## A nonconforming finite element method for fourth order curl equations in $\mathbb {R}^3$

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- by Bin Zheng, Qiya Hu and Jinchao Xu PDF
- Math. Comp.
**80**(2011), 1871-1886 Request permission

## 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 $(\nabla \times )^2$ and $(\nabla \times )^4$ 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.## References

- D. Biskamp, E. Schwarz, and J.F. Drake,
*Ion-controlled collisionless magnetic reconnection*, Phys. Rev. Lett., 75:3850-3853, 1995. - H. Blum and R. Rannacher,
*On the boundary value problem of the biharmonic operator on domains with angular corners*, Math. Methods Appl. Sci.**2**(1980), no. 4, 556–581. MR**595625**, DOI 10.1002/mma.1670020416 - Fioralba Cakoni and Houssem Haddar,
*A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media*, Inverse Probl. Imaging**1**(2007), no. 3, 443–456. MR**2308973**, DOI 10.3934/ipi.2007.1.443 - Philippe G. Ciarlet,
*The finite element method for elliptic problems*, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR**0520174** - Ramon Codina and Noel Hernández-Silva,
*Stabilized finite element approximation of the stationary magneto-hydrodynamics equations*, Comput. Mech.**38**(2006), no. 4-5, 344–355. MR**2246129**, DOI 10.1007/s00466-006-0037-x - J.-F. Gerbeau,
*A stabilized finite element method for the incompressible magnetohydrodynamic equations*, Numer. Math.**87**(2000), no. 1, 83–111. MR**1800155**, DOI 10.1007/s002110000193 - K. Germaschewski and R. Grauer,
*Longitudinal and transversal structure functions in two-dimensional electron magnetohydrodynamic flows*, Phys. Plasmas, 6:3788-3793, 1999. - Vivette Girault and Pierre-Arnaud Raviart,
*Finite element methods for Navier-Stokes equations*, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR**851383**, DOI 10.1007/978-3-642-61623-5 - J. Gopalakrishnan, L. E. García-Castillo, and L. F. Demkowicz,
*Nédélec spaces in affine coordinates*, Comput. Math. Appl.**49**(2005), no. 7-8, 1285–1294. MR**2141266**, DOI 10.1016/j.camwa.2004.02.012 - 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**(2003), no. 6, 709–731. MR**2009590**, DOI 10.1002/num.10067 - Qiya Hu and Jun Zou,
*Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions*, Math. Comp.**73**(2004), no. 245, 35–61. MR**2034110**, DOI 10.1090/S0025-5718-03-01541-2 - S.C. Jardin,
*A triangular finite element with first-derivative continuity applied to fusion MHD applications*, J. Comput. Phys., 200:133-152, 2004. - J. A. Breslau and S. C. Jardin,
*Global extended magnetohydrodynamic studies of fast magnetic reconnection*, Phys. Plasmas**10**(2003), no. 5, 1291–1298. MR**1978167**, DOI 10.1063/1.1566026 - K. S. Kang and D. E. Keyes,
*Implicit symmetrized streamfunction formulations of magnetohydrodynamics*, Internat. J. Numer. Methods Fluids**58**(2008), no. 11, 1201–1222. MR**2475392**, DOI 10.1002/fld.1755 - S.K. Krzeminski, M. Smialek, and M. Wlodarczyk,
*Finite element approximation of biharmonic mathematical model for MHD flow using $\Psi$ - An approach*, IEEE Trans. Magn., 36:1313-1318, 2000. - S. Lankalapalli, J. E. Flaherty, M. S. Shephard, and H. Strauss,
*An adaptive finite element method for magnetohydrodynamics*, J. Comput. Phys.**225**(2007), no. 1, 363–381. MR**2346682**, DOI 10.1016/j.jcp.2006.12.010 - 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**(1997), no. Dec., 198–210. Special issue on multilevel methods (Copper Mountain, CO, 1997). MR**1615165** - L. Morley,
*The triangular equilibrium problems in the solution of plate bending problems*, Aero. Quart., 19:149-169, 1968. - J.-C. Nédélec,
*Mixed finite elements in $\textbf {R}^{3}$*, Numer. Math.**35**(1980), no. 3, 315–341. MR**592160**, DOI 10.1007/BF01396415 - Trygve K. Nilssen, Xue-Cheng Tai, and Ragnar Winther,
*A robust nonconforming $H^2$-element*, Math. Comp.**70**(2001), no. 234, 489–505. MR**1709156**, DOI 10.1090/S0025-5718-00-01230-8 - 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**(2007), no. 2, 1919–1936. MR**2349689**, DOI 10.1016/j.jcp.2007.02.027 - R. Rannacher,
*Finite element approximation of simply supported plates and the Babuška paradox*, Z. Angew. Math. Mech.**59**(1979), no. 3, T73–T76. MR**533989** - Nizar Ben Salah, Azzeddine Soulaimani, and Wagdi G. Habashi,
*A finite element method for magnetohydrodynamics*, Comput. Methods Appl. Mech. Engrg.**190**(2001), no. 43-44, 5867–5892. MR**1848902**, DOI 10.1016/S0045-7825(01)00196-7 - Dominik Schötzau,
*Mixed finite element methods for stationary incompressible magneto-hydrodynamics*, Numer. Math.**96**(2004), no. 4, 771–800. MR**2036365**, DOI 10.1007/s00211-003-0487-4 - H. R. Strauss and D. W. Longcope,
*An adaptive finite element method for magnetohydrodynamics*, J. Comput. Phys.**147**(1998), no. 2, 318–336. MR**1663571**, DOI 10.1006/jcph.1998.6091 - 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. - G. Tóth,
*Numerical simulations of magnetohydrodynamic flows*, Invited review at the The Interaction of Stars with their Environment conference, 1996. - Ming Wang and Jinchao Xu,
*The Morley element for fourth order elliptic equations in any dimensions*, Numer. Math.**103**(2006), no. 1, 155–169. MR**2207619**, DOI 10.1007/s00211-005-0662-x - M. Wang and J. Xu,
*Minimal finite element spaces for $2m$-th order partial differential equations in $\mathbb {R}^n$*, submitted, 2006. - Ming Wang, Jin-chao Xu, and Yu-cheng Hu,
*Modified Morley element method for a fourth order elliptic singular perturbation problem*, J. Comput. Math.**24**(2006), no. 2, 113–120. MR**2204450** - Jon P. Webb,
*Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements*, IEEE Trans. Antennas and Propagation**47**(1999), no. 8, 1244–1253. MR**1711458**, DOI 10.1109/8.791939 - Matthias Wiedmer,
*Finite element approximation for equations of magnetohydrodynamics*, Math. Comp.**69**(2000), no. 229, 83–101. MR**1654014**, DOI 10.1090/S0025-5718-99-01146-1 - 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.

## 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
- MR Author ID: 228866
- Email: xu@math.psu.edu
- 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. - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**80**(2011), 1871-1886 - MSC (2010): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2011-02480-4
- MathSciNet review: 2813342