A local and parallel Uzawa finite element method for the generalized Navier–Stokes equations
Introduction
Let Ω be a bounded domain in with a Lipschitz continuous boundary ∂Ω, the generalized Navier–Stokes equations are modeled as follows:where α ≥ 0 represents a nonnegative real number, ν > 0 denotes the viscosity coefficient, and 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.
Section snippets
Preliminary
Let ( · , · ) and ‖ · ‖0 denote the usual L2-scalar product and L2-norm over L2(Ω), respectively. Define s as a non-negative integer, the s-order Sobolev space is given by . The standard norm and seminorm on Hs(Ω) are denoted by ‖ · ‖s and | · |s, respectively. The closure of under the norm ‖ · ‖s is expressed with and its dual space is written as .
For convenience, the following useful Sobolev spaces are introduced:
An Uzawa finite element method based on Oseen scheme
In [6], several Uzawa methods for solving the steady Navier–Stokes equations were proposed by Chen, and the proofs of their convergence rates were derived. In this section, we are going to put forward an Uzawa finite element method based on Oseen scheme for the problems (1)–(3).
Let Th denote a regular triangulation where h > 0 is a real positive parameter in [2], [3], for finite element κ ∈ Th, we defineas the finite element mesh size.
The problems (1)–(3) are approximated by
Convergence analysis
In this section, we aim to investigate the convergence rates of the Uzawa finite element method based on Oseen scheme. First, we will prove that the sequences and generated by (18) and (19) are consistently bounded, and then analyze the corresponding convergence rates. Lemma 4.1 Let (uh, ph) ∈ (Xh, Mh) and be the solutions to (16)and (17) and the iterative sequences generated by (18)and (19), respectively. If Λ satisfies (13) and ρ satisfies then we have
Full overlapping domain decomposition technique
Let D1, D2, ..., DJ be a set of non-overlapping subdomains of the domain Ω, Ωj can be derived from extending outward to a certain size and satisfies an overlapping domain decomposition of Ω is composed of these subdomains Ω1, Ω2, ..., ΩJ. Each processor is only responsible for a subdomain with a fine mesh size h. The remaining domain generates a coarse mesh size H ≫ h by adopting adaptive processing techniques to make coarse and fine grid compatible at the
Numerical experiments
In this section, numerical experiments are investigated the advantages and disadvantages of the local and parallel Uzawa finite element method, the Uzawa finite element method and the standard finite element method from two aspects of accuracy and convergence rates.
Suppose that the entire domain is the finite element subspace is divided by regular triangulations with the fine mesh size h and coarse mesh size H. Here, the finite element pair is used
Acknowledgments
Supported in part by NSF of China (No. 11771259), special support program to develop innovative talents in the region of Shaanxi province, and the Natural Science Basic Research Plan of Shaanxi Province (No. 2018JQ4039).
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