  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.
