A nonconforming finite element method for fourth order curl equations in

Authors:
Bin Zheng, Qiya Hu and Jinchao Xu

Journal:
Math. Comp. **80** (2011), 1871-1886

MSC (2010):
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-2011-02480-4

Published electronically:
March 25, 2011

MathSciNet review:
2813342

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both and operators. The element has a very small number of degrees of freedom, and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.

**1.**D. Biskamp, E. Schwarz, and J.F. Drake,*Ion-controlled collisionless magnetic reconnection*, Phys. Rev. Lett., 75:3850-3853, 1995.**2.**H. Blum and R. Rannacher,*On the boundary value problem of the biharmonic operator on domains with angular corners*, Math. Methods Appl. Sci., 2:556-581, 1980. MR**595625 (82a:35022)****3.**F. Cakoni and H. Haddar,*A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media*, Inverse Problems and Imaging, 1:443-456, 2007. MR**2308973 (2008d:35227)****4.**P.G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam New York, 1978. MR**0520174 (58:25001)****5.**R. Codina and N. Hernandez-Silva,*Stabilized finite element approximation of the stationary magneto-hydrodynamics equations*, Comput. Mech., 38:344-355, 2006. MR**2246129 (2007e:76186)****6.**J.-F. Gerbeau,*A stabilized finite element method for the incompressible magnetohydrodynamic equations*, Numer. Math., 87:83-111, 2000. MR**1800155 (2001j:76071)****7.**K. Germaschewski and R. Grauer,*Longitudinal and transversal structure functions in two-dimensional electron magnetohydrodynamic flows*, Phys. Plasmas, 6:3788-3793, 1999.**8.**V. Girault and P. Raviart,*Finite Element Methods for Navier-Stokes Equations*, Springer-Verlag, Berlin, Heidelberg, 1986. MR**851383 (88b:65129)****9.**J. Gopalakrishnan, L.E. Garcia-Castillo, and L.F. Demkowicz,*Nédélec spaces in affine coordinates*, Computers and Mathematics with Applications, 49:1285-1294, 2005. MR**2141266 (2006a:65160)****10.**J.L. Guermond and P.D. Minev,*Mixed finite element approximation of an MHD problem involving conducting and insulating regions: The 3D case*, Numer. Methods Partial Differential Equations, 19:709-731, 2003. MR**2009590 (2004h:65099)****11.**Q. Hu and J. Zou,*Substructuring preconditioners for saddle-point problems arising from Maxwell's equations in three dimensions*, Math. Comp., 73:35-61, 2004. MR**2034110 (2004m:65197)****12.**S.C. Jardin,*A triangular finite element with first-derivative continuity applied to fusion MHD applications*, J. Comput. Phys., 200:133-152, 2004.**13.**S.C. Jardin and J.A. Breslau,*Implicit solution of the four-field extended magnetohydrodynamic equations using higher-order high-continuity finite elements*, Phys. Plasmas, 12:056101.1-056101.10, 2005. MR**1978167 (2004c:76151)****14.**K.S. Kang and D.E. Keyes,*Implicit symmetrized streamfunction formulations of magnetohydrodynamics*, Int. J. Numer. Meth. Fluids, 58:1201-1222, 2008. MR**2475392 (2009m:76109)****15.**S.K. Krzeminski, M. Smialek, and M. Wlodarczyk,*Finite element approximation of biharmonic mathematical model for MHD flow using - An approach*, IEEE Trans. Magn., 36:1313-1318, 2000.**16.**S. Lankalapallia, J.E. Flaherty, M.S. Shephard, and H. Strauss,*An adaptive finite element method for magnetohydrodynamics*, J.Comput. Phys., 225:363-381, 2007. MR**2346682 (2009a:76093)****17.**W.J. Layton, A.J. Meir, and P.G. Schmidt,*A two-level discretization method for the stationary MHD equations*, Electron. Trans. Numer. Anal., 6:198-210, 1997. MR**1615165 (99c:76067)****18.**L. Morley,*The triangular equilibrium problems in the solution of plate bending problems*, Aero. Quart., 19:149-169, 1968.**19.