Nonlinear diffusion in type-II superconductors
In this paper we study a nonlinear evolution equation ∂ t ( σ ( | E | ) E ) + ∇ × ∇ × E = F in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field E in a conductive medium under the influence of a force F . It is an approximation...
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| Published in | Journal of computational and applied mathematics Vol. 215; no. 2; pp. 568 - 576 |
|---|---|
| Main Author | |
| Format | Journal Article Conference Proceeding |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
01.06.2008
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0377-0427 1879-1778 |
| DOI | 10.1016/j.cam.2006.03.055 |
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| Abstract | In this paper we study a nonlinear evolution equation
∂
t
(
σ
(
|
E
|
)
E
)
+
∇
×
∇
×
E
=
F
in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field
E
in a conductive medium under the influence of a force
F
. It is an approximation of Bean's critical-state model for type-II superconductors. We design a nonlinear numerical scheme for the time discretization. We prove the convergence of the proposed method. The proof is based on a generalization of
div–
curl lemma for transient problems. We also derive some error estimates for the approximate solution. |
|---|---|
| AbstractList | In this paper we study a nonlinear evolution equation
∂
t
(
σ
(
|
E
|
)
E
)
+
∇
×
∇
×
E
=
F
in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field
E
in a conductive medium under the influence of a force
F
. It is an approximation of Bean's critical-state model for type-II superconductors. We design a nonlinear numerical scheme for the time discretization. We prove the convergence of the proposed method. The proof is based on a generalization of
div–
curl lemma for transient problems. We also derive some error estimates for the approximate solution. |
| Author | Slodička, Marián |
| Author_xml | – sequence: 1 givenname: Marian surname: SLODICKA fullname: SLODICKA, Marian organization: Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Gent, Belgium |
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| Cites_doi | 10.1103/RevModPhys.36.39 10.1103/PhysRev.181.682 10.1007/BF02575899 10.1007/978-3-642-96379-7 10.1006/jcph.1996.0243 10.1103/RevModPhys.36.31 10.1007/BF02575898 10.3934/dcds.2002.8.781 10.1137/S0036144599371913 10.1016/S0362-546X(99)00147-9 10.1017/S0956792500002333 |
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| Keywords | 65M15 Superconductors Error estimates Compensated compactness 82D55 Time discretization 65M12 Approximation Initial condition Error estimation Evolution equation Numerical method Boundary condition Discretization method Convergence Non linear equation Numerical analysis Electric field Applied mathematics 65M12;65M15;82D55 |
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| References | Nochetto (bib14) 1985; 22 L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. Chapman (bib5) 2000; 42 J. Kačur, Method of Rothe in Evolution Equations, Teubner Texte zur Mathematik, vol. 80, Teubner, Leipzig, 1985. Nochetto (bib15) 1985; 22 Barrett, Prigozhin (bib2) 2000; 42A Fabrizio, Morro (bib8) 2003 F. London, Superfluids, Macroscopic Theory of Superfluid Helium, vol. II, Wiley, New York, 1954. Beasley, Labusch, Webb (bib4) 1969; 181 Prigozhin (bib17) 1996; 7 Bean (bib3) 1964; 36 Yin, Li, Zou (bib19) 2002; 8 Vajnberg (bib18) 1973 Anderson, Kim (bib1) 1964; 36 Prigozhin (bib16) 1996; 129 Mayergoyz (bib13) 1998 D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer, Berlin, 1977. F. London, Superfluids. Macroscopic Theory of Superconductivity, vol. I, Wiley, Chapman & Hall, New York, London, 1950. Dafermos (bib6) 2000 Yin (10.1016/j.cam.2006.03.055_bib19) 2002; 8 Barrett (10.1016/j.cam.2006.03.055_bib2) 2000; 42A Dafermos (10.1016/j.cam.2006.03.055_bib6) 2000 Mayergoyz (10.1016/j.cam.2006.03.055_bib13) 1998 Prigozhin (10.1016/j.cam.2006.03.055_bib17) 1996; 7 Beasley (10.1016/j.cam.2006.03.055_bib4) 1969; 181 Chapman (10.1016/j.cam.2006.03.055_bib5) 2000; 42 Bean (10.1016/j.cam.2006.03.055_bib3) 1964; 36 Anderson (10.1016/j.cam.2006.03.055_bib1) 1964; 36 10.1016/j.cam.2006.03.055_bib9 10.1016/j.cam.2006.03.055_bib12 10.1016/j.cam.2006.03.055_bib11 Fabrizio (10.1016/j.cam.2006.03.055_bib8) 2003 10.1016/j.cam.2006.03.055_bib7 10.1016/j.cam.2006.03.055_bib10 Nochetto (10.1016/j.cam.2006.03.055_bib14) 1985; 22 Prigozhin (10.1016/j.cam.2006.03.055_bib16) 1996; 129 Nochetto (10.1016/j.cam.2006.03.055_bib15) 1985; 22 Vajnberg (10.1016/j.cam.2006.03.055_bib18) 1973 |
