Mixed schemes for quad-curl equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 147-161.

In this paper, mixed schemes are presented for two variants of quad-curl equations. Specifically, stable equivalent mixed formulations for the model problems are presented, which can be discretized by finite elements of low regularity and of low degree. The regularities of the mixed formulations and thus equivalently the primal formulations are established, and some finite elements examples are given which can exploit the regularity of the solutions to an optimal extent.

DOI : 10.1051/m2an/2018005
Classification : 65N30, 35Q60, 76E25, 76W05
Mots-clés : Quad-curl equation, mixed scheme, regularity analysis, finite element method
Zhang, Shuo 1

1 
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Zhang, Shuo. Mixed schemes for quad-curl equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 147-161. doi : 10.1051/m2an/2018005. http://archive.numdam.org/articles/10.1051/m2an/2018005/

[1] D.N Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1–155. | DOI | MR | Zbl

[2] D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229–246. | DOI | MR | Zbl

[3] D. Boffi, F. Brezzi and M. Fortin. Mixed Finite Element Methods and Applications, Vol. 44. Springer (2013). | DOI | MR | Zbl

[4] S.C. Brenner, J. Sun and L.-Y. Sung, Hodge decomposition methods for a quad-curl problem on planar domains. J. Sci. Comput. 73 (2017) 495–513. | DOI | MR | Zbl

[5] F. Cakoni and H. Haddar, A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Probl. Imaging 1 (2007) 443. | DOI | MR | Zbl

[6] F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media. Inverse Probl. 26 (2010) 074004. | DOI | MR | Zbl

[7] L. Chacón, A.N. Simakov and A. Zocco, Steady-state properties of driven magnetic reconnection in 2D electron magnetohydrodynamics. Phys. Rev. Lett. 99 (2007) 235001. | DOI

[8] L. Chen, Multigrid methods for saddle point systems using constrained smoothers. Comput. Math. Appl. 70 (2015) 2854–2866. | DOI | MR | Zbl

[9] L. Chen, X. Hu, M. Wang and J. Xu, A multigrid solver based on distributive smoother and residual overweighting for Oseen problems. Numer. Math. Theory Methods Appl. 8 (2015) 237–252. | DOI | MR | Zbl

[10] L. Chen, Y. Wu, L. Zhong and J. Zhou, Multigrid Preconditioners for Mixed Finite Element Methods of Vector Laplacian. Preprint (2016). | arXiv | MR

[11] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. (1978). | Zbl

[12] P.G. Ciarlet and P.-A. Raviart, A mixed finite element method for the biharmonic equation, in Proc. of Symposium on Mathematical Aspects of Finite Elements in PDE (1974) 125–145. | MR | Zbl

[13] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Vol. 5. Springer (1986). | DOI | MR | Zbl

[14] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237–339. | DOI | MR | Zbl

[15] R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45 (2007) 2483–2509. | DOI | MR | Zbl

[16] Q. Hong, J. Hu, S. Shu and J. Xu, A discontinuous Galerkin method for the fourth-order curl problem. J. Comput. Math. 30 (2012) 565–578. | DOI | MR | Zbl

[17] F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Comput. Methods Appl. Mech. Eng. 64 (1987) 509–521. | DOI | MR | Zbl

[18] F. Kikuchi, Mixed formulations for finite element analysis of magnetostatic and electrostatic problems. Jpn. J. Appl. Math. 6 (1989) 209–221. | DOI | MR | Zbl

[19] Z. Li and S. Zhang, A stable mixed element method for biharmonic equation with first-order function spaces. Comput. Methods Appl. Math. 17 (2017) 601–616. | DOI | MR | Zbl

[20] P. Monk Finite Element Methods for Maxwell’s Equations. Oxford University Press (2003). | DOI | MR | Zbl

[21] P. Monk and J. Sun, Finite element methods for Maxwell’s transmission eigenvalues. SIAM J. Sci. Comput. 34 (2012) B247–B264. | DOI | MR | Zbl

[22] J.-C. Nédélec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl

[23] M. Neilan, Discrete and conforming smooth de Rham complexes in three dimensions. Math. Comput. 84 (2015) 2059–2081. | DOI | MR | Zbl

[24] S. Nicaise, Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 411–430. | DOI | MR | Zbl

[25] S. Nicaise, Singularities of the Quad Curl Problem. HALpreprint (2016). | MR

[26] S.K. Park and X.-L. Gao. Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 59 (2008) 904–917. | DOI | MR | Zbl

[27] T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems. SIAM J. Matrix Anal. Appl. 13 (1992) 887–904. | DOI | MR | Zbl

[28] J. Sun, A mixed fem for the quad-curl eigenvalue problem. Numer. Math. 132 (2016) 185–200. | DOI | MR | Zbl

[29] J. Sun and L. Xu, Computation of Maxwell’s transmission eigenvalues and its applications in inverse medium problems. Inverse Probl. 29 (2013) 104013. | DOI | MR | Zbl

[30] X.-C. Tai and R. Winther, A discrete de Rham complex with enhanced smoothness. Calcolo 43 (2006) 287–306. | DOI | MR | Zbl

[31] M. Wang and J. Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations. Math. Comput. 76 (2007) 1–18. | DOI | MR | Zbl

[32] M. Wang, Z.-C. Shi and J. Xu, A new class of Zienkiewicz-type non-conforming element in any dimensions. Numer. Math. 106 (2007) 335–347. | DOI | MR | Zbl

[33] M. Wang, Z.-C. Shi and J. Xu, Some n-rectangle nonconforming elements for fourth order elliptic equations. J. Comput. Math. 25 (2007) 408–420. | MR | Zbl

[34] J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581–613. | DOI | MR | Zbl

[35] J. Xu, Fast poisson-based solvers for linear and nonlinear PDEs, in Proc. of the International Congress of Mathematics (2010) Vol. 4, 2886–2912. | MR | Zbl

[36] F. Yang, A.C.M. Chong, D.C.C. Lam and P. Tong, Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39 (2002) 2731–2743. | DOI | Zbl

[37] A. Ženíšek, Polynomial approximation on tetrahedrons in the finite element method. J. Approx. Theory 7 (1973) 334–351. | DOI | MR | Zbl

[38] S. Zhang, A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl. Numer. Math. 59 (2009) 219–233. | DOI | MR | Zbl

[39] S. Zhang, Regular decomposition and a framework of order reduced methods for fourth order problems. Numer. Math. 138 (2018) 241–271. | DOI | MR | Zbl

[40] S. Zhang, Y. Xi and X. Ji, A multi-level mixed element method for the eigenvalue problem of biharmonic equation. J. Sci. Comput. (2017). | MR

[41] B. Zheng, Q. Hu and J. Xu, A nonconforming finite element method for fourth order curl equations in . Math. Comput. 80 (2011) 1871–1886. | DOI | MR | Zbl

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