Discontinuous finite element methods for a bi-wave equation modeling 𝑑-wave superconductors

Feng, Xiaobing; Neilan, Michael

doi:10.1090/S0025-5718-2010-02436-6

Discontinuous finite element methods for a bi-wave equation modeling $d$-wave superconductors
HTML articles powered by AMS MathViewer

by Xiaobing Feng and Michael Neilan PDF

Math. Comp. 80 (2011), 1303-1333 Request permission

Abstract:

This paper concerns discontinuous finite element approximations of a fourth-order bi-wave equation arising as a simplified Ginzburg-Landau-type model for $d$-wave superconductors in the absence of an applied magnetic field. In the first half of the paper, we construct a variant of the Morley finite element method, which was originally developed for approximating the fourth-order biharmonic equation, for the bi-wave equation. It is proved that, unlike the biharmonic equation, it is necessary to impose a mesh constraint and to include certain penalty terms in the method to guarantee convergence. Nearly optimal order (off by a factor $|\mathrm {ln} h|$) error estimates in the energy norm and in the $H^1$-norm are established for the proposed Morley-type nonconforming method. In the second half of the paper, we develop a symmetric interior penalty discontinuous Galerkin method for the bi-wave equation using general meshes and prove optimal order error estimates in the energy norm. Finally, numerical experiments are provided to gauge the efficiency of the proposed methods and to validate the theoretical error bounds.

References

Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882, DOI 10.1137/0719052
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, DOI 10.1090/S0025-5718-1977-0431742-5
D. G. Bourgin and R. Duffin, The Dirichlet problem for the virbrating string equation, Bull. Amer. Math. Soc. 45 (1939), 851–858. MR 729, DOI 10.1090/S0002-9904-1939-07103-6
Susanne C. Brenner, Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions, Electron. Trans. Numer. Anal. 18 (2004), 42–48. MR 2083293
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
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
Qiang Du, Studies of a Ginzburg-Landau model for $d$-wave superconductors, SIAM J. Appl. Math. 59 (1999), no. 4, 1225–1250. MR 1686712, DOI 10.1137/S0036139997329902
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
Jim Douglas Jr. and Todd Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 207–216. MR 0440955
D. L. Feder and C. Kallin, Microscopic derivation of the Ginzburg-Landau equations for a d-wave superconductor, Phys. Rev. B, 55:559–574, 1997.
Xiaobing Feng and Ohannes A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition, Math. Comp. 76 (2007), no. 259, 1093–1117. MR 2299767, DOI 10.1090/S0025-5718-07-01985-0
X. Feng and M. Neilan, Finite element methods for a bi-wave equation modeling $d$-wave superconductors, J. Comput. Math., 28(3):331–353, 2010.
V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh. Eksper. Teoret. Fiz. 20:1064–1082, 1950 (in Russian), in: L.D. Landau, and I.D. ter Haar (Eds.), Man of Physics, Pergamon, Oxford, 1965, pp. 138–167 (in English).
Ohannes A. Karakashian and Frederic Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. MR 2034620, DOI 10.1137/S0036142902405217
A. Lasis and E. Süli, Poincaré-type inequalities for broken Sobolev spaces, Oxford University Computing Laboratory, Numerical Analysis Technical Report, 03(10), 2003.
L. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart. 19, 149-169, 1968.
Y. Ren, J.-H. Xu, and C. S. Ting, Ginzburg-Landau equations for mixed $s+d$ symmetry superconductors, Phys. Rev. B, 53:2249–2252, 1996.
Béatrice Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations, Frontiers in Applied Mathematics, vol. 35, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Theory and implementation. MR 2431403, DOI 10.1137/1.9780898717440
Zhong Ci Shi, Error estimates for the Morley element, Math. Numer. Sinica 12 (1990), no. 2, 113–118 (Chinese, with English summary); English transl., Chinese J. Numer. Math. Appl. 12 (1990), no. 3, 102–108. MR 1070298
M. Tinkham, Introduction to Superconductivity, Second Edition, Dover Publications, 2004.
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
Mary Fanett Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), no. 1, 152–161. MR 471383, DOI 10.1137/0715010
J.-H. Xu, Y. Ren, and C. S. Ting, Ginzburg-Landau equations for a d-wave superconductor with nonmagnetic impurities, Phys. Rev. B, 53(18):12481-12495, 1996.