### ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS

Weifeng Zhang1, Shuo Zhang2

1. 1. Department of Engineering Mechanics and Department of Applied Mathematics, Kunming University of Science and Technology, Kunming 650500, China;
2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
• Received:2018-07-14 Revised:2018-12-02 Online:2020-07-15 Published:2020-07-15
• Supported by:

The work of the first author was supported in part by Yunnan Provincial Science and Technology Department Research Award: Interdisciplinary Research in Computational Mathematics and Mechanics with Applications in Energy Engineering (Grant No. 2009CI128) and Yunnan Provincial Science and Technology Project (Grant No. 2014RA071). The work of the second author was supported partially by the National Natural Science Foundation of China with Grant Nos 11471026 and 11871465 and National Centre for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.

Weifeng Zhang, Shuo Zhang. ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS[J]. Journal of Computational Mathematics, 2020, 38(4): 565-579.

Quad-curl equations with Navier type boundary conditions are studied in this paper. Stable order reduced formulations equivalent to the model problems are presented, and finite element discretizations are designed. Optimal convergence rates are proved.

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 [1] C. Amrouche, M. Dauge, V. Girault and C. Bernardi, Vector potentials in three-dimensional non-smooth domains, Mathematical Methods in the Applied Sciences, 21(1998), 823-864.[2] D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the reissner-mindlin plate, SIAM Journal on Numerical Analysis, 26:6(1989), 1276-1290.[3] D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, 15(2006), 1-155.[4] M.S. Birman and M.Z. Solomyak, L2-theory of the maxwell operator in arbitrary domains, Russian Mathematical Surveys, 42:6(1987), 75-96.[5] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer, Berlin, 2013.[6] A. Bossavit, Whitney forms:a class of finite elements for three-dimensional computations in electromagnetism, IEE Proceedings A, 135:8(1988), 493-500.[7] S. C. Brenner, J. Cui, F. Li and L. Y. Sung, A nonconforming finite element method for a twodimensional curl-curl and grad-div problem, Numerische Mathematik, 109(2008), 509-533.[8] S.C. Brenner, J.G. Sun and L.Y. Sung, Hodge decomposition methods for a quad-curl problem on planar domains, Journal of Scientific Computing, 73(2017), 495-513.[9] F. Cakoni, D. Colton, P. Monk and J.G. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems, 26:7(2010), 074004.[10] F. Cakoni and H. Haddar, A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media, Inverse Problems and Imaging, 1:3(2007), 443-456.[11] L. Chacón, A.N. Simakov and A. Zocco, Steady-state properties of driven magnetic reconnection in 2D electron magnetohydrodynamics. Physical Review Letters, 99:23(2007), 235001.[12] P.G. Ciarlet and P.A. Raviart, A mixed finite element method for the biharmonic equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, 24:4(1974), 125-145.[13] A.B. Dhia, C. Hazard and S. Lohrengel, A singular field method for the solution of Maxwell's equations in polyhedral domains, SIAM Journal on Applied Mathematics, 59:6(1999), 2028-2044.[14] C.S. Feng and S. Zhang, Optimal solver for Morley element discretization of biharmonic equation on shape-regular grids, Journal of Computational Mathematics, 34:2(2016), 159-173.[15] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms, Springer, Berlin, 1986.[16] L. Grasedyck, L. Wang and J.C. Xu, A nearly optimal multigrid method for general unstructured grids, Numerische Mathematik, 134:3(2016), 637-666.[17] R.F. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11(2002), 237-339.[18] R.F. Hiptmair and J.C. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM Journal on Numerical Analysis, 45:6(2007), 2483-2509.[19] Q.G. Hong, J. Hu, S. Shu and J.C. Xu, A discontinuous Galerkin method for the fourth-order curl problem, Journal of Computational Mathematics, 30:6(2012), 565-578.[20] F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Computer Methods in Applied Mechanics and Engineering, 64:1(1987), 509-521.[21] S. Nicaise, Singularities of the quad-curl problem, Journal of Differential Equations, 264:8(2018), 5025-5069.[22] S.K. Park and X.L. Gao, Variational formulation of a modified couple stress theory and its application to a simple shear problem, Zeitschrift für angewandte Mathematik und Physik, 59:5(2008), 904-917.[23] J.E. Pasciak and J. Zhao, Overlapping schwarz methods in H(curl) on polyhedral domains, Journal of Numerical Mathematics, 10:3(2002), 221-234.[24] T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM Journal on Matrix Analysis and Applications, 13:3(1992), 887-904.[25] J.G. Sun, A mixed FEM for the quad-curl eigenvalue problem, Numerische Mathematik, 132:1(2016), 185-200.[26] J.C. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56:3(1996), 215-235.[27] J.C. Xu, Fast Poisson-based solvers for linear and nonlinear PDEs, Proceedings of the International Congress of Mathematics, 4(2010), 2886-2912.[28] F.Yang, A.C.M. Chong, D.C.C. Lam and P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39:10(2002), 2731-2743.[29] S. Zhang, Mixed schemes for quad-curl equations, ESAIM:Mathematical Modelling and Numerical Analysis, 52:1(2018), 147-161.[30] S. Zhang, Regular decomposition and a framework of order reduced methods for fourth order problems, Numerische Mathematik, 138:1(2018), 241-271.[31] S. Zhang and J.C. Xu, Optimal solvers for fourth-order PDEs discretized on unstructured grids, SIAM Journal on Numerical Analysis, 52(2014), 282-307.[32] B. Zheng, Q.Y. Hu and J.C. Xu, A nonconforming finite element method for fourth order curl equations in R3, Mathematics of Computation, 80:276(2011), 1871-1886.
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