. 5(1), 53-80 (2017) · Zbl 1360.76217
[11] Capiński, M., Peszat, S.: On the existence of a solution to stochastic Navier-Stokes equations. Nonlinear Anal. Theory Methods Appl. 44(2), 141-177 (2001) · Zbl 0976.60063 · doi:10.1016/S0362-546X(99)00255-2
[12] Carelli, E., Hausenblas, E., Prohl, A.: Time-splitting methods to solve the stochastic incompressible Stokes equation. SIAM J. Numer. Anal. 50(6), 2917-2939 (2012) · Zbl 1262.60065 · doi:10.1137/100819436
[13] Carelli, E., Prohl, A.: Rates of convergence for discretizations of the stochastic incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 50(5), 2467-2496 (2012) · Zbl 1426.76231 · doi:10.1137/110845008
[14] Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Semin. Mat. Univ. Padova 31, 308-340 (1961) · Zbl 0116.18002
[15] Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49(1), 1-23 (1943) · Zbl 0063.00985 · doi:10.1090/S0002-9904-1943-07818-4
[16] de Bouard, A., Debussche, A.: A semi-discrete scheme for the stochastic nonlinear Schrödinger equation. Numer. Math. 96(4), 733-770 (2004) · Zbl 1055.65008 · doi:10.1007/s00211-003-0494-5
[17] Debussche, A., Printems, J.: Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete Contin. Dyn. Syst. Ser. B 6(4), 761-781 (2006) · Zbl 1132.35076 · doi:10.3934/dcdsb.2006.6.761
[18] Fernando, B.P.W., Rüdiger, B., Sritharan, S.S.: Mild solutions of stochastic Navier-Stokes equation with jump noise in \[L^p\] Lp-spaces. Math. Nachr. 288(14-15), 1615-1621 (2015) · Zbl 1446.35097 · doi:10.1002/mana.201300248
[19] Flandoli, F.: Dissipativity and invariant measures for stochastic Navier-Stokes equations. NoDea Nonlinear Differ. Equ. Appl. 1(4), 403-423 (1994) · Zbl 0820.35108 · doi:10.1007/BF01194988
[20] Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Field 102(3), 367-391 (1995) · Zbl 0831.60072 · doi:10.1007/BF01192467
[21] Flandoli, F., Schmalfuß, B.: Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force. J. Dyn. Differ. Equ. 11(2), 355-398 (1999) · Zbl 0931.35124 · doi:10.1023/A:1021937715194
[22] Flandoli, F., Tortorelli, V.M.: Time discretization of Ornstein-Uhlenbeck equations and stochastic Navier-Stokes equations with a generalized noise. Stochastics 55(1-2), 141-165 (1995) · Zbl 0886.60057
[23] Giga, Y., Miyakawa, T.: Solutions in \[{L}^rLr\] of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 89(3), 267-281 (1985) · Zbl 0587.35078 · doi:10.1007/BF00276875
[24] Guermond, J.-L., Minev, P.D., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44-47), 6011-6045 (2006) · Zbl 1122.76072 · doi:10.1016/j.cma.2005.10.010
[25] Hou, T.Y., Luo, W., Rozovskii, B., Zhou, H.-M.: Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J. Comput. Phys. 216(2), 687-706 (2006) · Zbl 1095.76047 · doi:10.1016/j.jcp.2006.01.008
[26] Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. Springer, New York (2014) · Zbl 0638.60065
[27] Langa, J.A., Real, J., Simon, J.: Existence and regularity of the pressure for the stochastic Navier-Stokes equations. Appl. Math. Optim. 48(3), 195-210 (2003) · Zbl 1049.60058 · doi:10.1007/s00245-003-0773-7
[28] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires, volume 31. Dunod Paris (1969) · Zbl 0189.40603
[29] Manna, U., Menaldi, J.-L., Sritharan, S.S.: Stochastic 2-D Navier-Stokes equation with artificial compressibility. Commun. Stoch. Anal. 1(1), 123-139 (2007) · Zbl 1328.76056
[30] Milstein, G.N., Tretyakov, M.: Layer methods for stochastic Navier-Stokes equations using simplest characteristics. J. Comput. Appl. Math. 302, 1-23 (2016) · Zbl 1334.65016 · doi:10.1016/j.cam.2016.01.051
[31] Nečas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (2011)
[32] Pope, S.: Turbulent Flows. Cambridge University Press, Cambridge (2000) · Zbl 0966.76002 · doi:10.1017/CBO9780511840531
[33] Printems, J.: On the discretization in time of parabolic stochastic partial differential equations. ESAIM Math. Model. Numer. Anal. Model. Math. Anal. Numer. 35(6), 1055-1078 (2001) · Zbl 0991.60051 · doi:10.1051/m2an:2001148
[34] Prohl, A.: Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Advances in Numerical Mathematics. Vieweg+Teubner Verlag, Berlin (2013) · Zbl 0874.76002
[35] Shen, J.: On error estimates of projection methods for Navier-Stokes equations: first-order schemes. SIAM J. Numer. Anal. 29(1), 57-77 (1992) · Zbl 0741.76051 · doi:10.1137/0729004
[36] Shen, J.: On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations. Numer. Math. 62(1), 49-73 (1992) · Zbl 0739.76017 · doi:10.1007/BF01396220
[37] Shen, J.: On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32(2), 386-403 (1995) · Zbl 0822.35008 · doi:10.1137/0732016
[38] Témam, R.: Une méthode d’approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 96, 115-152 (1968) · Zbl 0181.18903 · doi:10.24033/bsmf.1662
[39] Témam, R.: Navier-Stokes equations and nonlinear functional analysis. In: CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (1983) · Zbl 0522.35002
[40] Témam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and Its Applications, North-Holland (1984) · Zbl 0568.35002
[41] Yin, H.: Stochastic Navier-Stokes equations with artificial compressibility in random durations. Int. J. Stoch. Anal. (2010) 2010:Article ID 730492 · Zbl 1202.60107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.