Analysis and Approximation of the Ginzburg-Landau Model of ... · Theauthors consider the...

SIAM REVIEWVol. 34, No. 1, pp. 54-81, March 1992

() 1992 Society for Industrial and Applied Mathematics003

ANALYSIS AND APPROXIMATION OF THE GINZBURG-LANDAU MODEL OFSUPERCONDUCTIVITY*

QIANG DUt, MAX D. GUNZBURGER*, AND JANET S. PETERSON

Abstract. The authors consider the Ginzburg-Landau model for superconductivity. First some well-known features of superconducting materials are reviewed and then various results concerning the model,the resultant differential equations, and their solution on bounded domains are derived. Then, finite elementapproximations of the solutions of the Ginzburg-Landau equations are considered and error estimates of op-timal order are derived.

Key words, superconductivity, Ginzburg-Landau equations, finite element approximations

AMS(MOS) subject classifications. 81J05, 65N30, 35J60

1. Introduction. The superconductivity of certain metals (such as mercury, lead,and tin) at very low temperatures was discovered by H. Kamerlingh-Onnes in 1908. Heobserved that electrical resistance disappeared completely below some critical tempera-ture. Indeed, closed currents in a ring of superconducting material have been observedto flow without decay for over two years, and the resistivity of some of these materialshas been estimated to be no greater than 10-23 ohm-cm!

In addition to this perfect conductivity property, superconductors are also charac-terized by the property of perfect diamagnetism. This phenomenon was discovered in1933 by W. Meissner and R. Ochsenfeld, and is also known as the Meissner effect. Theyobserved that not only is a magnetic field excluded from a superconductor, i.e., if a mag-netic field is applied to a superconducting material at a temperature below the criticaltemperature, it does not penetrate into the material, but also that a magnetic field isexpelled from a superconductor, i.e., if a superconductor subject to a magnetic field iscooled through the critical temperature, the magnetic field is expelled from the mate-rial. Of course, sufficiently large magnetic fields cannot be excluded from the material,so that the Meissner effect also predicts the existence of a critical magnetic field abovewhich the material ceases to be superconducting, even at temperatures below the criticaltemperature. Furthermore, passage through the critical temperature is reversible. Then,simple thermodynamic arguments can be used (see, e.g., [33]) to show that the transitionfrom the normal to the superconducting state at zero applied magnetic field is not ac-companied by any release of latent heat; thus, we have what is known as a second-ordertransition.

A good theoretical understanding of low-temperature superconductivity was not ar-rived at until the 1950s. Indeed, a completely acceptable microscopic theory did not existuntil Bardeen, Cooper, and Schrieffer [4] published their landmark paper in 1957. How-ever, even earlier, various macroscopic theories were proposed, most notably the homo-geneous theory of London and London [29] in 1935, the nonlocal theory of Pippard [32]in 1950, and the theory of Ginzburg and Landau [19], also in 1950, a full 7 years beforethe Bardeen, Cooper, Schrieffer (BCS) theory! The Ginzburg-Landau (GL) theory was

Received by the editors September 26, 1990; accepted for publication (in revised form) July 29, 1991.tDepartment of Mathematics, University of Chicago, Illinois 60637. Present address, Department of

Mathematics, Michigan State University, East Lansing, Michigan 48224.*Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia

24061. This research was supported by the Department of Energy at the Los Alamos National Laboratory,and by the U.S. Air Force Office of Scientific Research grant AFOSR-90-0179.

Department of Mathematics, Virgina Polytechnic Institute and State University, Blacksburg, Virginia24061. This research was supported by the Department of Energy at the Los Alamos National Laboratory.

54

GINZBURG-LANDAU MODEL OF SUPERCONDUCTIVITY 55

itself based on a general theory, introduced by Landau in 1937, for second-order phasetransitions in fluids that was based on minimizing the Helmholtz free energy. Ginzburgand Landau thought of the conducting electrons as being a "fluid" that could appearin two phases, namely superconducting and normal (nonsuperconducting.) Through astroke of intuitive genius, Ginzburg and Landau added to the theory of phase transitionscertain effects, motivated by quantum-mechanical considerations, to account for the factthat the electron "fluid" motion is affected by the presence of magnetic fields.

The GL theory was not widely accepted immediately, mainly due to its phenomeno-logical character. However, in 1959, Gor’kov [21] showed that, in the appropriate limit,the macroscopic GL theory can be derived from the microscopic BCS theory. After thiswork of Gor’kov, the GL theory became accepted as a valid (macroscopic) model forlow-temperature superconducting effects.

At the time that Ginzburg and Landau proposed their theory, it was thought thatthe transition between the superconducting and normal phases is always accompaniedby positive surface energy, so that the minimum energy principle would lead to relativelyfew such transitions in a sample of material. Indeed, this agreed with experimental ob-servations in what is now known as type I superconductors. Then, in 1957 (the sameyear as BCS), Abrikosov [1] investigated what would happen if the surface energy ac-companying phase transitions was negative. The GL theory then predicts that, in orderto minimize the energy, there would be relatively many phase transitions in a materialsample, and that indeed the normal and superconducting state could coexist in what isknown as the mixed state. About ten years later, such type II superconductors were ob-served experimentally. It is another remarkable feature of the GL theory that it allowedfor such materials, even before their existence was known! From a technological stand-point, type II superconductors are the ones of greatest interest, mainly because they canretain superconductivity properties in the presence of large applied magnetic fields.

Due to the extremely low temperature necessary for known materials, e.g., metals, tobecome superconducting, their practical usefulness was very limited and therefore gen-eral interest in superconductivity waned. However, after the recent advances in cryogen-ics and, even more so, after the recent discovery of high-temperature superconductors,there has naturally been a resurgence in interest. One question that arises is the appli-cability of the GL theory, or some variant of it, to high-temperature superconductors.In this regard, no general consensus has been reached.

