Parallel iterative stabilized finite element methods based on the quadratic equal-order elements for incompressible flow
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Parallel iterative stabilized finite element methods based on the quadratic equal‑order elements for incompressible flows Bo Zheng1 · Yueqiang Shang1 Received: 1 June 2019 / Revised: 26 March 2020 / Accepted: 22 September 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract Combining the quadratic equal-order stabilized method with the approach of local and parallel finite element computations and classical iterative methods for the discretization of the steady-state Navier–Stokes equations, three parallel iterative stabilized finite element methods based on fully overlapping domain decomposition are proposed and compared in this paper. In these methods, each processor independently computes an approximate solution in its own subdomain using a global composite mesh that is fine around its own subdomain and coarse elsewhere, making the methods be easy to implement based on existing codes and have low communication complexity. Under some (strong) uniqueness conditions, stability and convergence theory of the parallel iterative stabilized methods are derived. Numerical tests are also performed to demonstrate the stability, convergence orders and high efficiency of the proposed methods. Keywords Navier–Stokes equations · Iterative stabilized method · Finite element · Parallel algorithm · Domain decomposition method Mathematics Subject Classification 76D05 · 35Q30 · 65N55 · 65N30
1 Introduction Let 𝛺 ⊂ ℝd ( d = 2 or 3) be a bounded domain with Lipschitz-continuous boundary 𝛤 = 𝜕𝛺 and satisfy a regularity condition stated in (A0) below. We consider the following steady-state incompressible Navier–Stokes equations: * Yueqiang Shang [email protected]; [email protected] Bo Zheng [email protected] 1
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
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− 𝜈𝛥u + (u ⋅ ∇)u + ∇p = f , ∇ ⋅ u = 0, u = 0,
in 𝛺, on 𝛤 ,
in 𝛺,
(1) (2) (3)
where u = (u1 , u2 , ..., ud ) is the velocity vector, p the pressure, f = (f1 , f2 , ..., fd ) the prescribed body force, and 𝜈 > 0 the viscosity coefficient. For the above incompressible Navier–Stokes system, much numerical work has been devoted to the development of efficient numerical schemes, and many numerical discretization methods have been proposed and studied for the Navier–Stokes equations and related flow problems (cf. [1–3]). Among these successful numerical approximation methods, finite element methods have gained great popularity in the computational fluid dynamics community, since they are flexible in approximating the solution domain and their well-developed theories in analysis. It is universally acknowledged that the mixed finite element methods used for the discretization of the Navier–Stokes equations require the finite element spaces of the velocity and pressure satisfying the so-called discrete inf-sup condition. Frequently, they may generate non-physical oscillations for the pressure if the finite element spaces do not satisfy this con
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