**J.C. Nédélec,*Mixed finite elements in*, Numer. Math., 35:315-341, 1980. MR**592160 (81k:65125)****20.**T.K. Nilssen, X.-C. Cai, and R. Winther,*A robust nonconforming element*, Math. Comp., 70:489-505, 2001. MR**1709156 (2001g:65158)****21.**S. Ovtchinnikov, F. Dobrian, X.-C. Cai, and D.E. Keyes,*Additive Schwarz-based fully coupled implicit methods for resistive Hall Magnetohydrodynamic problems*, J.Comput. Phys., 225:1919-1936, 2007. MR**2349689 (2008f:76138)****22.**R. Rannacher,*Finite element approximation of simply supported plates and the Babuska paradox*, ZAMM, 59:73-76, 1979. MR**533989 (80d:65122)****23.**N.B. Salah, A. Soulaimani, and W.G. Habashi,*A finite element method for magnetohydrodynamics*, Comput. Methods Appl. Mech. Engrg., 190:5867-5892, 2001. MR**1848902 (2002e:76032)****24.**D. Schötzau,*Mixed finite element methods for stationary incompressible magnetohydrodynamics*, Numer. Math., 96:771-800, 2004. MR**2036365 (2005b:76088)****25.**H.R. Strauss and D.W. Longcope,*An adaptive finite element method for magnetohydrodynamics*, J.Comput. Phys., 147:318-336, 1998. MR**1663571 (99h:76062)****26.**D. Sun,*Substructuring preconditioners for high order edge finite element discretizations to Maxwell's equations in three-dimensions*, Ph.D. Thesis, Chinese Academy of Sciences, 2008.**27.**G. Tóth,*Numerical simulations of magnetohydrodynamic flows*, Invited review at the The Interaction of Stars with their Environment conference, 1996.**28.**M. Wang and J. Xu,*The Morley element for fourth order elliptic equations in any dimensions*, Numer. Math., 103:155-169, 2006. MR**2207619 (2006i:65205)****29.**M. Wang and J. Xu,*Minimal finite element spaces for -th order partial differential equations in*, submitted, 2006.**30.**M. Wang, J. Xu, and Y. Hu,*Modified Morley element method for a fourth elliptic singular perturbation problem*, J. Comput. Math., 24:113-120, 2006. MR**2204450 (2006k:65340)****31.**Jon P. Webb,*Hierarchical vector basis functions of arbitrary order for triangular and tetrahedral finite elements*, IEEE Trans. Antennas Propag., 47:1244-1253, 1999. MR**1711458 (2000g:78031)****32.**M. Wiedmer,*Finite element approximation for equations of magnetohydrodynamics*, Math. Comp., 69:83-101, 1999. MR**1654014 (2000i:65195)****33.**U. Ziegler,*Adaptive mesh refinement in MHD modeling, realization, tests and application*, in Edith Falgarone and Thierry Passot, editors, Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics 614, pages 127-151, Springer-Verlag, Berlin Heidelberg, 2003.

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Additional Information

**Bin Zheng**

Affiliation:
Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Address at time of publication:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912

Email:
bin_zheng@brown.edu

**Qiya Hu**

Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

Email:
hqy@lsec.cc.ac.cn

**Jinchao Xu**

Affiliation:
Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Email:
xu@math.psu.edu

DOI:
https://doi.org/10.1090/S0025-5718-2011-02480-4

Received by editor(s):
January 29, 2010

Received by editor(s) in revised form:
July 20, 2010

Published electronically:
March 25, 2011

Additional Notes:
The second author was supported by The Key Project of Natural Science Foundation of China G11031006, National Basic Research Program of China No. G2011309702 and Natural Science Foundation of China G10771178.

The third author was supported by the National Science Foundation under contracts DMS-0609727 and DMS-0915153 and by the Center for Computational Mathematics and Applications, Pennsylvania State University.

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.