Our short introduction by no means does justice to the history of superconductivity,nor do we intend to give a full description of even the GL theory. There are, however,many excellent references that may be consulted for detailed descriptions of both themicroscopic and macroscopic theories of low-temperature superconductivity. Amongthese are [13], [26], [33], and [38]. There has also been substantial interest in the GLmodel within the mathematical physics community, e.g., see [5], [8], [10], [16], [24], [31],[34], [35], [40], and [41]. For another recent survey, see [9].

Approximations of solutions of the GL model for superconductivity have been ob-tained by many authors. These approximations are usually obtained by using series solu-tions of one type or another; see, e.g., [1], [6], [16], [23], [25], [27], [28], and [39]. MonteCarlo-simulated annealing simulations have also been obtained by [15].

A main goal of our ongoing work is to develop robust and efficient codes that can beused to help determine, through comparisons with experimental observations, the extentto which GL models can be applied to high-temperature superconductors. Althoughthe results and algorithms of this paper are presented in the context of GL models forlow-temperature superconductivity, many of these apply equally well to GL models for

56 QIANG DU, MAX D. GUNZBURGER, AND JANET S. PETERSON

high-temperature superconductivity.In 2, we briefly describe the GL model for superconductivity. Then, in 3, we ob-

tain results concerning the model, some of which are new, and often verify that themodel agrees with experimental observations; some of the results are generalizationsof known mathematical results. In 4, we develop and analyze finite element algorithmsfor approximating solutions to the model. Finally, in 5, we briefly describe a periodicGinzburg-Landau model that is actually used, instead of the boundary value model of2, in most computational studies of superconductivity.

2. The Ginzburg-Landau model for superconductivity. We trust that many of ourreaders are not familiar with the Ginzburg-Landau model of superconductivity. There-fore, in this section we discuss the derivation of the GL model, and of some well-knownfeatures of superconductors that are well described by the model. Necessarily, our pre-sentation will be rather sketchy. For details concerning the material of this section, anyof the many books on superconductivity, e.g., [13], [26], [33], and [38] may be consulted.

2.1. The Ginzburg-Landau free energy. Ginzburg and Landau postulated that theHelmholtz free energy per unit volume of a superconducting material is given by

[h[ 1 ( esA) l(2.1) f n + 112 + I]4 + + -ihv- cHere, the constant f is the free energy of the normal (nonsuperconducting) state in theabsence of magnetic fields, is the (complex-valued) order parameter, A is the magneticpotential, h curl A is the magnetic field, a and/ are constants (with respect to thespace variable x) whoe values depend on the temperature, c is the speed of light, e andm are the charge and mass, respectively, of the superconducting charge-carriers, and27rh is Planck’s constant.

The microscopic electron pairing theory of superconductivity implies that e 2e,where e is the electron charge. The value of m is somewhat more arbitrary since anychange in its definition can be compensated by an attendant change in the magnitude of. However, it is customary to either set m m or m 2m, where m is the electronmass. We will adopt the latter value. In this case, [1 n, where n is the densityof superconducting charge carriers (equaling half the density of electrons) in the sam-ple; thus the magnitude of the order parameter gives the density of superconductingelectron pairs.

The appearance of an order parameter is a part of the Landau theory of second-order phase transitions. The need for a complex-valued order parameter is somewhat dif-ficult to explain. Ginzburg and Landau thought of their order parameter as an "averagedwave-function of the superconducting electrons." The connection, made by Gor’kov, be-tween the microscopic BCS theory and the macroscopic GL theory justified the need for

to be complex-valued. However, recall that the GL theory preceded the BCS theoryby seven years, so that the arguments of Ginzburg and Landau were based on physicalintuition. Adopting the wave-function view of, one may, as was done above, normalizethe order parameter so that the square of its magnitude is proportional to the densityof superconducting charge-carriers. The phase of the order parameter b can then beshown to be related to the current in the superconductor. (More on this later.)

Let us briefly discuss the various terms appearing in (2.1). First, note that f, +[hlZ/87r is the free energy density of the normal state in the presence of the magneticfield h. Next, recall that the Landau theory of second-order phase transitions was basedon the following three assumptions: (a) there exists an order parameter that goes to

GINZBURG-I.ANDAU MODEL OF SUPERCONDUCTIVITY 57

zero at the transition; (b) the free energy may be expanded as a power series in theorder parameter; and (c) the coefficients in the expansion are regular functions of thetemperature. Various physical arguments can be invoked to deduce that the expansionpostulated in (b) is in even powers of [[. Thus, recalling that in the present context thetransition is one between the normal and superconducting states, the first four terms onthe right-hand side of (2.1) represent a truncation of this power series for small valuesof I1, i.e., near the transition.

In the GL theory, the density of superconducting charge-carriers, and thus the orderparameter, is allowed to be spatially varying. Then, another consequence of the inter-pretation of as a wave-function is the existence of a kinetic energy density associatedwith spatial variations of that must be accounted for in the free energy density. Vari-ations in the order parameter should penalize the energy, so that it is natural to add tothe free energy density a term proportional to IV!2. On the other hand, the free en-ergy should be gauge-invariant. Here, Ginzburg and Landau, through another strokeof intuitive genius, postulated that the last term in (2.1) is the added energy density, ingauge-invariant form, due to the spatial variations in . Following [38], we can elucidatethis point by noting that the last term in (2.1) may be rewritten in the form

(2.2) 2ml [/2where is the phase of , i.e.,

(2.3)

The first term in (2.2) clearly penalizes the energy when l1 varies, and the second termmay be interpreted as a gauge-invariant kinetic energy density associated with currentsin the superconductor.

