Title: Analysis of weak Galerkin mixed FEM based on the velocity--pseudostress formulation for Navier--Stokes equation on polygonal meshes Show Chinese title

Title: 基于速度-伪应力公式在多边形网格上对纳维-斯托克斯方程的弱伽辽金混合有限元方法分析

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 $ \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. Show Chinese abstract

Abstract: 本文介绍了、数学分析并数值验证了一种基于巴拿赫空间的新型弱伽辽金（WG）混合有限元方法，用于伪应力-速度形式的定常纳维-斯托克斯方程。 更精确地说，在所提出的技术中引入了一个依赖于压力以及扩散项和对流项的修改后的伪应力张量，称为$ \boldsymbol{\sigma} $，并推导出一个双混合变分公式，其中上述伪应力张量和速度是系统的主未知量，而压力则通过后处理公式计算。 因此，只需为张量变量提供一个WG空间，并为速度提供一个总次数最多为 'k' 的分片多项式向量空间。 此外，为了定义弱离散双线性形式，其连续版本涉及经典的散度算子，提出了一个弱散度算子作为经典散度算子在适当离散子空间中的已知替代方法。 使用不动点方法以及 Babuška-Brezzi 理论和 Banach-Nečas-Babuška 定理的离散版本证明了数值解的适定性。 此外，还为所提出的方法推导了一个先验误差估计。 最后，给出了几个数值结果，以说明该方法的良好性能并确认理论收敛速率。