Analysis of weak Galerkin mixed FEM based on the velocity--pseudostress formulation for Navier--Stokes equation on polygonal meshes

TL;DR

This work develops a weak Galerkin mixed finite element method for the stationary Navier–Stokes equations using a velocity–pseudostress formulation, introducing a modified pseudostress σ that depends on pressure and employing pressure post-processing. The method forms a dual-mixed variational problem with σ and the velocity u as primary unknowns, and discretizes it with a WG space for σ and a piecewise polynomial velocity space, using a weak divergence operator and a stabilization term to ensure stability. The authors prove well-posedness via a fixed-point argument and discrete Babuška–Brezzi theory under a small-data assumption, and derive a priori error estimates, including suboptimal and optimal convergence rates, with an accompanying pressure error bound. Numerical results on polygonal meshes verify the theoretical rates and demonstrate the method’s effectiveness for problems such as lid-driven cavities and flows in heterogeneous porous domains, highlighting accuracy, robustness, and applicability to complex geometries. The approach offers direct computation of quantities like stress without velocity derivatives and provides a flexible framework for polygonal meshes and nonstandard Banach spaces, with potential impact on stable, accurate simulations of incompressible flows in challenging geometries.

Abstract

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 $ \boldsymbolσ $, 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 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.

Figures (5)

• Figure 1: Example 1, samples of the kind of meshes utilized.
• Figure 2: Example 1, snapshots of the numerical stress components (first row, left to right), the velocity components and pressure (second row, left to right), computed with $k = 0$ in the mesh made of hexagons with $h = 3.030$e-2.
• Figure 3: Example 2. The numerical velocity and pressure for $\nu =1$ (first row) and $\nu=0.01$ (second row) .
• Figure 4: Example 4. (a). Schematic of of the computational domain $\Omega=\Omega_{p}\cup\Omega_{f}$ , where $\Omega_{f}$ is the large porosity subdomain and $\Omega_{p}$ is subdomain with the small porosity. (b). The polygonal mesh of domain.
• Figure 5: Example 4. The numerical velocity and the first two components of stress from left to right.

Theorems & Definitions (35)

• Theorem 2.1
• proof
• Definition 3.1: Wang13
• Definition 3.2: Wang13
• Lemma 3.3
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3