In the presence of an applied magnetic field H, the Gibbs free energy density gdiffers from f due to the work done by the electromagnetic force induced by the appliedfield. This work (per unit volume) is given by -h. H/47r so that the Gibbs free energydensity is given by

471-ihV esA )c

2 Ihl 2 h. H87r 47r

If f denotes the region occupied by the superconducting sample, the Gibbs free energy6 of the sample is then given by

The basic thermodynamic postulate of the GL theory is that the superconducting sampleis in a state such that its Gibbs free energy is a minimum.

The dependence of the value of the constants c and/3 on the temperature can beobtained from the microscopic theory. The constant/3 is positive; otherwise, 6 would nothave a minimum value. If c is positive as well, then one easily sees that 0 and h Hminimizes 6. This is the normal, or nonsuperconducting state. On the other hand, if cis negative, and in the absence of surface effects, it can be shown that is minimized by


A 0 (so that h 0) and (--Cg//) 1/2. This is the ideal superconductor with aconstant density of superconducting charge-carriers and an excluded magnetic field, i.e.,a superconductor with a perfect Meissner effect. With the presence of surface effects,minimizers of are not so trivial.

Remark. It will, at times, be convenient to consider the functional , defined by

c H. H 1i

[ 1 (eA)+ +

instead of . Clearly, (, A) is a minimizer of if and only if it is also a minimizer of 8.The main virtue of the functional 8 is that it is nonnegative.

Remark. High-temperature superconductors are generally anisotropic and inhomo-geneous, as opposed to low-temperature superconductors. e flee ener densities andfunctionals given above correspond to low-temperature superconductors. In principle,there is no dicul in eending the GL model to high-temperature superconductors.For example, the constants and fl, whose value depends on the temperature, are re-placed by spatially vaing scalar-valued functions, and the constant m is replaced bya matr, with possibly spatially vaing entries. (Of course, we would replace 1/mwith mX.) However, in practice, there are diculties. In the first place, the functionalform and values of , fl, and m are not own. In addition, the justification of theGL theo as an appropriate limit of a microscopic theo is not available in the high-temperature case. For these, and other reasons, the ju is still out as to whether or nothigh-temperature superconductors can be modeled by anisotropic and inhomogeneousgeneralizations of the GL theol.

2.2. The Ginzburg-Landau equations and bounda conditions. Ifwe use standardtechniques from the calculus of variations, the minimization of with respect to varia-tions in and A yields the celebrated Ginzburg-Landau equations

(2.6)1 (_ihv_eA)

and

2iesh 4e(2.7) curlcurlA + (*V V*) + I[eA curlH inmsC msc2

where (.)* denotes the complex conjugate. Note that the current is given by

j (c/4r)curl h (c/4zr)curl curl A,

so that

(2.8)

Note the dependence of the current j on the phase of the order parameter.


Candidate minimizers of g are not a priori constrained to satisfy any boundary con-ditions. Thus, the minimization process also yields the natural boundary conditions

( es )(2.9) i/V+--A .n=0 onF

and

(2.10) curlAn=Hn onF,

where F denotes the boundary of and n the unit outer normal vector to F.More general boundary conditions have also been suggested. First, note that the GL

free energy functionals are not valid near the boundary of the superconducting sample.(On the other hand, the GL differential equations are usually retained, even in the vicin-ity of the boundary.) Aboundary condition is then determined by requiring that the nor-mal component of the current be continuous at the boundary, i.e., j. n (c/47r)curl H. non F, or even more commonly, that the normal component of the current vanishes, i.e.,j. n (c/47r)curl H. n 0 on F. In either case, (2.8) then yields that

* (-ihV eA) n + (ilV A*) n=0. onF.

It is easily seen that this relation implies that

(2.11) (-ihV- A).n i7 on F

for some real valued function 7. The microscopic theory ofsuperconductivity yields that

3’ # 0 for a superconductor-normal metal interface, and that 7 0 for a superconductor-insulator (or vacuum) interface. Note that in the latter case, (2.9) and (2.11) coincide.We will, for convenience, adopt the boundary condition (2.9), although no real difficul-ties would be engendered by instead considering the more general boundary condition(2.11).

Remark. We have derived the GL equations (2.6) and (2.7) by minimizing the GLform of the Gibbs free energy. As has been already noted, these equations may alsobe derived as an appropriate limit of the BCS microscopic theory of superconductivity.Likewise, the boundary condition (2.9) may be derived from the microscopic theory.

Remark. The boundary condition (2.11) can also result directly from setting the firstvariation of G to zero ifwe add to the definition (2.4) the term fr i7]12 dF. We do notknow if there is any physical justification for the addition of this term to the free energy.

Remark. The Ginzburg-Landau equations can be easily generalized to the case ofhigh-temperature superconductors For example, (2.6) is replaced by

l(-ihV -eA) m- (-ihV -)+c+/31,2 0 inf,,2 c

where e and c and 3 are scalar-valued functions and ms is a matrix-valued function.

2.3. Fluxoid quantization; some important scales and parameters. Let 0 denotea closed curve lying in the material sample such that I1 : 0 everywhere on the curve,i.e., the curve 0E nowhere intersects a normal region. Let E denote a surface boundedby this closed curve. Consider the expression

mc fo J d(OE)


The first term on the right is the magnetic flux through the surface E, and ’ is knownas thefluxoid. Using Stokes’ theorem and h curl A, we then have that

mc j ).d(O)so that, using (2.3), (2.8), and the fact that 1] is single valued,

(2.12) ,_ hcfo V d(OF_,)-Oons El

where (I)0 27rhc/e8 and n is an integer. Thus, the fluxoid (I)’ is quantized.For type I superconductors without holes or nonsuperconducting material inclu-

sions, i.e., f is simply connected, fluxoid quantization is of no importance since in thiscase E always lies in the superconductor and we may choose n 0. On the other hand,if there are holes or such inclusions in the sample, i.e., f is multiply connected, and if0 encloses such a hole, then (2.12) implies that ’ must be an integer multiple of I’0.

