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local and parallel Finite Element algorithm Based on Oseen-Type Iteration for the Stationary Incompressible MHD Flow

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JOURNAL OF SCIENTIFIC COMPUTING 2017年第1期70卷 149-174页

作者： Tang, Qili Huang, Yunqing Xiangtan Univ Sch Math & Computat Sci Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Hunan Peoples R China Henan Univ Sci & Technol Sch Math & Stat Luoyang 471023 Henan Peoples R China

In this work, we are concerned with the local and parallel finite element algorithm based on the Oseen-type iteration for solving the stationary incompressible magnetohydrodynamics. Under the uniqueness condition, the error estimates with respect to iterative step m and small mesh sizes H and of the proposed method are derived. Finally, some numerical experiments are provided to show the high efficiency of our algorithm.

关键词： local and parallel algorithm Finite element Oseen iteration Stationary incompressible magnetohydrodynamics

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Analysis of the local and parallel space-time algorithm for the heat equation

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2021年 100卷 167-181页

作者： Xue, Dandan Hou, Yanren Li, Yi Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Northwest Univ Sch Math Xian 710127 Shaanxi Peoples R China

In this paper, a second-order local and parallel space-time algorithm is proposed and analyzed for the heat equation. This scheme is based on the parareal with spectral deferred correction method in time and the expandable local and parallel finite element method in space. It realizes the parallelism both in the temporal as well as in the spatial direction. We prove its stability and the optimal error estimation in L-2-norm. At last, several numerical experiments are presented to demonstrate the effectiveness of our parallel scheme.

关键词： Time-dependent Parareal Spectral deferred correction local and parallel algorithm Superposition principle

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local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2017年第5期33卷 1513-1539页

作者： Zhang, Yuhong Hou, Yanren Shan, Li Dong, Xiaojing Xi An Jiao Tong Univ Sch Math & Stat Dept Comp Sci Xian 710049 Shaanxi Peoples R China Xi An Jiao Tong Univ Ctr Computat Geosci Xian 710049 Shaanxi Peoples R China Liaoning Tech Univ Dept Comp Sci Coll Sci Fuxin 123000 Peoples R China

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.(c) 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1513-1539, 2017

关键词： local a priori estimates local and parallel algorithm stationary incompressible magnetohydrodynamics two-grid discretization

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local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2018年第4期44卷 1295-1319页

作者： Dong, Xiaojing He, Yinnian Wei, Hongbo Zhang, Yuhong Inst Appl Phys & Computat Math Beijing 100191 Peoples R China Xiangtan Univ Hunan Key Lab Computat & Simulat Sci & Engn Key Lab Intelligent Comp & Informat Proc Sch Math & Computat SciMinist Educ Xiangtan 411105 Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China China Elect Technol Grp Corp Res Inst 20 Xian 710068 Shaanxi Peoples R China

Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments.

关键词： local and parallel algorithm Finite element Partition of unity method Stationary incompressible magnetohydrodynamics

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An expandable local and parallel two-grid finite element scheme

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2016年第12期71卷 2541-2556页

作者： Hou, Yanren Du, GuangZhi Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China

An expandable local and parallel two-grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of Poisson equation. Compared with the usual local and parallel finite element schemes, the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of CPUs. Convergence results based on H-1 and L-2 a priori error estimation of the scheme are obtained, which show that the scheme can reach the optimal convergence orders within vertical bar In H vertical bar(2) or vertical bar In H vertical bar two-grid iterations if the coarse mesh size H and the fine mesh size h are properly configured in 2-D or 3-D case, respectively. Some numerical results are presented at the end of the paper to support our analysis. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Two-grid finite element method Domain decomposition method local and parallel algorithm Error estimation

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parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

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Applied Mathematics and Mechanics(English Edition) 2022年第1期43卷 141-154页

作者： Qili TANG Yunqing HUANG Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan Province411105China

Based on local algorithms,some parallel finite element(FE)iterative methods for stationary incompressible magnetohydrodynamics(MHD)are *** approaches are on account of two-grid skill include two major phases:find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution,and then locally solve linearized residual subproblems by one of three iterations(Stokes-type,Newton,and Oseen-type)on subdomains with fine grid in parallel to approximate the high frequency *** error estimates with regard to two mesh sizes and iterative steps of the proposed algorithms are *** numerical examples are implemented to verify the algorithm.

关键词： local and parallel algorithm finite element(FE)method iteration stationary incompressible magnetohydrodynamics(MHD)

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Newton Iterative parallel Finite Element algorithm for the Steady Navier-Stokes Equations

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JOURNAL OF SCIENTIFIC COMPUTING 2010年第1期44卷 92-106页

作者： He, Yinnian Mei, Liquan Shang, Yueqiang Cui, Juan Xi An Jiao Tong Univ Fac Sci State Key Lab Multiphase Flow Power Engn Xian 710049 Peoples R China Guizhou Normal Univ Sch Math & Comp Sci Guiyang 550001 Peoples R China

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Omega and computing a linear problem on a fine grid in some subdomains Omega (j) aS,Omega with j=1,aEuro broken vertical bar,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and ha parts per thousand(a)H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.

关键词： Navier-Stokes equations Newton iterative method Two-grid method local and parallel algorithm

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CONVERGENCE ANALYSIS OF SOME FINITE ELEMENT parallel algorithmS FOR THE STATIONARY INCOMPRESSIBLE MHD EQUATIONS

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Journal of Computational Mathematics 2024年第1期42卷 49-70页

作者： Xiaojing Dong Yinnian He Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China Institute of Applied Physics and Computational Mathematics Beijing 100088China School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China

By combination of iteration methods with the partition of unity method(PUM),some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics(MHD)with different physical parameters are presented and *** algorithms are highly *** first,a global solution is obtained on a coarse grid for all approaches by one of the iteration *** parallelized residual schemes,local corrected solutions are calculated on finer meshes with overlapping *** subdomains can be achieved flexibly by a class of *** proposed algorithm is proved to be uniformly stable and ***,one numerical example is presented to confirm the theoretical findings.

关键词： Partition of unity method local and parallel algorithm Finite element method Iteration methods Magnetohydrodynamics

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A parallel finite element method for incompressible magnetohydrodynamics equations

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APPLIED MATHEMATICS LETTERS 2020年 102卷 106076-000页

作者： Dong, Xiaojing He, Yinnian Xiangtan Univ Key Lab Intelligent Comp & Informat Proc Hunan Key Lab Computat & Simulat Sci & Engn Minist EducSch Math & Computat Sci Xiangtan 411105 Peoples R China Inst Appl Phys & Computat Math POB 8009 Beijing 100088 Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Peoples R China

Based on the partition of unity method (PUM), a parallel finite element method (FEM) is designed for stationary incompressible magnetohydrodynamics (MHD) equations. The nonlinear problem is solved globally on a coarse grid, and then correction subproblems on corresponding subdomains with fine meshes are computed in parallel by Picard iteration. The subdomains are generated by a class of partition of unity, which gives a flexible and controllable way to decompose the domain. The optimal error estimate is proved. The validity of the proposed algorithm is testified by numerical examples. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Partition of unity method local and parallel algorithm Finite element Picard iteration Stationary incompressible magnetohydrodynamics

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