Analysis of Weak Galerkin Mixed Finite Element Method Based on the Velocity–Pseudostress Formulation for Navier–Stokes Equation on Polygonal Meshes

Abstract

The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin mixed finite element method based on Banach spaces for the stationary Navier–Stokes equation in pseudostress–velocity formulation. Specifically, a modified pseudostress tensor, which depends on the pressure as well as the diffusive and convective terms, is introduced as an auxiliary unknown, and the incompressibility condition is then used to eliminate the pressure, which is subsequently computed using a postprocessing formula. Consequently, to discretize the resulting mixed formulation, it is sufficient to provide a tensorial weak Galerkin space for the pseudostress and a space of piecewise polynomial vectors of total degree at most ’k’ for the velocity. Moreover, the weak gradient/divergence operator is utilized to propose the weak discrete bilinear forms, whose continuous version involves the classical gradient/divergence operators. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška 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.