Fluxoid quantization can be used to explain the persistence of currents in a super-conducting ring. (Our explanation is drawn from [38].) The current cannot decreaseby arbitrarily small amounts, but only in finite jumps such that the fluxoid decreases byone or more integer multiples of 0. If only a single, or a few electrons were involved,this could be easily accomplished. However, for a superconductor, we are requiring aquantumjump in (the phase of) . Such a macroscopic change requires the simultaneousquantum jump by a very large (> 10) number of particles. Such an event is, of course,extremely improbable, so that the current in the superconductor persists.

For type II superconductors, (2.12), i.e., the quantization of the fluxoid, has someother important consequences, even in simply connected domains. In the mixedsuperconducting-normal state, we may place 0E in a superconducting region. Any re-gion where the field penetrates that is enclosed by 0E must have a fluxoid value that isan integer multiple of 0. This implies that such regions cannot be arbitrarily small, sothat, even though the surface energy associated with the transition from one state to theother is negative, we do not have arbitrarily small scale transitions; they are limited bythe requirement that the fluxoid is an integer multiple of 0.

In the introduction, we mentioned that above a critical value of the field, super-conductivity is destroyed. For a superconductor with perfect Meissner effect this criti-cal field can be computed. First, observe that with a perfect Meissner effect, Ih] 0and o (-o//) 1/2 so that from (2.4), g f,, a2/(2/). On the otherhand, above the critical field, superconductivity is lost, i.e., 0 and h H, so thatgn fn IH12/(87r). At the critical field Hc IHI, at which the transition from thesuperconducting to the normal state occurs, the value of the Gibbs free energy in eachof the states must be the same, i.e., at IHI n we have that g gn, so that

(2.13) Hc 14a2We will useH as a fundamental scale for magnetic fields. It should be noted that for typeI superconductors, H is the field at which superconductivity is lost. However, due to theexistence of the mixed superconducting-normal state, type II superconductors are not ina perfect Meissner state and retain some superconductivity properties at fields above,and in some cases much above, H. This fact makes type II superconductors interestingfrom technological and design points of view. We will discuss this in more detail in 2.4.


Two length scales play important roles in the theory and understanding of supercon-ductors. First, assume that the applied field H is constant. Assuming that the sample isperfectly conducting, i.e., 0 (-all3) :/2, let us see how far the field penetratesinto the sample. With the use of the relation h curl A we deduce that

(2.14)1

curl curl h + -h 0,

where

):/2msc2 :/2

Equation (2.14) is known as the London equation, and the parameter A, whose valuedepends on the temperature, is known as the (London)penetration depth. The latterterminology can be easily justified by examining, for example, a one-dimensional versionof (2.14). Here we let h denote the single nontrivial component of the field and z thedistance from the boundary of the sample. Then, (2.14) and (2.10) reduce to d2h/dz2(1/,2)h 0 for z > 0 and h(0) H, where H denotes the applied field. Then, hHe-z/ so that A clearly gives a measure ofhow far the field h penetrates into the sample.

Now, let us determine what is the length scale of variations of the order parameter.We now assume that there is a perfect Meissner effect, i.e., A 0. Then, the coefficientsin (2.6) become real, and we may take to be real as well, so that (2.6) reduces to

(2.15) A-2 (+-fl3)=0’awhere

The parameter is known as the (Ginzburg-Landau) coherence length. This terminologycan be justified by examining, for example, a one-dimensional, linearized (about theperfect conducting state) version of (2.15). Thus, letting z denote the single coordinate,setting 0(1 + #) with 0 (-c//3) :/2, and neglecting nonlinear terms in #, we

have that d2#/dx2 (2/2)/z 0. Then,/z is proportional to e+’// so that a smalldisturbance of from 0 will decay in a characteristic length of order .

The ratio of the penetration length to the coherence length, i.e.,

is known as the Ginzburg-Landau parameter. Its importance lies in the fact that thissingle parameter, along with the given data, i.e., the applied field, completely determinesthe solution of the GL equations. In particular, as we shall see in 3, the value of t

determines whether the superconductor is of type I or II.The central role that n plays can be made evident by nondimensionalizing lengths

by A, fields by x/Hc (the factor x/ is customary and convenient), currents by (cHc)/(2x/TrA), the magnetic potential by x/-AH, the order parameter by 0 (-a/) :/2,


free energy densities by c9//, and the Gibbs free energy by (cA3)/fl. In this case, theGibbs free energy, in nondimensionalized form, is given by

jf( 1 ( i )2

(2.16) G(,A) A-I12 + 114 + --xTn A + Ihl 2 2h. H

where we have used the same notation to denote nondimensionalized variables used todenote variables with dimension. Likewise, the GL equations and boundary conditions(2.6)-(2.10) are, in nondimensional form, given by

(2.17)iV A

i(2. lS) j curl h curl curl A (*V V* I12A + curl H in f,

(2.19) V4-A .n=0 onr

and

(2.20) curlAn=Hn onF.

Now it is clear that , along with the data H, completely determines minimizers of theGibbs free energy, or solutions of the GL equations. It is important to note that high-temperature superconductors, and in general, inhomogeneous or anisotropic supercon-ductors, cannot be characterized by a single, constant parameter such as , as is the casefor low-temperature, homogeneous, isotropic superconductors.

2.4. Some properties of superconductors. We close this section by discussing somefurther properties of superconductors. Space limitations preclude us from giving any ofthe formal arguments that are used in the literature to explain these properties. Anyof the references cited at the beginning of 2 may be consulted for all details. We domention that all of these properties, as well as the ones discussed previously, have beenverified experimentally.

