A quadratic C0 interior penalty method for the quad-curl problem
Abstract
In this paper we study the C0 interior penalty method for a quad-curl problem arising from magnetohydrodynamics model on bounded polygons or polyhedrons. We prove the well-posedness of the numerical scheme and then derive the optimal error estimates in a discrete energy norm. A post-processing procedure that can produce C1 approximations is also presented. The performance of the method is illustrated by numerical experiments.
Keyword : C0 interior penalty method, MHD, quad-curl problem, error analysis
How to Cite
Sun, Z., Gao, F., Wang, C., & Zhang, Y. (2020). A quadratic C0 interior penalty method for the quad-curl problem. Mathematical Modelling and Analysis, 25(2), 208-225. https://doi.org/10.3846/mma.2020.9796
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
D. Biskamp. Magnetic reconnection in plasmas. Number 3. Cambridge University Press, 2005.
S.C. Brenner. C0 interior penalty methods. In Frontiers in Numerical AnalysisDurham 2010, pp. 79–147. Springer, 2011. https://doi.org/10.1007/978-3-642-23914-4_2
S.C. Brenner, J. Cui, T. Gudi and L.-Y. Sung. Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes. Numerische Mathematik, 119(1):21–47, 2011. https://doi.org/10.1007/s00211-011-0379-y
S.C. Brenner and R. Scott. The mathematical theory of finite element methods, volume 15. Springer Science and Business Media, 2007.
S.C. Brenner, J. Sun and L.-Y. Sung. Hodge decomposition methods for a quadcurl problem on planar domains. Journal of Scientific Computing, 73(2):495–513, 2017. https://doi.org/10.1007/s10915-017-0449-0
S.C. Brenner and L.-Y. Sung. C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. Journal of Scientific Computing, 22(1-3):83–118, 2005. https://doi.org/10.1007/s10915-004-4135-7
S.C. Brenner and L.-Y. Sung. Multigrid algorithms for C0 interior penalty methods. SIAM Journal on Numerical Analysis, 44(1):199–223, 2006. https://doi.org/10.1137/040611835
S.C. Brenner, L.-Y. Sung, H. Zhang and Y. Zhang. A quadratic C0 interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates. SIAM Journal on Numerical Analysis, 50(6):3329–3350, 2012. https://doi.org/10.1137/110845926
F. Cakoni, D. Colton, P. Monk and J. Sun. The inverse electromagnetic scattering problem for anisotropic media. Inverse Problems, 26(7):074004, 2010. https://doi.org/10.1088/0266-5611/26/7/074004
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):443–456, 2007. https://doi.org/10.3934/ipi.2007.1.443
L. Chacón, A.N. Simakov and A. Zocco. Steady-state properties of driven magnetic reconnection in 2D electron magnetohydrodynamics. Physical Review Letters, 99(23):235001, 2007. https://doi.org/10.1103/PhysRevLett.99.235001
G. Chen, W. Qiu and L. Xu. Analysis of a mixed finite element method for the quad-curl problem. arXiv preprint arXiv:1811.06724, 2018.
Q. Hong, J. Hu, S. Shu and J. Xu. A discontinuous Galerkin method for the fourth-order curl problem. Journal of Computational Mathematics, 30(6):565– 578, 2012. https://doi.org/10.4208/jcm.1206-m3572
W. Ming and J. Xu. The Morley element for fourth order elliptic equations in any dimensions. Numerische Mathematik, 103(1):155–169, 2006. https://doi.org/10.1007/s00211-005-0662-x
P. Monk and J. Sun. Finite element methods for Maxwell’s transmission eigenvalues. SIAM Journal on Scientific Computing, 34(3):B247–B264, 2012. https://doi.org/10.1137/110839990
L.S.D. Morley. The triangular equilibrium element in the solution of plate bending problems. The Aeronautical Quarterly, 19(2):149–169, 1968. https://doi.org/10.1017/S0001925900004546
J.-C. Nédélec. Mixed finite elements in R3. Numerische Mathematik, 35(3):315– 341, 1980. https://doi.org/10.1007/BF01396415
J.-C. Nédélec. A new family of mixed finite elements in R3. Numerische Mathematik, 50(1):57–81, 1986. https://doi.org/10.1007/BF01389668
S. Nicaise. Singularities of the quad curl problem. Journal of Differential Equations, 264(8):5025–5069, 2018. https://doi.org/10.1016/j.jde.2017.12.032
J. Sun. A mixed FEM for the quad-curl eigenvalue problem. Numerische Mathematik, 132(1):185–200, 2016. https://doi.org/10.1007/s00211-015-0708-7
Z. Sun, J. Cui, F. Gao and C. Wang. Multigrid methods for a quad-curl problem based on C0 interior penalty method. Computers & Mathematics with Applications, 76(9):2192–2211, 2018. https://doi.org/10.1016/j.camwa.2018.07.048
Alexander Ženíšek. Tetrahedral finite Cm-elements. Acta Universitatis Carolinae. Mathematica et Physica, 15(1):189–193, 1974.
