Abstract
In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg–Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform \(L^{3+\delta }\) regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a \(\ell ^2(W^{1,3+\delta })\) estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge.
Similar content being viewed by others
Notes
Since (1.10) implies \(\partial _t\mathbf{A}\cdot \mathbf{n}=0\), (1.8) and (1.10) imply \(\mathrm{Re}\big [\overline{\psi }\big (\frac{i}{\kappa } \nabla + \mathbf {A}\big ) \psi \big ]\cdot \mathbf{n}=0\) and (1.9) implies \([\nabla \times (\nabla \times \mathbf{A}-\mathbf{H})]\cdot \mathbf{n}=0\) (if a vector field \(\mathbf{u}\) satisfies \(\mathbf{n} \times \mathbf{u} = 0\) on \(\partial \Omega \), then \((\nabla \times \mathbf{u}) \cdot \mathbf{n}= 0\) on \(\partial \Omega \)), it follows from (1.7) that \(\nabla \phi \cdot \mathbf{n}=-\nabla (\nabla \cdot \mathbf{A})\cdot \mathbf{n}=0\) on each smooth piece of \(\partial \Omega \). Hence, (1.8)–(1.10) imply (1.5).
The monotonicity makes use of the fact that \((|{\mathscr {S}}_h|^{2}{\mathscr {S}}_h-|\widetilde{\mathscr {S}}_h|^{2}\widetilde{\mathscr {S}}_h, {\mathscr {S}}_h-\widetilde{\mathscr {S}}_h)\ge 0\) for all \({\mathscr {S}}_h,\widetilde{\mathscr {S}}_h \in {\mathbb S}_{h}^r\).
By identifying the vector fields with the 2-forms, in terms of the notation of [3, decomposition (2.18)], we have \(\mathbf{C}(\Omega )\cong {\mathfrak Z}^{*2}\), \(\mathbf{C}(\Omega )^\perp \cong \mathring{\mathfrak B}^2\), \(\mathbf{G}(\Omega )\cong {\mathfrak B}^{*2}\) and \(\mathbf{X}(\Omega )\cong \mathring{\mathfrak H}^2\).
By identifying the vector fields with the 2-forms, in terms of the notation of [3, definition (2.12)], we have \(\widetilde{\mathbf{X}}(\Omega )={\mathfrak H}^2\).
By identifying the vector fields with the 2-forms, in terms of the notation of [3, definition (2.12)], we have \(\widetilde{\mathbf{X}}(\Omega )\cong {\mathfrak H}^2\) and \(\widetilde{\mathbf{Y}}(\Omega )\cong H\Lambda ^2(\Omega )\cap \mathring{H}^*\Lambda ^2(\Omega )\cap {\mathfrak H}^{2\perp }\). Then, by using [3, Theorem 2.2 on page 23] and the Lax–Milgram lemma, one can show that the problem (3.13)–(3.15) has a unique weak solution in \(\widetilde{\mathbf{Y}}(\Omega )\).
If \(\mathbf{v}\in \mathbf{H}(\mathrm{div})\) then \(\mathbf{v}\cdot \mathbf{n}\) is well defined on \(\partial \Omega \). In this case, the divergence-free part \(\nabla \times \mathbf{u}\) satisfies \((\nabla \times \mathbf{u})\cdot \mathbf{n}=0\) on \(\partial \Omega \), due to the boundary conditions implicitly imposed in the weak formulations (3.16) and (3.17).
See footnote 6 on this boundary condition.
This is a immediate consequence of Lemma 3.3 and the following decomposition proved in [8]:
$$\begin{aligned} \mathbf{H}(\mathrm{curl,div})= \mathbf{H}^1+ \{\nabla \varphi :\varphi \in H^1,\,\, \Delta \varphi \in L^2,\,\,\nabla \varphi \cdot \mathbf{n} =0\,\,\text{ on }\,\,\partial \Omega \}. \end{aligned}$$By identifying the vector fields with the 1-forms, in terms of the notation of [3, Theorem 5.11 on page 74], we have \(\mathbf{C}(\Omega )\cong {\mathfrak Z}^{1}\) and \(\mathbf{C}(\Omega )^\perp \cong {\mathfrak Z}^{1\perp }\).
