Numerical Analysis of 2D Stochastic Navier--Stokes Equations with Transport Noise: Regularity and Spatial Semidiscretization

This paper establishes strong convergence rates for the spatial finite element discretization of a two-dimensional stochastic Navier–Stokes system with transport noise and no-slip boundary conditions on a convex polygonal domain. The main challenge arises from the lack of spatial (D(A))-regularity of the solution (where (A) is the Stokes operator), which prevents the application of standard error analysis techniques. Under a small-noise assumption, we prove that the weak solution satisfies [ u \in L^2\bigl(Ω; C([0,T]; \dot{H}_σ^{\varrho}) \cap L^2(0,T; \dot{H}σ^{1+\varrho})\bigr) ] for some (\varrho \in (0,\tfrac{1}{2})). To address the low regularity in the numerical analysis, we introduce a novel smoothing operator (J{h,α} = A_h^α\mathcal{P}h A^{-α}) with (α\in (0,1)), where (A_h) is the discrete Stokes operator and (\mathcal{P}h) the discrete Helmholtz projection. This tool enables a complete error analysis for a MINI-element spatial semidiscretization, yielding the mean-square convergence estimate [ |u - u_h|{L^2(Ω; C([0,T]; L^2(\mathcal O;\mathbb{R}^2)))} + |\nabla(u - u_h)|{L^2(Ω\times (0,T); L^2(\mathcal{O};\mathbb{R}^{2\times2}))} \leqslant c, h^{\varrho} \log\big(1 + \frac{1}{h}\big). ] The framework can be extended to broader stochastic fluid models with rough noise and Dirichlet boundary conditions.

💡 Research Summary

This paper presents a groundbreaking numerical analysis for the two-dimensional stochastic Navier-Stokes equations (SNSEs) driven by transport noise under physical no-slip boundary conditions on a convex polygonal domain. The central challenge addressed is the lack of spatial D(A)-regularity of the solution, where A is the Stokes operator, which previously invalidated standard error analysis techniques for spatial discretizations in this realistic setting.

The authors’ approach is twofold. First, under a key “small-noise” assumption requiring the sum of the squared W¹,∞ norms of the divergence-free transport noise vector fields (ζ_n) to be sufficiently small, they establish enhanced spatial regularity for the weak solution. Specifically, Proposition 3.1 proves that for some ϱ ∈ (0, 1/2), the solution u belongs to L²(Ω; C(