四阶定常 Bi-Wave 奇异摄动问题的协调Galerkin 有限元拟一致超收敛分析
Quasi-Uniform Superconvergence Analysis of Conforming GalerkinFinite Element Method for the Fourth Order Stationary Bi-Wave Singular Perturbation Problem
摘要: 在本文中，我们利用协调 Galerkin 有限元方法求解模拟高温超导体 d 波现象的四阶定常 Bi- wave 奇异摄动问题。首先，分析其线性问题的变分格式下解的有界性；其次，利用 Brouwer 不动点定理证明非线性 Bi-wave 问题逼近解的存在唯一性；进而，基千 Bogner-Fox-Schmit 单元的高精度性质，得到在能量模意义下不依赖于参数负次幕的拟一致超逼近和超收敛误差估计；最后，我们通过相应的数值算例验证理论分析的正确性。
Abstract: In this paper, the conforming Galerkin finite element method is presented to solve the fourth order stationary Bi-wave singular perturbation problem simulating high tem- perature superconductor d wave phenomenon. Firstly, the boundedness of the solution under the variational scheme of its linear problem is analyzed; Secondly, the existence and uniqueness of the approximate solution for the nonlinear Bi-wave problem are proved by using Brouwer fixed point theorem; Furthermore, based on the high ac- curacy property of Bogner-Fox-Schmit element, quasi-uniform superconvergence and superclose error estimates independent of the negative power of the parameter in the energy norm are obtained; Finally, the corresponding numerical examples are provided to verify the correctness of the theoretical analysis.
文章引用：吴颜眯, 石东洋. 四阶定常 Bi-Wave 奇异摄动问题的协调Galerkin 有限元拟一致超收敛分析[J]. 应用数学进展, 2022, 11(3): 1389-1399. https://doi.org/10.12677/AAM.2022.113151

参考文献

[1] Joynt, R. (1990) Upward Curvature of Hc2 in High-Tc Superconductors: Possible Evidence for s + d Pairing. Physical Review B, 41, 4271-4277.
https://doi.org/10.1103/PhysRevB.41.4271
[2] Ren, Y., Xu, J.H. and Ting, C.S. (1996) Ginzburg-Landau Equations for Mixed s+d Symmetry Superconductors. Physical Review B, 53, 2249-2252.
https://doi.org/10.1103/PhysRevB.53.2249
[3] Feder, D.L. and Kallin, C. (1997) Microscopic Derivation of the Ginzburg-Landau Equations for a d-Wave Superconductor. Physical Review B, 55, 559-574.
https://doi.org/10.1103/PhysRevB.55.559
[4] Dai, M.C. and Yang, T.J. (1999) The Anomalous Hall Effect for a Mixed s-Wave and d-Wave Symmetry Superconductor. Solid State Communications, 110, 425-430.
https://doi.org/10.1016/S0038-1098(99)00092-7
[5] Xu, J.H., Ren, Y. and Ting, C.S. (1996) Structures of Single Vortex and Vortex Lattice in a d-Wave Superconductor. Physical Review B, 53, 2991-2993.
https://doi.org/10.1103/PhysRevB.53.R2991
[6] Du, Q. (1999) Studies of Ginzburg-Landau Model for d-Wave Superconductors. SIAM Journal on Applied Mathematics, 59, 1225-1250.
https://doi.org/10.1137/S0036139997329902
[7] Feng, X.B. and Neilan, M. (2010) Finite Element Methods for a Bi-Wave Equation Modeling d-Wave Superconductors. Journal of Computational Mathematics, 28, 331-353.
https://doi.org/10.4208/jcm.2009.10-m1011
[8] Fushchych, W.I. and Roman, O.V. (1996) Symmetry Reduction and Some Exact Solutions of Nonlinear Bi-Wave Equations. Reports on Mathematical Physics, 37, 267-281.
https://doi.org/10.1016/0034-4877(96)89767-9
[9] Feng, X.B. and Neilan, M. (2010) Discontinuous Finite Element Methods for a Bi-Wave Equa- tion Modeling d-Wave Superconductors. Mathematics of Computation, 80, 1303-1333.
https://doi.org/10.1090/S0025-5718-2010-02436-6
[10] Shi, D.Y. and Wu, Y.M. (2018) Uniform Superconvergence Analysis of Ciarlet-Raviart Scheme for Bi-Wave Singular Perturbation Problem. Mathematical Methods in the Applied Sciences, 41, 7906-7914.
https://doi.org/10.1002/mma.5254
[11] Shi, D.Y. and Wu, Y.M. (2020) Uniformly Superconvergent Analysis of an Efficient Two- Grid Method for Nonlinear Bi-Wave Singular Perturbation Problem. Applied Mathematics and Computation, 367, Article ID: 124772.
https://doi.org/10.1016/j.amc.2019.124772
[12] Wu, Y.M. and Shi, D.Y. (2021) Quasi-Uniform Convergence Analysis of a Modified Penal- ty Finite Element Method for Nonlinear Singularly Perturbed Bi-Wave Problem. Numerical Methods for Partial Differential Equations, 37, 1766-1780.
https://doi.org/10.1002/num.22607
[13] Shi, D.Y. and Wu, Y.M. (2021) Quasi-Uniform Convergence Analysis of Rectangular Morley Element for the Singularly Perturbed Bi-Wave Equation. Applied Numerical Mathematics, 161, 169-177.
https://doi.org/10.1016/j.apnum.2020.11.002
[14] 林群, 严宁宁. 高效有限元构造与分析[M]. 保定: 河北大学出版社, 1996.
[15] Browder, F.E. and Finn, R. (1965) Existence and Uniqueness Theorems for Solutions of Non- linear Boundary Value Problems. Proceedings of Symposia in Applied Mathematics, 17, 24-49.
https://doi.org/10.1090/psapm/017/0197933
[16] Lin, J.F. and Luo, Q. (2004) Superconvergence for the Bogner-Fox-Schmit Element. Compu- tational Mathematics, 26, 47-50.
https://doi.org/10.3321/j.issn:0254-7791.2004.01.006

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