A new streamline diffusion finite element method for the generalized Oseen problem

Liao, Xin; Shi, Dongyang; Xu, Chuanju

doi:10.1007/s10483-018-2296-6

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…Many finite element spaces are known to satisfy the above mentioned Assumptions (B1)-(B4). The readers are referred to [3,7,12,30,40] for some of these finite element spaces. For instance, the MINI finite elements [46] and the P 2 −P 0 finite elements [47] satisfy Assumptions (B1)-(B4) when k = 1, while the Taylor-Hood elements [48] and the augmented P 2 −P 1 elements [49,50] satisfy Assumptions (B1)-(B4) when k = 2.…”

Section: (B4) Stabilitymentioning

confidence: 99%

“…Therefore, it is reasonable to study the Oseen problem first. For the Oseen problem, some numercial methods were proposed and analyzed such as the weak Galerkin finite element methods [27], the Nitsche cut finite element methods [28], the stabilized Nitsche cut finite element methods [29] and the new streamline diffusion finite element methods [30]. However, to the authors' best knowledge, there is no study on local and parallel finite element discretizations for the unsteady Oseen problems.…”

Section: Introductionmentioning

confidence: 99%

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A Parallel Finite Element Algorithm for the Unsteady Oseen Equations

sci¹

2021

AAMM

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Based on fully overlapping domain decomposition, a parallel finite element algorithm for the unsteady Oseen equations is proposed and analyzed. In this algorithm, each processor independently computes a finite element approximate solution in its own subdomain by using a locally refined multiscale mesh at each time step, where conforming finite element pairs are used for the spatial discretizations and backward Euler scheme is used for the temporal discretizations, respectively. Each subproblem is defined in the entire domain with 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 using an existing sequential solver without extensive recoding. The algorithm is easy to implement and has low communication cost. Error bounds of the parallel finite element approximate solutions are estimated. Numerical experiments are also given to demonstrate the effectiveness of the algorithm.

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Section: (B4) Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Parallel Finite Element Algorithm for the Unsteady Oseen Equations

sci¹

2021

AAMM

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“…In the past decades, traditional methods have been development greatly, e.g., researchers used FEM, [10] the parallel FEM, [11] the weak Galerkin FEM, [12] and other traditional methods to solve the Oseen equations. [13][14][15] Although some good results are obtained by these methods, the generation of grid often has a certain influence on the computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

Meshfree-based physics-informed neural networks for the unsteady Oseen equations

Peng¹,

Yuan²,

Zhang³

et al. 2023

Chinese Phys. B

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In this paper, we propose the meshfree-based Physics-informed neural networks (PINNs) for solving the unsteady Oseen equations. Firstly, based on the ideas of meshfree and small sample learning(SSL), we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh. Specifically, we optimize the neural network by minimizing the loss function to satisfy the differential operators, initial condition and boundary condition. Then, we prove the convergence of the loss function and the convergence of the neural network. In addition, the feasibility and effectiveness of the method are verified by the results of numerical experiments, and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.

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Local and parallel finite element algorithms for the time-dependent Oseen equations

Zheng

Shang

2020

Numer Algor

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A new streamline diffusion finite element method for the generalized Oseen problem

Cited by 4 publications

References 13 publications

A Parallel Finite Element Algorithm for the Unsteady Oseen Equations

A Parallel Finite Element Algorithm for the Unsteady Oseen Equations

Meshfree-based physics-informed neural networks for the unsteady Oseen equations

Local and parallel finite element algorithms for the time-dependent Oseen equations

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