| References_xml | – volume: 181 start-page: 682 year: 1969 end-page: 700 ident: bib4 article-title: Flux creep in type-II superconductors publication-title: Phys. Rev. – volume: 22 start-page: 501 year: 1985 end-page: 534 ident: bib15 article-title: Error estimates for two-phase Stefan problems in several space variables, II: nonlinear flux conditions publication-title: Calcolo – volume: 36 start-page: 39 year: 1964 end-page: 43 ident: bib1 article-title: Hard superconductivity: theory of the motion of Abrikosov flux lines publication-title: Rev. Mod. Phys. – volume: 36 start-page: 31 year: 1964 end-page: 39 ident: bib3 article-title: Magnetization of high-field superconductors publication-title: Rev. Mod. Phys. – volume: 42 start-page: 555 year: 2000 end-page: 598 ident: bib5 article-title: A hierarchy of models for type-II superconductors publication-title: SIAM Rev. – reference: L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. – volume: 8 start-page: 781 year: 2002 end-page: 794 ident: bib19 article-title: A degenerate evolution system modeling Bean's critical-state type-II superconductors publication-title: Discrete Contin. Dyn. Syst. – year: 2003 ident: bib8 article-title: Electromagnetism of Continuous Media, Mathematical Modelling and Applications – reference: F. London, Superfluids, Macroscopic Theory of Superfluid Helium, vol. II, Wiley, New York, 1954. – reference: D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer, Berlin, 1977. – year: 1998 ident: bib13 article-title: Nonlinear Diffusion of Electromagnetic Fields with Applications to Eddy Currents and Superconductivity – volume: 22 start-page: 457 year: 1985 end-page: 499 ident: bib14 article-title: Error estimates for two-phase Stefan problems in several space variables, I: linear boundary conditions publication-title: Calcolo – volume: 7 start-page: 237 year: 1996 end-page: 247 ident: bib17 article-title: On the bean critical-state model in superconductivity publication-title: European J. Appl. Math. – reference: F. London, Superfluids. Macroscopic Theory of Superconductivity, vol. I, Wiley, Chapman & Hall, New York, London, 1950. – year: 1973 ident: bib18 article-title: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations – reference: J. Kačur, Method of Rothe in Evolution Equations, Teubner Texte zur Mathematik, vol. 80, Teubner, Leipzig, 1985. – volume: 129 start-page: 190 year: 1996 end-page: 200 ident: bib16 article-title: The bean model in superconductivity: variational formulation and numerical solution publication-title: J. Comput. Phys. – volume: 42A start-page: 977 year: 2000 end-page: 993 ident: bib2 article-title: Bean's critical-state model as the publication-title: Nonlinear Anal. Theory Methods Appl. – year: 2000 ident: bib6 article-title: Hyperbolic Conservation Laws in Continuum Physics – volume: 36 start-page: 39 year: 1964 ident: 10.1016/j.cam.2006.03.055_bib1 article-title: Hard superconductivity: theory of the motion of Abrikosov flux lines publication-title: Rev. Mod. Phys. doi: 10.1103/RevModPhys.36.39 – volume: 181 start-page: 682 year: 1969 ident: 10.1016/j.cam.2006.03.055_bib4 article-title: Flux creep in type-II superconductors publication-title: Phys. Rev. doi: 10.1103/PhysRev.181.682 – volume: 22 start-page: 501 year: 1985 ident: 10.1016/j.cam.2006.03.055_bib15 article-title: Error estimates for two-phase Stefan problems in several space variables, II: nonlinear flux conditions publication-title: Calcolo doi: 10.1007/BF02575899 – ident: 10.1016/j.cam.2006.03.055_bib9 doi: 10.1007/978-3-642-96379-7 – volume: 129 start-page: 190 issue: 1 year: 1996 ident: 10.1016/j.cam.2006.03.055_bib16 article-title: The bean model in superconductivity: variational formulation and numerical solution publication-title: J. Comput. Phys. doi: 10.1006/jcph.1996.0243 – volume: 36 start-page: 31 year: 1964 ident: 10.1016/j.cam.2006.03.055_bib3 article-title: Magnetization of high-field superconductors publication-title: Rev. Mod. 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Syst. doi: 10.3934/dcds.2002.8.781 – year: 1973 ident: 10.1016/j.cam.2006.03.055_bib18 – volume: 42 start-page: 555 issue: 4 year: 2000 ident: 10.1016/j.cam.2006.03.055_bib5 article-title: A hierarchy of models for type-II superconductors publication-title: SIAM Rev. doi: 10.1137/S0036144599371913 – year: 2003 ident: 10.1016/j.cam.2006.03.055_bib8 – volume: 42A start-page: 977 issue: 6 year: 2000 ident: 10.1016/j.cam.2006.03.055_bib2 article-title: Bean's critical-state model as the p→∞ limit of an evolutionary p-Laplacian equation publication-title: Nonlinear Anal. Theory Methods Appl. doi: 10.1016/S0362-546X(99)00147-9 – year: 2000 ident: 10.1016/j.cam.2006.03.055_bib6 – ident: 10.1016/j.cam.2006.03.055_bib10 – volume: 7 start-page: 237 issue: 3 year: 1996 ident: 10.1016/j.cam.2006.03.055_bib17 article-title: On the bean critical-state model in superconductivity publication-title: European J. Appl. Math. doi: 10.1017/S0956792500002333 |
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| Snippet | In this paper we study a nonlinear evolution equation
∂
t
(
σ
(
|
E
|
)
E
)
+
∇
×
∇
×
E
=
F
in a bounded domain subject to appropriate initial and boundary... |
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| StartPage | 568 |
| SubjectTerms | Combinatorics Combinatorics. Ordered structures Compensated compactness Designs and configurations Error estimates Exact sciences and technology Global analysis, analysis on manifolds Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Sciences and techniques of general use Superconductors Time discretization Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| Title | Nonlinear diffusion in type-II superconductors |
| URI | https://dx.doi.org/10.1016/j.cam.2006.03.055 |
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