Parallel iterative finite element algorithms based on full domain partition for the stationary Navier–Stokes equations

Shang, Yueqiang; He, Yinnian

doi:10.1016/j.apnum.2010.03.013

Cited by 61 publications

(17 citation statements)

References 15 publications

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“…Proof. The proof follows the framework of [11,12]. Let s be an integer such that s ≥ max{2γ − 1, γ + 1}, D j , and j (j = 1, 2 .…”

Section: A Local Finite Element Variational Multiscale Methodsmentioning

confidence: 99%

“…The idea that use a locally refined global mesh to compute a local finite element solution in a given interested subdomain was first proposed by Xu and Zhou for a class of linear and nonlinear elliptic boundary value problems. This discretization approach was subsequently applied to solve numerically the elliptic eigenvalue problems , the Stokes equations , and the Navier–Stokes equations . In particular, by combing this discretization approach with classical iterative methods for the Navier–Stokes equations, several parallel iterative algorithms for the numerical solution of laminar incompressible Navier–Stokes equations were developed and analyzed in .…”

Section: Introductionmentioning

confidence: 99%

“…This discretization approach was subsequently applied to solve numerically the elliptic eigenvalue problems , the Stokes equations , and the Navier–Stokes equations . In particular, by combing this discretization approach with classical iterative methods for the Navier–Stokes equations, several parallel iterative algorithms for the numerical solution of laminar incompressible Navier–Stokes equations were developed and analyzed in . It was also combined with the defect‐correction method and the subgrid stabilization method .…”

Section: Introductionmentioning

confidence: 99%

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A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

Shang

2014

Numerical Methods Partial

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Based on fully overlapping domain decomposition and a recent variational multiscale method, a parallel finite element variational multiscale method for convection dominated incompressible flows is proposed and analyzed. In this method, each processor computes a local finite element solution in its own subdomain using a global mesh that is locally refined around its own subdomain, where a stabilization term based on two local Gauss integrations is adopted to stabilize the numerical form of the Navier-Stokes equations. Using the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method.

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“…Proof. The proof follows the framework of [11,12]. Let s be an integer such that s ≥ max{2γ − 1, γ + 1}, D j , and j (j = 1, 2 .…”

Section: A Local Finite Element Variational Multiscale Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

Shang

2014

Numerical Methods Partial

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“…For any

(v, T, q) \in {boldX}_{boldh} (D) \times {boldW}_{boldh} (D) \times M_{h} (D)

, there holds

‖ v ‖_{1, D} \leq c h^{- 1} ‖ v ‖_{0, D}, ‖ T ‖_{1, D} \leq c h^{- 1} ‖ T ‖_{0, D}, ‖ q ‖_{0, D} \leq c h^{- 1} ‖ q ‖_{- 1, D} .

A3. Superapproximations . Let

ω \in C^{\infty} (Ω)

with

supp (ω) \subset \subset D

.…”

Section: Preliminariesmentioning

confidence: 99%

Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics

Zhang

Hou

Shan

et al. 2017

Numerical Methods Partial

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This article presents a local and parallel finite element method for the stationary incompressible magnetohydrodynamics problem. The key idea of this algorithm comes from the two‐grid discretization technique. Specifically, we solve the nonlinear system on a global coarse mesh, and then solve a series of linear problems on several subdomains in parallel. Furthermore, local a priori estimates are obtained on a general shape regular grid. The efficiency of the algorithm is also illustrated by some numerical experiments.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1513–1539, 2017

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“…And then, we apply three kinds of iterative methods, called Stokes iteration, Newton iteration and Oseen iteration (cf. References 11,12), to deal with the nonlinear convective term. The combination of Uzawa iterative method and three iterative methods is one of the innovations of this article.…”

Section: Introductionmentioning

confidence: 99%

Parallel iterative stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

Zhou

Shang

2020

Numerical Methods in Fluids

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Based on full domain partition technique, some parallel iterative pressure projection stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions are designed and analyzed. In these algorithms, the lowest equal‐order P1 − P1 elements are used for finite element discretization and a local pressure projection stabilized method is used to counteract the invalidness of the discrete inf‐sup condition. Each subproblem is solved on a global composite mesh with the vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. We estimate the optimal error bounds of the approximate solutions with the use of some (strong) uniqueness conditions. Numerical results are also given to demonstrate the effectiveness of the parallel algorithms.

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Parallel iterative finite element algorithms based on full domain partition for the stationary Navier–Stokes equations

Cited by 61 publications

References 15 publications

A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics

Parallel iterative stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

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