The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin (WG) mixed-FEM based on Banach spaces for the stationary Navier--Stokes equation in pseudostress-velocity formulation. More precisely, a modified pseudostress tensor, called $ \boldsymbol{\sigma} $, depending on the pressure, and the diffusive and convective terms has been introduced in the proposed technique, and a dual-mixed variational formulation has been derived where the aforementioned pseudostress tensor and the velocity, are the main unknowns of the system, whereas the pressure is computed via a post-processing formula. Thus, it is sufficient to provide a WG space for the tensor variable and a space of piecewise polynomial vectors of total degree at most 'k' for the velocity. Moreover, in order to define the weak discrete bilinear form, whose continuous version involves the classical divergence operator, the weak divergence operator as a well-known alternative for the classical divergence operator in a suitable discrete subspace is proposed. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babu\v{s}ka-Brezzi theory and the Banach-Ne\v{c}as-Babu\v{s}ka theorem. Additionally, an a priori error estimate is derived for the proposed method. Finally, several numerical results illustrating the method's good performance and confirming the theoretical rates of convergence are presented.

翻译：本文提出、数学分析并数值验证了一种基于Banach空间的弱Galerkin混合有限元新方法，用于求解稳态Navier-Stokes方程的伪应力-速度格式。具体而言，该方法引入了一个依赖于压力、扩散项与对流项的修正伪应力张量$ \boldsymbol{\sigma} $，并推导出双混合变分公式，其中上述伪应力张量与速度为系统的主要未知量，而压力则通过后处理公式计算得出。因此，只需为张量变量提供弱Galerkin空间，为速度提供总次数不超过'k'的分片多项式向量空间即可。此外，为定义连续版本涉及经典散度算子的弱离散双线性形式，本文在合适的离散子空间中提出弱散度算子作为经典散度算子的已知替代方案。通过不动点方法以及Babu\v{s}ka-Brezzi理论和Banach-Ne\v{c}as-Babu\v{s}ka定理的离散版本，证明了数值解的良好适定性。同时，推导了所提方法的前验误差估计。最后，通过多个数值结果展示了该方法的高效性能，并验证了理论收敛阶。