A local and parallel Uzawa finite element method for the generalized Navier–Stokes equations

Abstract

In this paper, we propose and develop a local and parallel Uzawa finite element method for the generalized Navier–Stokes equations. The Uzawa finite element method is no need to deal with the saddle point problem, and only solves one vector-valued elliptic equation and one simple scalar-valued equation. It has the geometric convergence with a crispation number γ what has nothing to do with the mesh size h. As for the local and parallel Uzawa finite element method, each subproblem is a global problem, but most of degrees of freedom originate from the subdomain. Moreover, the presented method is easy to be applied with less communication requirements and has good parallelism. Finally, numerical results verify the performance of the proposed method.

Introduction

Let Ω be a bounded domain in $R^d(d=2,3)$ with a Lipschitz continuous boundary ∂Ω, the generalized Navier–Stokes equations are modeled as follows:

$$\alpha u-\nu \Delta u+(u·\nabla )u+\nabla p=f,\text{in}\Omega ,$$

$$\nabla ·u=0,\text{in}\Omega ,$$

$$u=0,\text{on}\partial \Omega ,$$

where α ≥ 0 represents a nonnegative real number, ν > 0 denotes the viscosity coefficient, $u\left(x\right)=(u_1\left(x\right),u_2\left(x\right),\dots ,u_d\left(x\right))$ and $p=p\left(x\right)$ are corresponding to the velocity vector and pressure, respectively, and f(x) is the external force.

Until now, numerical methods for the Navier–Stokes problems have been largely investigated in the past several decades, an imperfect list of references is referred to [7], [11], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [27], [32], [33]. The Uzawa method [4], [5], [6], [7], [10], [32] is identified as a very interesting method in numerous numerical methods. Originally, Arrow et al. proposed the Uzawa method [1] in 1958, and Nochetto obtained the convergence rates of the Uzawa method for the Stokes equations [26] in 2004. Then, the Uzawa method for solving the steady Navier–Stokes equations was developed and the convergence had been proved by Temam in [32]. Furthermore, the convergence rates of the Uzawa method for the steady Navier–Stokes equations were presented in [4], [5], [6], [10], and this method converged geometrically with a crispation number γ unrelated to the mesh size h.

As we know, the local and parallel finite element methods are eventful in modern scientific and engineering computing. Historically, a local and parallel access to finite element discretizations was put forward to solve a class of linear and nonlinear elliptic boundary value problems in [34], [35], [36], [38]. Then, this method was adopted for the Stokes equations [8]. Whereafter, the local and parallel finite element methods using full domain decomposition techniques were designed and developed for the stationary Stokes equations [29]. Furthermore, many kindsof local and parallel finite element algorithms were proposed for the stationary and non-stationary Navier-Stokes equations referred to [9], [28], [30], [31], [37]. This approach involved less communication than current standard methods. Moreover, the existing sequential PDE codes can be efficiently used in a parallel system with a little recoding.

However, there are few results discussing the Uzawa finite element method together with the local and parallel finite element method based on full overlapping domain decomposition techniques for the problems (1)–(3). In this paper, we will bring in a new local and parallel Uzawa finite element method to solve the problems (1)–(3). Similar to [4], [29], we apply the Uzawa finite element element method based on Oseen scheme to solve the problems (1)–(3) on the subdomains instead of the entire domain. Among the presented method, the Uzawa finite element method and the standard finite element method, the presented method has achieved the same accuracy as the other two methods. Numerical results show that the presented method can drastically reduce computation time and is the most efficient among these three methods.

This paper is structured as follows: some preliminary knowledge is introduced in Section 2. In Section 3, the finite element spaces and their properties are given, the Uzawa finite element method based on Oseen scheme for the problems (1)–(3) is proposed, and the well-posedness of this method is proved. In Section 4, convergence rates of the proposed Uzawa method are derived. In Section 5, the presented method is studied. Finally, numerical results are shown in Section 6.