We have already mentioned that if the applied field is sufficiently strong, then su-perconductivity is destroyed. (Here, we assume that the applied field is constant withmagnitude H.) However, in general this critical field is not given by (2.13), which wasderived assuming an ideal Meissner effect. It can be shown that if

(2.21) H > H2 x/H,

then superconductivity is lost.Remark. It should be noted that the derivation of this result, as given in various

books on superconductivity, is somewhat formal. First, the derivation presupposes theexistence of such a critical field in order to derive its value. In addition, the derivationalso assumes that the order parameter is continuous at the transition from the normal tothe superconducting states; this certainly is not true for type I superconductors. In spiteof these apparent shortcomings, we proceed, as is customary, to apply this result.

For ideal type I superconductors (real ones behave in a somewhat more complexmanner), < 1/x/ and (2.21) imply that as the field is reduced, the sample will remain


normal until a field He2 < Hc is reached, at which superconductivity will be total. This,in turn, implies a nonreversible loop, since as we increase the field, superconductivity isretained until H H. In any case, the transition from the normal to the superconduct-ing state at H or He2 is a first-order one, i.e., at the critical field the order parameterjumps from zero to b0 (-a/)/2. This phenomenon is sketched in Fig. 2.1. (Inreal type I superconductors, there is field penetration at the surface of the sample; thiscomplicates the idealized hysteresis loop just described.)

He2 H Hc c2

FIG. 2.1. The transitionfrom the normalto the superconductingstate. Arrowspoint in the direction ofchangein the applied field H.) (a) Hysteresis loop for type superconductors ( < 1//). (b) Reversible, second-ordertransition for type II superconductors ( > 1/x).

For type II superconductors, > 1/x/ and we see that He2 > H, i.e., supercon-ductivity is not completely lost even at fields higher than the thermodynamic critical fieldH. Unlike type I superconductors, the transition at H2 is a second-order one, and theorder parameter is continuous there; the situation is sketched in Fig. 2.1. (Due to sur-face effects, which are neglected in the derivation of (2.21), real type II superconductorsretain some superconductivity properties at fields even higher than H2.)

In the type II setting, there also exists a second critical field HI. For H < HI,the superconductor behaves very much like a type I superconductor, i.e., with a perfectMeissner effect, except near the surfaces. For an ideal superconductor, superconductiv-ity is lost for H > He2. For H < H < H2, we have the appearance of vortices, orfilaments. The superconducting and normal states coexist. At the center of the vortices,the order parameter vanishes, i.e., we have the normal state. Near H, these vortex fila-ments are well isolated, but as the field is increased, the vortices become more numerousand come closer and closer. At the same time, as the field is increased, the maximumvalue of the magnitude of the order parameter decreases, until at H2 it vanishes. It canbe shown that each vortex has associated with it one fluxoid, i.e., for each vortex I,’ 0.

This stark contrast betweeen type I and type II superconductors can be furtherdemonstrated by examining the magnetization curves for each type. Here, we considerthe magnetization M of the sample, given by 47rM B H. (These are all bulk, or av-eraged quantities. In particular, B is the average value of the field and H the magnitudeof the (constant) applied field.) For type I superconductors (neglecting any hysteresis ef-fects), and for H < He, the sample is in the superconducting state, B 0, and therefore-47rM H; for H > H, the sample is normal, B H, and therefore 47rM 0. Thetransition from -47rM H to -47rM 0 at H H is discontinuous. Figure 2.2 givesa sketch of the magnetization curve for an ideal type I superconductor, i.e., n < 1/x/.

For type II superconductors and H < H < H, we again have the superconductingstate B 0 and -47rM H, while for H > He2 > H, we have the normal stateB H and -4rM 0. However, for ncl ( n ( He2 we have a mixed state where0 < B < H and therefore 0 < -47rM < H. The transition between-47rM H atH Hc to -47rM 0 at H Hc2 is continuous. Figure 2.2 also gives a sketch of


4xM 4M

H Hcl Hc c2

FIG. 2.2. Magnetization M vs. applied field H. (a) Type superconductors ( < 1/x/-). (b) Type IIsuperconductors > 1/x/).

the magnetization curve for an ideal type II superconductor. Note that regardless of thevalue of n, the area under the magnetization curve is the same, i.e.,

(-4’M)dH H8’"

We close this review section with some comments on the factors which limit thevalidi of the GL model. For example, we have assumed that higher powers can be ne-glected in the expansion of f in terms of powers of [[. so, the GL theo assumes alocal relation beeen the current and the vector potential. In general, this relation is notlocal. It can be shown, both from the microscopic theo and from nonlocal macroscopictheories, that in order for the GL model to be valid, we must have that the temperatureis close to the critical temperature at which the transition from the normal to the super-conducting states occurs. It can also be shown that the temperature range over which theGL model is valid is larger for pe II than it is for pe I superconductors. Nevertheless,it should be emphasized that for temperatures sufficiently close to the critical transitiontemperature, the GLmodel is valid for bothpe I and II superconductors. Furthermore,experimental evidence suggests that, in fact, the GL model remains useful in describingcorrect physics even for temperatures so different from the critical temperature that amathematical justification cannot be provided.