C. Wang, Z. Sun and J. Cui. A new error analysis of a mixed finite element method for the quad-curl problem. Applied Mathematics and Computation, 349:23–38, 2019. https://doi.org/10.1016/j.amc.2018.12.027
Q. Zhang, L. Wang and Z. Zhang. An H2(curl)-conforming finite element in 2D and its applications to the quad-curl problem. SIAM Journal on Scientific Computing, 41(3):A1527–A1547, 2019. https://doi.org/10.1137/18M1199988
S. Zhang. Mixed schemes for quad-curl equations. ESAIM: Mathematical Modelling and Numerical Analysis, 52(1):147–161, 2018. https://doi.org/10.1051/m2an/2018005
B. Zheng, Q. Hu and J. Xu. A nonconforming finite element method for fourth order curl equations in R3. Mathematics of Computation, 80(276):1871–1886, 2011. https://doi.org/10.1090/S0025-5718-2011-02480-4
S.C. Brenner. C0 interior penalty methods. In Frontiers in Numerical AnalysisDurham 2010, pp. 79–147. Springer, 2011. https://doi.org/10.1007/978-3-642-23914-4_2
S.C. Brenner, J. Cui, T. Gudi and L.-Y. Sung. Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes. Numerische Mathematik, 119(1):21–47, 2011. https://doi.org/10.1007/s00211-011-0379-y
S.C. Brenner and R. Scott. The mathematical theory of finite element methods, volume 15. Springer Science and Business Media, 2007.
S.C. Brenner, J. Sun and L.-Y. Sung. Hodge decomposition methods for a quadcurl problem on planar domains. Journal of Scientific Computing, 73(2):495–513, 2017. https://doi.org/10.1007/s10915-017-0449-0
S.C. Brenner and L.-Y. Sung. C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. Journal of Scientific Computing, 22(1-3):83–118, 2005. https://doi.org/10.1007/s10915-004-4135-7
S.C. Brenner and L.-Y. Sung. Multigrid algorithms for C0 interior penalty methods. SIAM Journal on Numerical Analysis, 44(1):199–223, 2006. https://doi.org/10.1137/040611835
S.C. Brenner, L.-Y. Sung, H. Zhang and Y. Zhang. A quadratic C0 interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates. SIAM Journal on Numerical Analysis, 50(6):3329–3350, 2012. https://doi.org/10.1137/110845926
F. Cakoni, D. Colton, P. Monk and J. Sun. The inverse electromagnetic scattering problem for anisotropic media. Inverse Problems, 26(7):074004, 2010. https://doi.org/10.1088/0266-5611/26/7/074004
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):443–456, 2007. https://doi.org/10.3934/ipi.2007.1.443
L. Chacón, A.N. Simakov and A. Zocco. Steady-state properties of driven magnetic reconnection in 2D electron magnetohydrodynamics. Physical Review Letters, 99(23):235001, 2007. https://doi.org/10.1103/PhysRevLett.99.235001
G. Chen, W. Qiu and L. Xu. Analysis of a mixed finite element method for the quad-curl problem. arXiv preprint arXiv:1811.06724, 2018.
Q. Hong, J. Hu, S. Shu and J. Xu. A discontinuous Galerkin method for the fourth-order curl problem. Journal of Computational Mathematics, 30(6):565– 578, 2012. https://doi.org/10.4208/jcm.1206-m3572
W. Ming and J. Xu. The Morley element for fourth order elliptic equations in any dimensions. Numerische Mathematik, 103(1):155–169, 2006. https://doi.org/10.1007/s00211-005-0662-x
P. Monk and J. Sun. Finite element methods for Maxwell’s transmission eigenvalues. SIAM Journal on Scientific Computing, 34(3):B247–B264, 2012. https://doi.org/10.1137/110839990
L.S.D. Morley. The triangular equilibrium element in the solution of plate bending problems. The Aeronautical Quarterly, 19(2):149–169, 1968. https://doi.org/10.1017/S0001925900004546
J.-C. Nédélec. Mixed finite elements in R3. Numerische Mathematik, 35(3):315– 341, 1980. https://doi.org/10.1007/BF01396415
J.-C. Nédélec. A new family of mixed finite elements in R3. Numerische Mathematik, 50(1):57–81, 1986. https://doi.org/10.1007/BF01389668
S. Nicaise. Singularities of the quad curl problem. Journal of Differential Equations, 264(8):5025–5069, 2018. https://doi.org/10.1016/j.jde.2017.12.032
J. Sun. A mixed FEM for the quad-curl eigenvalue problem. Numerische Mathematik, 132(1):185–200, 2016. https://doi.org/10.1007/s00211-015-0708-7
Z. Sun, J. Cui, F. Gao and C. Wang. Multigrid methods for a quad-curl problem based on C0 interior penalty method. Computers & Mathematics with Applications, 76(9):2192–2211, 2018. https://doi.org/10.1016/j.camwa.2018.07.048
Alexander Ženíšek. Tetrahedral finite Cm-elements. Acta Universitatis Carolinae. Mathematica et Physica, 15(1):189–193, 1974.
C. Wang, Z. Sun and J. Cui. A new error analysis of a mixed finite element method for the quad-curl problem. Applied Mathematics and Computation, 349:23–38, 2019. https://doi.org/10.1016/j.amc.2018.12.027
Q. Zhang, L. Wang and Z. Zhang. An H2(curl)-conforming finite element in 2D and its applications to the quad-curl problem. SIAM Journal on Scientific Computing, 41(3):A1527–A1547, 2019. https://doi.org/10.1137/18M1199988
S. Zhang. Mixed schemes for quad-curl equations. ESAIM: Mathematical Modelling and Numerical Analysis, 52(1):147–161, 2018. https://doi.org/10.1051/m2an/2018005
B. Zheng, Q. Hu and J. Xu. A nonconforming finite element method for fourth order curl equations in R3. Mathematics of Computation, 80(276):1871–1886, 2011. https://doi.org/10.1090/S0025-5718-2011-02480-4