References
Adams, R.A.: Sobolev spaces. Academic Press, New York (1975)
Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth Domains. Math. Meth. Appl. Sci. 21, 823–864 (1998)
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica, pp. 1–155 (2006)
Alstrøm, T.S., Sørensen, M.P., Pedersen, N.F., Madsen, F.: Magnetic flux lines in complex geometry type-II superconductors studied by the time dependent Ginzburg-Landau equation. Acta Appl. Math. 115, 63–74 (2011)
Ashyralyev, A., Piskarev, S., Weis, L.: On well-posedness of difference schemes for abstract parabolic equations in \(L_p([0, T];E)\) spaces. Numer. Funct. Anal. Optim. 23, 669–693 (2002)
Baelus, B.J., Kadowaki, K., Peeters, F.M.: Influence of surface defects on vortex penetration and expulsion in mesoscopic superconductors. Phys. Rev. B 71, 024514 (2005)
Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction, Springer-Verlag Berlin Heidelberg 1976, Printed in Germany
Birman, M., Solomyak, M.: \(L^2\)-theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42, 75–96 (1987)
Chen, Z.: Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity. Numer. Math. 76, 323–353 (1997)
Chen, Z., Dai, S.: Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J. Numer. Anal. 38, 1961–1985 (2001)
Chen, Z., Hoffmann, K.H., Liang, J.: On a non-stationary Ginzburg-Landau superconductivity model. Math. Methods Appl. Sci. 16, 855–875 (1993)
Christiansen, S.H., Scheid, C.: Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation. ESAIM: M2AN 45, 739–760 (2011)
Christiansen, S. H., Munthe-Kaas, H. Z., Owren, B.: Topics in structure-preserving discretization. Acta Numerica, pp. 1-119 (2011)
Costabel, M.: A remark on the regularity of solutions of Maxwells equations on Lipschitz domains. Math. Methods Appl. Sci. 12, 365–368 (1990)
Dauge, M.: Elliptic Boundary Value Problems in Corner Domains. Springer-Verlag, Berlin Heidelberg (1988)
Dauge, M.: Neumann and mixed problems on curvilinear polyhedra. Integr. Equat. 0per Th. 15, 227–261 (1992)
Dauge, M.: Regularity and singularities in polyhedral domains. The case of Laplace and Maxwell equations. Slides d’un mini-cours de 3 heures, Karlsruhe, 7 avril 2008. https://perso.univ-rennes1.fr/monique.dauge/publis/Talk_Karlsruhe08.html
De Gennes, P.G.: Superconductivity of Metal and Alloys. Advanced Books Classics, Westview Press (1999)
Du, Q.: Discrete gauge invariant approximations of a time dependent ginzburg-landau model of superconductivity. Math. Comp. 67, 965–986 (1998)
Du, Q.: Numerical approximations of the Ginzburg-Landau models for superconductivity. J. Math. Phys. 46, 095109 (2005)
Du, Q., Ju, L.: Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations. Math. Comp. 74, 1257–1280 (2005)
Frahm, H., Ullah, S., Dorsey, A.: Flux dynamics and the growth of the superconducting phase. Phys. Rev. Letters 66, 3067–3072 (1991)
Gao, H., Li, B., Sun, W.: Optimal error estimates of linearized Crank-Nicolson-Galerkin FEMs for the time-dependent Ginzburg-Landau equations. SIAM J. Numer. Anal. 52, 1183–1202 (2014)
Gao, H., Sun, W.: An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg-Landau equations of superconductivity. J. Comput. Physics 294, 329–345 (2015)
Gao, H., Sun, W.: Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg–Landau equations of superconductivity. Preprint. arXiv:1508.05601
Ginzburg, V., Landau, L.: Theory of Superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950)
Gropp, W.D., Kaper, H.G., Leaf, G.K., Levine, D.M., Palumbo, M., Vinokur, V.M.: Numerical simulation of vortex dynamics in type-II superconductors. J. Comput. Phys. 123, 254–266 (1996)
Gor’kov, L.P., Eliashberg, G.M.: Generalization of the Ginzburg–Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Sov. Phys. JETP 27, 328–334 (1968)
Gunter, D., Kaper, H., Leaf, G.: Implicit integration of the time-dependent Ginzburg–Landau equations of superconductivity. SIAM J. Sci. Comput. 23, 1943–1958 (2002)
Kovács, B., Li, B., Lubich, Ch.: \(A\)-stable time discretizations preserve maximal parabolic regularity. SIAM J. Numer. Anal. 54, 3600–3624 (2016)
Kozono, H., Yanagisawa, T.: \(L^r\)-variational inequality for vector fields and the Helmholtz–Weyl decomposition in bounded domains. Indiana Univ. Math. J. 58, 1853–1920 (2009)
Li, B., Zhang, Z.: A new approach for numerical simulation of the time-dependent Ginzburg–Landau equations. J. Comput. Phys. 303, 238–250 (2015)
Li, B., Zhang, Z.: Mathematical and numerical analysis of time-dependent Ginzburg–Landau equations in nonconvex polygons based on Hodge decomposition. Math. Comp. 86, 1579–1608 (2017)
Li, B., Yang, C.: Global well-posedness of the time-dependent Ginzburg–Landau superconductivity model in curved polyhedra. J. Math. Anal. Appl. 451, 102–116 (2017)
Li, B.: Maximum-norm stability and maximal \(L^p\) regularity of FEMs for parabolic equations with Lipschitz continuous coefficients. Numer. Math. 131, 489–516 (2015)