3. Minimizers ofthe Ginzburg-Landau functional. In this sectionwe examine somequestions about minimizers of the GL functional (2.4), or equivalently, of (2.5). The lat-ter, in terms of nondimensionalized variables, is given by

g(,A) (lel 1) + -V A e + [curiA- HI2

where fl is a bounded, open subset ofNa, d =2 or 3. Unless otheise noted, we will alsoassume that fl is a domain with "smooth" bounda or is a convex polyhedral domain.The bounda of fl will be denoted by r. We will also make use of the functional

(, A) e(, A) + j Idi AI

Throughout, for any nonnegative integer s, H(fl) will denote the Sobolev space ofreal-valued functions having square integrable derivatives of order up to s. The corre-sponding spaces of complex-valued functions will be denoted by (fl). Corresponding


spaces of vector-valued functions, each of whose d components belong to HS(f), willbe denoted by H8 (f), i.e., H (f) [H (gt)] a. Norms of functions belonging to H (f),H(f), and 7-/8(f) will all be denoted, without any possible ambiguity, by I1" I1, Fordetails concerning these spaces, consult [2]. A similar notational convention will holdfor the Lebesgue spaces L’(f) and their complex and vector-valued counterpartsand Lp(f), respectively.

We will make use of the following subspaces of I-I (f):

HI(f)={QHl(f) Q.n=0onr}

and

Hl(div ;dr) { Q e Hl(f) div Q 0 in f and Q. n 0 on F }.

We note that (lldivQIl + Ilcurl QI]o)/2 and I[curl Ql[0 define norms on H(f) andH(div; f), respectively, that are equivalent to the standard Hl(f)-norm [IQl[1; see,e.g., [20l.

3.1. Gauge invariance. We begin by giving a precise definition of gauge invariance.For any He(f), let the linear transformation G from 7-/1(f) I-I (f) into itself bedefined by

G(, A) (, Q) e 7-/1 (gt) x H (f) V (, A) e 7-/1 (gt) H (f)

and Q=A+V.

Note that if (, Q) G(, A), then (, A) G_(, Q).DEFINITION. (, A) and (, Q) are said to be gauge equivalent if and only if there

exists a E U2() such that (, A) G(, Q).We have the following two results about gauge invariance with respect to 7-/l(f)

Hn(div f).LEMMA 3.1. Any (, Q) TI (f) H1 (2) is gauge equivalent to an element of

7-/1 () Hn(div ;).Proof. Let be defined byA div Q in ft and O/On Q. n. Then, H2(f)

and clearly G(, Q) e 7-/1(f) H(div; f).LEMMA 3.2. (, A) and (, Q) 7-/1 (f) Hn(div f) are gauge equivalent if and

only ifA Q and Ceic for some real constant aProof. The proof is obvious.Of course, Ginzburg and Landau constructed their free energy functional so that it

is gauge invariant. This is stated in the following proposition whose proof is merely asimple calculation.

PROPOSITION 3.3. For all H2(f)and (,A) E ../1() HI(), (,A)(G(, A)), i.e., is invariant under the gauge transformation G

Thus, for example, if (, A) is a minimizer of , then so is G(, A), i.e., so is any(, Q) that can be obtained from (, A) through a gauge transformation. In this way, itis possible to define equivalence classes of minimizers, i.e., minimizers that may be ob-tained from each other through gauge transformations. By choosing a particular gauge,one extracts a particular member of its equivalence class.


3.2. Existence of minimizers. We now give some results concerning the existenceof minimizers of - and in various spaces, and the relation between the various mini-mizers.

PROPOSITION 3.4. has at least one minimizer belonging to 7-/ (f) Hn(div f).Proof. It is easy to check that is nonnegative, is continuous in the strong topology,

and lower semicontinuous in the weak topology. Then the proofproceeds using standardarguments; see, e.g., [24]. First, we have the existence of a minimizing sequence, i.e.,(CA, An) E 7-/1 (’) X I-Kin (div ft) such that (CA, An) inf > 0. Then, due to theequivalence of norms, we have the b0undedness of (CA, An) E 7-/l(ft) x Hl(ft). Afterextracting a subsequence, we see that the weak limit will be a minimizer. Cl

THEOREM 3.5. has at least one minimizer belonging to 7-i (t2) x H1(t2). Moreover,

min $ minnx(a) xH(a) nx(a) xH(div

Proof. The results follow from Propositions 3.3 and 3.4. Cl

COROLLARY 3.6. Any minimizer of in 7-/1 (f) x H1, (div ft) is also a minimizer ofin 7-/1(f/) x Hi(f/). Any minimizer of in 7-/l(ft) x HI(fI) is gauge equivalent to a

minimizer of in 7-/1 (’) X H(div f).Proof. The results follow from the fact that 7-/1(9t) x HX(div; f) c 7-[1(f) x

HI(’-). [-]

LEMMA 3.7. For any (, A) 7-/l(f) Hl(f), let H9 (f) \ IR satisfy

Then,

A=divA in f and O/On A. n oAF.

’(G(, A)) 9r(, A)- Ja IdivAle (, A).

Proof. The results follow from Proposition 3.3 and the definitions of and ’. [3

THEOREM 3.8..T has at least one minimizer in 7-[ (f) H (ft). Moreover, all mini-mizers (, A) of " satisfy div A 0.

Proof. The proof is similar to that of Proposition 3.4. If (CA, An) is a minimiz-ing sequence, so is, by Lemma 3.7, Gcn (CA, A,). The boundedness of the sequenceGCn (n, An) is easily seen. Thus, we can proceed to obtain the existence of a minimizer.The fact that div A 0 for minimizers also follows from Lemma 3.7.

COROLLARY 3.9.

min U min .T,?-/l(f) xH (fl) n(a) xH(div ;a)

and

min .7" min 9r7-/1(fl) xH (fl) 7-/1(fl) x H1. (a)

min .7" min g.

Proof. The first equality follows from Theorem 3.8 and the fact that if div A 0,then -(Go(, A)) ’(, A), where is defined as in Lemma 3.7. Then, the secondequality easily follows from the first. Finally, the third equality follows from the fact that-(, A) g(, A) for (,A) 6 7-/(f) HX(div f). ]

Thus, we have shown thatwe can locate minimizers ofg in _/1 (")XH (’2) by lookingfor minimizers of " in 7-/1 (f) x H(f). We shall see that the latter problem has somedefinite advantages, especially from a computational point of view.


