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 $$u \in L^2\bigl(\Omega; C([0,T]; \dot{H}_{\sigma}^{\varrho}) \cap L^2(0,T; \dot{H}_{\sigma}^{1+\varrho})\bigr)$$ for some $\varrho \in (0,\tfrac{1}{2})$. To address the low regularity in the numerical analysis, we introduce a novel smoothing operator $J_{h,\alpha} = A_h^{\alpha}\mathcal{P}_h A^{-\alpha}$ 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 $$\|u - u_h\|_{L^2(\Omega; C([0,T]; L^2(\mathcal O;\mathbb{R}^2)))} + \|\nabla(u - u_h)\|_{L^2(\Omega \times (0,T); L^2(\mathcal{O};\mathbb{R}^{2\times2}))} \leqslant c\, h^{\varrho} \log\big(1 + \frac{1}{h}\big).$$ The framework can be extended to broader stochastic fluid models with rough noise and Dirichlet boundary conditions.