Li, B.: Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp. (2017). doi:10.1090/mcom/3316
Li, B., Sun, W.: Maximal \(L^p\) analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra. Math. Comp. 86, 1071–1102 (2017)
Liu, F., Mondello, M., Goldenfeld, N.: Kinetics of the superconducting transition. Phys. Rev. Lett. 66, 3071–3074 (1991)
Mu, M.: A linearized Crank–Nicolson–Galerkin method for the Ginzburg–Landau model. SIAM J. Sci. Comput. 18, 1028–1039 (1997)
Mu, M., Huang, Y.: An alternating Crank-Nicolson method for decoupling the Ginzburg–Landau equations. SIAM J. Numer. Anal. 35, 1740–1761 (1998)
Nédélec, J.C.: Mixed finite element in \({\mathbb{R}}^3\). Numer. Math. 35, 315–341 (1980)
Nédélec, J.C.: A new family of mixed finite elements in \({\mathbb{R}}^3\). Numer. Math. 50, 57–81 (1986)
Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains. J. Lond. Math. Soc. 60, 237–257 (1999)
Showalter, R.E.: A decoupled mixed FEM for Ginzburg-Landau equations. Math. Surv. Monogr. 49, (1997)
Tinkham, M.: Introduction to Superconductivity, 2nd edn. McGraw-Hill, New York (1994)
Richardson, W., Pardhanani, A., Carey, G., Ardelea, A.: Numerical effects in the simulation of Ginzburg–Landau models for superconductivity. Int. J. Numer. Eng. 59, 1251–1272 (2004)
Vodolazov, D.Y., Maksimov, I.L., Brandt, E.H.: Vortex entry conditions in type-II superconductors. Effect of surface defects. Physica C 384, 211–226 (2003)
Winiecki, T., Adams, C.: A fast semi-implicit finite difference method for the TDGL equation. J. Comput. Phys. 179, 127–139 (2002)
Weis, L.: A new approach to maximal \(L^p\)-regularity. In: Evolution Equ. and Appl. Physical Life Sci., Lecture Notes in Pure and Applied Mathematics 215, Marcel Dekker, New York, pp. 195–214 (2001)
Weck, N.: Maxwells boundary value problems on Riemannian manifolds with nonsmooth boundaries. J. Math. Anal. Appl. 46, 410–437 (1974)
Yang, C.: A linearized Crank-Nicolson-Galerkin FEM for the time-dependent Ginzburg–Landau equations under the temporal gauge. Numer. Methods Partial Differ. Equ. 30, 1279–1290 (2014)
Acknowledgements
I would like to express my gratitude to Prof. Christian Lubich for the helpful discussions on the time discretization, and thank Prof. Weiwei Sun for the email communications on this topic. I also would like to thank Prof. Qiang Du for the communications in CSRC, Beijing, on the time-independency of the external magnetic field and the incompatibility of the initial data with the boundary conditions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by a Grant from the Germany/Hong Kong Joint Research Scheme sponsored by the Research Grants Council of Hong Kong and the German Academic Exchange Service of Germany (Ref. No. G- PolyU502/16). The research stay of the author at Universität Tübingen was partially supported by the Alexander von Humboldt Foundation.
Appendix: Well-posedness of the PDE problem (1.6)–(1.11)
Appendix: Well-posedness of the PDE problem (1.6)–(1.11)
Theorem A.1
There exists a unique weak solution of (1.6)–(1.11) with the following regularity:
Proof
Global well-posedness of time-dependent Ginzburg–Landau equations in curved polyhedra was proved in [34]. The convergence of numerical solutions proved in this paper yields an alternative proof.
In fact, from (3.95) and (3.98) we see that there exists a weak solution \((\Psi ,{\varvec{\Lambda }})\) of (1.6)–(1.11) with the regularity above. It remains to prove the uniqueness of the weak solution.
Suppose that there are two weak solutions \((\psi ,\mathbf{A})\) and \((\Psi ,{\varvec{\Lambda }})\) for the system (1.6)–(1.11). Then we define \(e=\psi -\Psi \) and \(\mathbf{E}=\mathbf{A}-{\varvec{\Lambda }}\) and consider the difference equations
and
which hold for any \(\varphi \in L^2(0,T;{\mathcal H}^1)\) and \(\mathbf{a}\in L^2(0,T;\mathbf{H}(\mathrm{curl},\mathrm{div}))\). Choosing \(\varphi (x,t)=e(x,t)1_{(0,t')}(t)\) in (A.1) and considering the real part, we obtain
where \(\epsilon \) can be arbitrarily small. By choosing \(\mathbf{a}(x,t)=\mathbf{E}(x,t)1_{(0,t')}(t)\) in (A.2), we get
where \(\epsilon \) can be arbitrarily small. By choosing \(\epsilon <\frac{1}{4} \min (1, \kappa ^{-2} )\) and summing up the two inequalities above, we have
which implies
via Gronwall’s inequality. Uniqueness of the weak solution is proved. \(\square \)
Rights and permissions
About this article
Cite this article
Li, B. Convergence of a decoupled mixed FEM for the dynamic Ginzburg–Landau equations in nonsmooth domains with incompatible initial data. Calcolo 54, 1441–1480 (2017). https://doi.org/10.1007/s10092-017-0237-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10092-017-0237-0
Keywords
- Ginzburg–Landau
- Superconductivity
- Finite element method
- Convergence
- Incompatible data
- Nonconvex polyhedra