3.3. The Ginzburg-Landau equations. We now derive the GL equations which wehave formally introduced in 2.2. At the same time, by looking for minimizers of " in7-/ (f) HI(f) we will be automatically choosing a gauge for (, A).

Taking the Frech6t derivative of ’(, A) with respect to (f) we obtain

(3.1)

+ (112 -])(* + *)] da 0 V e 7-/(f)

and taking the derivative with respect to A E H(f) we get

(3.2)[div Adiv/ + curl A. curl A + 112A. A + (*V- Cwp*). A] df

f H. curl A. df V . e Hn(f).

Integration by parts in (3.1) yields, should (, A) be sufficiently smooth, the first GLequation (2.17), and as a natural boundarycondition

(3.3) V.n=0 onF.

Of course, since A. n 0 on F, (3.3) is the same as (2.19).Let A Vq, where q H2(f) satisfies Aq divA in f and Oq/On 0 on F.

Then, A H(f). We substitute this A_ in (3.2) and integrate by parts to obtain

(divA)2 qdiv 112A + --(*V V*) 0,

where we have used (3.3) in the last equality. Thus, we have that div A 0 almosteverywhere in f, so that we are employing the (London or Coulomb) gauge, along with,of course, A. n 0 on F. Furthermore, (3.2) also yields the second GL equation (2.18)and, as a natural boundary condition, (2.20). Note that since divA 0, (2.18) may beexpressed in the form

i-AA + ---(*V V*) + 112A curlH in f.

This second-order system of elliptic partial differential equations is supplemented by theboundary conditions (2.20) and A. n 0.

3.4. Some properties of minimizers of the GL free energy and solutions of the GLequations. We now derive a series of properties possessed by minimizers of the GL freeenergy, or of solutions of the GL equations.

3.4.1. Nonexistence of local maxima. We first show that solutions of the GL equa-tions cannot correspond to local maxima of the GL free energy.

PROPOSITION 3.10. A solution ofthe Ginzburg-Landau equations (2.17), (2.18) can-not be a local maximum of, or of.

Proof. It is obvious that for fixed 7(f), : is a convex functional in H (f).Moreover, if (, A) is a solution of the GL equations, and if #(e) Y’(e, A), we have


that #(e) is locally convex at e 1. Indeed, we need only check d21z/de2. A simplecalculation yields that

ff (611a 2112 + 2 iv

Since (, A) is presumed to be a solution of the GL equations, we may then show that

d2#d2

e--1

with equality holding if and only if 0. [3

3.4.2. Boundedness of the order parameter. We next show that the magnitude ofthe order parameter is bounded, and in fact, is bounded by the ideal superconductingvalue, i.e., in nondimensionalized form, -< 1.

PROPOSITION 3.11. If (, A) is a solution of the Ginzburg-Landau equations, thenI1 < I almost everywhere.

Proof. We follow, in the present context of bounded domains, the proof in [40] forthis same result in the case of f IRa.

Set (11 X)+f, where f /11 and where q+ q if q > 0 and q+ 0 ifq < O. Let f+ {x f I1 > 1}. Then, on f+,

/V*n A@* f*Vll + (11- 1) vf* Af*

and

so that

Then, since (3.1) implies that

we have that

--Vf Af + I1(11 + 1)(11- 1)2 da O.

Then, since the integrand is positive, meas(f+) 0 and therefore, I1 I almosteverywhere. []

3.4.3. Constant solutions and the existence of mixed-state superconducting solu-tions for H < H. The normal solution is characterized by 0. In this case, the


potential A fit, where

(3.4) curl curl

and, in the gauge we are employing,

(3.5) divfit=0 inf and A.n=0 onF,

where H is the applied field. Note that (0, A) is a solution of the GL equations for anyvalue of .

If the applied field vanishes, i.e., H 0, then the ideal superconducting solution(with perfect Meissner effect) is the unique global minimizer of the GL free energy.

LEMMA 3.12. IfH O, thenProof. If H 0, then g(1,0) 0. However, for any (,A) e 7-/(f) x H(f),

(, A) _> 0, so that for any such (, A), g(, A) _> g(1, 0), with equality holding if andonly if 1 and A 0.

It should be noted that when H 0, this result does not rule out the existence ofother critical points; see [24].

Next, we show that if the applied field does not vanish, then, in most cases, the ?nlysolution of the GL equations with constant is the normal state 0 and A A.

LEMMA 3.13. Let H # 0 and, if f c IRa with a continuous normal vector, let itsboundary F have nonzero Euler number Then, the only solution of the Ginzburg-Landa uequations with constant and A continuous is the normal solution 0 and A A.

Proof. If constant, say k, then, from (2.18), (2.20), and our choice of gauge,we have that

(3.6) curl curl A + k2A curlH and div A 0 in

and

(3.7) curlAxn=Hxn and A.n=0 onI’.

Also, from (2.17), we have that (IAI 2 1 + Ikl2)k 0 so that either k 0 or Ikl 21 AI 2. In the first case, we easily deduce that A . Now, in the second case, if

Ikl 2 1, then A 0, which leads to It 0, a contradiction. We are left with thecase Ikl 2 < 1. In this case A defines a continuous vector field on f that has constantmagnitude IAI v/i k2 < 1. Meanwhile, since A. n 0 on I’, A also defines acontinuous tangential vector field on F, again with gons,tant magnitude IAI v/1 k2.If the normal vector to r is discontinuous, the fact that A, restricted to F, is a continuoustangential vector field on r implies that A 0, which again leads to the contradictoryresult H 0.

