Rates of Convergence for Discretizations of the Stochastic Incompressible Navier--Stokes Equations

Mathematical and Numerical Foundations of Turbulence Models and Applications

“…3.3 Basic Similarity Setting 57 We will not give much attention to a rigorous proof of such convergence. However, it is well known since the works of Stokes [43] that the Stokes equations describe well "slow flows," which are not turbulent and therefore not in the scope of this book.…”

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

“…(b) We also note that the error estimate for the velocity approximations of divergencefree finite element methods do not have the "bad" factor k − 1 2 (cf. [13]) at the expense of using divergence-free finite element spaces and not approximating the pressure.…”

“…A fully discrete mixed finite element scheme was also considered and strong convergence with rates was also proved for the velocity approximation. As noted in [13], the interaction of Lagrange multipliers with the stochastic forcing in the scheme limits the accuracy of general discretely LBB-stable space discretizations. Strategies to overcome this difficulty were also proposed in [13] although the convergence of the pressure approximation was not addressed.…”

“…As noted in [13], the interaction of Lagrange multipliers with the stochastic forcing in the scheme limits the accuracy of general discretely LBB-stable space discretizations. Strategies to overcome this difficulty were also proposed in [13] although the convergence of the pressure approximation was not addressed. Other recent works for stochastic Navier-Stokes equations include an iterative splitting scheme which was proposed in [6], a strong convergence in probability was established in the 2-D case for the velocity approximation.…”

Layer methods for stochastic Navier–Stokes equations using simplest characteristics

“…In [20] algorithms based on Wiener Chaos expansion for SNSE are considered, these algorithms can work on relatively short time intervals only. In [6,4] implicit and semi-discrete Euler time and …nite element based space-time discretizations are studied, convergence is proved in the mean-square (strong) sense. The work [14] deals with a time-splitting scheme combined with a Galerkin approximation in the space variable for SNSE exploiting the semi-group and cubature techniques, a weak convergence is proved for the proposed method.…”