RATES OF CONVERGENCE FOR DISCRETIZATIONS OF THE STOCHASTIC INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

We show strong convergence with rates for an implicit time discretization, a semiimplicit time discretization, and a related finite element based space-time discretization of the incompressible Navier-Stokes equations with multiplicative noise in two space dimensions. We use higher moments of comput...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2012-01, Vol.50 (5), p.2467-2496
Main Authors: CARELLI, ERICH, PROHL, ANDREAS
Format: Article
Language:English
Subjects:
Airy equation
Algorithms
Approximation
Convection
Error analysis
Error rates
Navier Stokes equation
Navier-Stokes equations
Noise
Nonlinearity
Perceptron convergence procedure
Random variables
Spacetime
Velocity
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Summary:We show strong convergence with rates for an implicit time discretization, a semiimplicit time discretization, and a related finite element based space-time discretization of the incompressible Navier-Stokes equations with multiplicative noise in two space dimensions. We use higher moments of computed iterates to optimally bound the error on a subset Ω κ of the sample space Ω, where corresponding paths are bounded in a proper function space, and ℙ[Ω κ ] → 1 holds for vanishing discretization parameters. This implies convergence in probability with rates, and motivates a practicable acception/rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity. It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.
ISSN:0036-1429
1095-7170
DOI:10.1137/110845008