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A fully divergence-free finite element method for magnetohydrodynamic equations
Hiptmair, Ralf1; Li, Lingxiao2; Mao, Shipeng2; Zheng, Weiying2
2018-04-01
Source PublicationMATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
ISSN0218-2025
Volume28Issue:4Pages:659-695
AbstractWe propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magnetic induction. We employ second-order semi-implicit timestepping, for which we rigorously establish an energy law and, as a consequence, unconditional stability. We prove unique solvability of the linear systems of equations to be solved in every timestep. For those we design an efficient preconditioner so that the number of preconditioned GMRES iterations is uniformly bounded with respect to the number of degrees of freedom. As both meshwidth and timestep size tend to zero, we prove that the discrete solutions converge to a weak solution of the continuous problem. Finally, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.
KeywordMagnetohydrodynamic equations divergence-free finite element method preconditioner magnetic vector potential driven cavity flow
DOI10.1142/S0218202518500173
Language英语
Funding ProjectNational Center for Mathematics and Interdisciplinary Sciences of China ; State Key Laboratory of Scientific and Engineering Computing of China ; National Magnetic Confinement Fusion Science Program of China[2015GB110003] ; National Key Research and Development Program of China[2016YFB0201304] ; China NSF[11471329] ; China NSF[91430215] ; Youth Innovation Promotion Association of CAS[2016003] ; National Science Fund for Distinguished Young Scholars[11725106]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000432721600002
PublisherWORLD SCIENTIFIC PUBL CO PTE LTD
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Cited Times:2[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/30360
Collection计算数学与科学工程计算研究所
Affiliation1.Swiss Fed Inst Technol, Seminar Appl Math, CH-8092 Zurich, Switzerland
2.Chinese Acad Sci, Sch Math Sci, Acad Math & Syst Sci, NCMIS,LSEC,ICMSEC, Beijing 100190, Peoples R China
Recommended Citation
GB/T 7714
Hiptmair, Ralf,Li, Lingxiao,Mao, Shipeng,et al. A fully divergence-free finite element method for magnetohydrodynamic equations[J]. MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES,2018,28(4):659-695.
APA Hiptmair, Ralf,Li, Lingxiao,Mao, Shipeng,&Zheng, Weiying.(2018).A fully divergence-free finite element method for magnetohydrodynamic equations.MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES,28(4),659-695.
MLA Hiptmair, Ralf,et al."A fully divergence-free finite element method for magnetohydrodynamic equations".MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES 28.4(2018):659-695.
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