Electronic Communications in Probability

An approximation scheme of stochastic Stokes equations

Hanbing Liu and Juan Yang

Full-text: Open access

Abstract

This work is concerned with the approximation to the solutions of the stochastic Stokes equations by the splitting up method. We apply the resolvent operator to evaluate the solution of the deterministic equations at the endpoints of every small interval, and the error is estimated. <br />

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 21, 10 pp.

Dates
Accepted: 22 March 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315560

Digital Object Identifier
doi:10.1214/ECP.v18-2374

Mathematical Reviews number (MathSciNet)
MR3037219

Zentralblatt MATH identifier
1329.60239

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Liu, Hanbing; Yang, Juan. An approximation scheme of stochastic Stokes equations. Electron. Commun. Probab. 18 (2013), paper no. 21, 10 pp. doi:10.1214/ECP.v18-2374. https://projecteuclid.org/euclid.ecp/1465315560


Export citation

References

  • Asiminoaei, Ioan; Rascanu, Aurel. Approximation and simulation of stochastic variational inequalities—splitting up method. Numer. Funct. Anal. Optim. 18 (1997), no. 3-4, 251–282.
  • Barbu, Viorel. A product formula approach to nonlinear optimal control problems. SIAM J. Control Optim. 26 (1988), no. 3, 497–520.
  • Barbu, V. Approximation of the Hamilton-Jacobi equations via Lie-Trotter product formula. Control Theory Adv. Tech. 4 (1988), no. 2, 189–208.
  • Barbu, V. The fractional step method for a nonlinear distributed control problem. Differential equations and control theory (IaÅŸi, 1990), 7–16, Pitman Res. Notes Math. Ser., 250, Longman Sci. Tech., Harlow, 1991.
  • Beale, J. Thomas; Greengard, Claude. Convergence of Euler-Stokes splitting of the Navier-Stokes equations. Comm. Pure Appl. Math. 47 (1994), no. 8, 1083–1115.
  • Bensoussan, A.; Glowinski, R.; Răşcanu, A. Approximation of some stochastic differential equations by the splitting up method. Appl. Math. Optim. 25 (1992), no. 1, 81–106.
  • Bensoussan, A.; Glowinski, R.; Răşcanu, A. Approximation of the Zakai equation by the splitting up method. SIAM J. Control Optim. 28 (1990), no. 6, 1420–1431.
  • Doersek, P. and Teichmann, J.: Efficient simulation and calibration of general HJM models by splitting schemes. pharXiv:1112.5330v(2011).
  • Flandoli, Franco. Dirichlet boundary value problem for stochastic parabolic equations: compatibility relations and regularity of solutions. Stochastics Stochastics Rep. 29 (1990), no. 3, 331–357.
  • Gyöngy, István; Krylov, Nicolai. On the rate of convergence of splitting-up approximations for SPDEs. Stochastic inequalities and applications, 301–321, Progr. Probab., 56, Birkhäuser, Basel, 2003.
  • Grecksch, W.; Kloeden, P. E. Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54 (1996), no. 1, 79–85.
  • Gyöngy, István; Nualart, David. Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Anal. 7 (1997), no. 4, 725–757.
  • Germani, A.; Piccioni, M. Semidiscretization of stochastic partial differential equations on ${\bf R}^ d$ by a finite-element technique. Stochastics 23 (1988), no. 2, 131–148.
  • Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 (2012), no. 4, 1611–1641.
  • Jentzen, Arnulf; Kloeden, Peter. Taylor expansions of solutions of stochastic partial differential equations with additive noise. Ann. Probab. 38 (2010), no. 2, 532–569.
  • Kloeden, P. and Neuenkrich, A.: Convergence of numerical methods for stochastic differential equations in mathematical finance. pharXiv:1204.6620v(2012).
  • Kloeden, Peter E.; Platen, Eckhard. Numerical solution of stochastic differential equations. Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. xxxvi+632 pp. ISBN: 3-540-54062-8
  • Marchuk, G. I. Methods of numerical mathematics. Translated from the Russian by JiÅ™i Ru̇žička. Applications of Mathematics, No. 2. Springer-Verlag, New York-Heidelberg, 1975. xii+316 pp.
  • Pazy, A. Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp. ISBN: 0-387-90845-5
  • Popa, Cătălin. Feedback laws for nonlinear distributed control problems via Trotter-type product formulae. SIAM J. Control Optim. 33 (1995), no. 4, 971–999.
  • Popa, Cătălin. On the convergence of Euler-Stokes splitting of the Navier-Stokes equations. Differential Integral Equations 15 (2002), no. 6, 657–670.
  • Popa, Cătălin. Trotter product formulae for Hamilton-Jacobi equations in infinite dimensions. Differential Integral Equations 4 (1991), no. 6, 1251–1268.
  • Trotter, H. F.: On the products of semigroups of operators. phPro. Amer. Math. Soc. 10(1959), 545-551.
  • Teman, R.: Sur la stabilité et la convergence de la méthode des pas fractionnaires. Thèse(1967), Paris.