Local and parallel finite element algorithms for the stokes problem

He, Ya‐Ling; Xu, Jinchao; Zhou, Aihui; Li, Jian

doi:10.1007/s00211-008-0141-2

Cited by 116 publications

(80 citation statements)

References 24 publications

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“…This is why we extend each disjoint subdomain with a certain distance to construct an enlarged subdomain, and use a global mesh that is fine around the enlarged subdomain to compute a local finite element solution in the disjoint subdomain, so as to alleviate the effect of coarse grid regions. Moreover, theory shows that by taking a suitable ratio of coarse mesh size H to fine mesh size h, we can obtain an asymptotically optimal error locally (see, e.g., [13,35,40]). …”

Section: Hh Jmentioning

confidence: 99%

“…In [14], by using a two-grid method and local refinement technique, He et al proposed a similar algorithm for the Stokes problem. Under some assumptions on the mesh and the mixed finite element spaces (namely, approximation, inverse estimate, superapproximation and stability), they derived the following error estimate for their algorithm:…”

Section: Error Estimatementioning

confidence: 99%

“…Under similar assumptions, following the framework of [14] and making some simple modifications, we can also derive the following error estimate for our parallel algorithm:…”

Section: Error Estimatementioning

confidence: 99%

“…Here we skip the proof details of (12); the interesting readers are referred to [14] for a reference. Instead, we put our emphasis on the numerical verification of the proposed algorithm.…”

Section: Error Estimatementioning

confidence: 99%

“…This approach was then applied to the standard steady Stokes equations [14,34] and the steady Navier-Stokes equations [13] by He et al In [35], by combining this approach with classical iterative methods for the steady Navier-Stokes equations, Shang and He designed several parallel iterative algorithms for the NavierStokes equations. It was also combined with the defect-correction method and the subgrid stabilization method in [33] and [32], respectively.…”

Section: Introductionmentioning

confidence: 99%

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A Parallel Finite Element Algorithm for Simulation of the Generalized Stokes Problem

Shang¹

2016

Bulletin of the Korean Mathematical Society

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Abstract. Based on a particular overlapping domain decomposition technique, a parallel finite element discretization algorithm for the generalized Stokes equations is proposed and investigated. In this algorithm, each processor computes a local approximate solution in its own subdomain by solving a global problem on a mesh that is fine around its own subdomain and coarse elsewhere, and hence avoids communication with other processors in the process of computations. This algorithm has low communication complexity. It only requires the application of an existing sequential solver on the global meshes associated with each subdomain, and hence can reuse existing sequential software. Numerical results are given to demonstrate the effectiveness of the parallel algorithm.

show abstract

Section: Hh Jmentioning

confidence: 99%

Section: Error Estimatementioning

confidence: 99%

“…Under similar assumptions, following the framework of [14] and making some simple modifications, we can also derive the following error estimate for our parallel algorithm:…”

Section: Error Estimatementioning

confidence: 99%

“…Here we skip the proof details of (12); the interesting readers are referred to [14] for a reference. Instead, we put our emphasis on the numerical verification of the proposed algorithm.…”

Section: Error Estimatementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Parallel Finite Element Algorithm for Simulation of the Generalized Stokes Problem

Shang¹

2016

Bulletin of the Korean Mathematical Society

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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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The physics informed neural networks for the unsteady Stokes problems

Yuan

2022

Numerical Methods in Fluids

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In this article, we develop the physics informed neural networks (PINNs) coupled with small sample learning for solving the transient Stokes equations. Specifically, the governing equations are encoded into the networks to construct the loss function, which involves the residual of differential equations, the initial/boundary conditions, and the residual of a handful of observations. The approximate solution was obtained by optimizing the loss function. Few sample data can rectify the network effectively and improve predictive accuracy.Moreover, the method can simultaneously solve each variable of the equations separately in a parallel framework. The information of the numerical data is compiled into the networks to enhance efficiency and accuracy in practice. Therefore, this method is a meshfree and fusion method that combined data-driven with model-driven. Inspired by the Galerkin method, the paper proves the convergence of the loss function and the capability of neural networks. Furthermore, numerical experiments are performed and discussed to demonstrate the performance of the method.

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Local and parallel finite element algorithms for the stokes problem

Cited by 116 publications

References 24 publications

A Parallel Finite Element Algorithm for Simulation of the Generalized Stokes Problem

A Parallel Finite Element Algorithm for Simulation of the Generalized Stokes Problem

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

The physics informed neural networks for the unsteady Stokes problems

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