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 $$u\in L^2(\mathrm{\Omega };C([0,T];{\dot{H}}_{\sigma }^{\varrho })\cap L^2(0,T;{\dot{H}}_{\sigma }^{1+\varrho }))$$ for some $\varrho \in (0,\frac{1}{2})$. To address the low regularity in the numerical analysis, we introduce a novel smoothing operator $J_{h,\alpha }=A_h^{\alpha }{\mathcal{P}}_h{A}^{-\alpha }$ with $\alpha \in (0,1)$, 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 $$\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_{L^2(\mathrm{\Omega };C([0,T];L^2(\mathcal{O};{\mathbb{R}}^2)))}+\mathrm{\parallel }\mathrm{\nabla }(u-u_h){\mathrm{\parallel }}_{L^2(\mathrm{\Omega }\times (0,T);L^2(\mathcal{O};{\mathbb{R}}^{2\times 2}))}⩽c{h}^{\varrho }\log (1+\frac{1}{h})\mathrm{.}$$ The framework can be extended to broader stochastic fluid models with rough noise and Dirichlet boundary conditions.