Abstract
Numerical solutions of the stochastic Stokes equations driven by white noise perturbed forcing terms using finite element methods are considered. The discretization of the white noise and finite element approximation algorithms are studied. The rate of convergence of the finite element approximations is proved to be almost first order (h|ln h|) in two dimensions and one half order (\(h^{\frac{1}{2}}\)) in three dimensions. Numerical results using the algorithms developed are also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Method. Springer, New York (1994)
Cao, Y., Yang, H., Yin, L.: Finite element methods for semilinear elliptic stochastic partial differential equations. Numer. Math. 106, 181–198 (2007)
Weinan, E., Mattingly, J., Sinai, Y.: Gibbsian dynamics and ergodicity for the stochastic forced Navier-Stokes equation. Comm. Math. Phys. 224(1), 83–106 (2001)
Flandoli, F.: Dissipativity and invariant measures for stochastic Navier-Stokes equations. NoDEA 1, 403–426 (1994)
Gunzburger, M.: Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic, London (1989)
Hou, T., Luo, W., Rozovskii, B., Zhou, H.: Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J. Comput. Phys. 216, 687–706 (2006)
Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flows, 2nd edn. Gordon and Breach, New York (1969)
Mattingly, J., Pardoux, E.: Malliavin calculus for the stochastic 2d Navier-Stokes equation. Comm. Pure Appl. Math. 59, 1742–1790 (2006)
Girault, V., Raviart, P.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986)
Walsh, J.: An introduction to stochastic partial differential equations. In: Lecture Notes in Mathematics 1180, pp. 265–439. Springer, New York (1986)
Xiu, D., Karniadakis, G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Martin Stynes.
Research of Yanzhao Cao was supported by the National Science Foundation under grant number DMS0609918 and the Air Force Office for Scientific Research under grant number FA550-07-1-0154.
Research of Max Gunzburger was supported by the Air Force Office for Scientific Research under grant number FA9550-08-1-0415.
Rights and permissions
About this article
Cite this article
Cao, Y., Chen, Z. & Gunzburger, M. Error analysis of finite element approximations of the stochastic Stokes equations. Adv Comput Math 33, 215–230 (2010). https://doi.org/10.1007/s10444-009-9127-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-009-9127-6