A stabilized finite element method for stochastic incompressible Navier–Stokes equations

Abstract

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier–Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier–Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for the benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier–Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations.

Keywords:

• stochastic Navier–Stokes equations
• finite element method
• polynomial chaos
• modified method of characteristics
• conjugate-gradient algorithm
• stochastic simulation

2000 AMS Subject Classifications :

• 35L04
• 60G51
• 65N30
• 65N75
• 76D05
• 60H15
• 76M10

Acknowledgements

This work was partly performed while the second author was a visiting professor at the Department of Mathematics, University of Taibah at Madinah. M. Seaïd is deeply grateful to Taibah University for their hospitality during a research visit there. M. El-Amrani acknowledges the support from the proyecto del plan nacional de I+D+i grant by Ministerio de ciencia e innovación de España under the contract No. MTM2008-03255/MTM.