At this point it is convenient to separate the cases of f c IR2 and f c IRa, sincea straightforward calculus proof is available in the former case. For f c IR2, let IZIH curl A so that (3.6) and (3.7) yield that

(3.8) curl I:I k2A in f

and

(3.9) I:t x n 0 on r.

Of course, we also have that

(3.10) div I:t 0 in f.


It is easy to show that I7/=/1, where 1 denotes a unit vector pe_rpendicular to the plane.Then, (3.8)-(3.10) imp_ly that Igrad[ constant > ~0, so that H has no interior maximaor minima. But since H 0 on F, this implies that H 0 in f as well, a contradiction.

We now turn to the only remaining case, namely k2 < 1 and f c IRa such thatthe normal to F is continuous. In this case, if the Euler number of F is not zero, it iswell known that one cannot have a nonvanishing continuous tangential vector field on Fwith constant magnitude. Thus, again, we are led to A 0 and the contradictory resultH =0:

Remark. The necessity of the continuity ofA is demonstrated by the following coun-terexam_ple, for which we thank A. J. Meir. Let f in IR be a square, and let IZI 1,where H is an Egyptian pyramid defined on f, i.e., a continuous piecewise linear func-tion vanishing on F and having an appropriate maximum value at the centroid of thesquare. Then, from (3.8) we deduce that the components of A are piecewise constantsover f, and that A satisfies all the requisite conditions.

Remark. Surfaces in IRa with Euler number zero are torus-like; see, e.g., [30]. Thenecessity of having a nonvanishing Euler number is demonstrated by the following coun-terexample, for which we thank W. Floyd and Y. Rong. Let f be a solid torus, let theorigin be located at the center of the torus, and let the z-axis be the axis of rotation forthe torus. Then, let

It is easily verified that A satisfies all the requisite conditions.Remark. Whenever Lemma 3.13 fails to hold, it is due to the fact that for 0 <

Ikl < 1 it is possible to define fields H such that there exists a potential A such that (3.6)and (3.7) hold, and such that IAI x/1 k on . However, given an arbitrary field H,it is quite likely that the solution ofA of (3.6) and (3.7) will not have constant magnitude,even when f is a solid torus.

We next show that for H < Hc and for any value of , the normal solution cannot bea global minimizer. (Note that in terms of nondimensionalized variables, Hc 1/x/-.)

PROPOSITION 3.14. Suppose the external field is smooth, and that H maxn IHI.Then, ifH < 1/x/, the normal solution 0 and A f where fit is defined by (3.4),(3.5), is not a global minimizer of.

Proof. Note that g(0,fit) V/2, where V measure(f). On the other hand,g(1,0) f [HI2 _< HZV. Thus, if H < 1/x/, (1,0) < ’(0, fit), and thus the normalsolution cannot be a global minimizer. V1

Lemma 3.13 and Proposition 3.14 have the following immediate consequence. Note,through an examination of the proof of Lemma 3.13, that here we need not require thatF have a nonvanishing Euler number.

COROLLARY 3.15. For any value of n, if the applied field H 0 is such that 0 <maxc IHI < 1/x/, then the Ginzburg-Landau equations have a solution such that bconstant.

Thus, if (in dimensional form) 0 < H < He, then the GL equations have a solutionsuch that b is not constant. This is a mixed-state solution, since there are places where bdiffers from both zero and one. Of course, the mixed-states of type I and II supercon-ductors can be quite different. In type I superconductors, for H < H, the mixed-state ispresent near boundaries; away from boundaries we have basically the perfect Meissnerstate. For type II superconductors, there is the possibility of vortex-like solutions wherea mixed-state can exist anywhere in 9t.


Remark. We close this section with a remark concerning the regularity of solutionsof the GL equations. It is easily shown that this regularity is completely determined bythe regularity of the linear problem

(3.11) A=0 and AA=-curlH inf

and

V.n=0, A.n=0, and curlAxn=Hn onF.

For sufficiently smooth F, it may be shown that solutions (, A) of (3.11), (3.12) belongto 7-/’+1(f) I-I’+l(f) whenever H Hm(f). This result may be obtained, e.g., form I and F of class Ca, using the theory of [3]. However, for less smooth boundaries,e.g., convex polyhedral domains in]R or IRa, it is not known if solutions of (3.11), (3.12)belong to 7-/ (f) x H(f), even for smooth H; it seems that at least additional compati-bility conditions along edges and at vertices are needed. The culprit is the "nonstandard"boundary condition for A found in (3.12).

4. Finite element approximations. Finite element approximations to the solutionsof the Ginzburg-Landau equations are defined in the usual manner. In order to keepthe exposition simple, we assume that f is a convex polyhedral domain. We choose fam-ilies of finite-dimensional subspaces Sh c 7-/1 (f) and Vh c I-I (f), parametrized by aparameter h that tends to zero. These spaces are constructed, in a standard way, frompartitions of f into finite elements; h is then some measure of the size of the finite ele-ments in a partition. We assume that the subspaces satisfy the following approximationproperties:

inf 1[- hllx -, 0 as h 0 V

(4.2) inf IIA--/hlll0 ash0 VAeH(f),

(4.3) inf I[- hl}l < Chm[lJllm+l /3 e .m+l(),

and

(4.4) infAeV

IIA-/hll < ChmllAIl,+l V A E Hm+l(f)n H(f).

We may consult [11] for conditions on the finite element partitions such that (4.1)-(4.4)are satisfied.

Finite element approximations are then defined as follows: seekh sh and Ah

Vh such that

(4.5)


and

(4.6)[div

Ahdiv fith + curl Ah. curlh + [hlAh "Ah

+--d ((h)*vh v(h)*)

fa H. curl .hdf V h Vh.

Note that (4.5), (4.6) is merely a discretization of (3.1), (3.2); as was seen in 3.3, thelatter is a weak formulation of the GL equation with the gauge div A 0 in 9t andA.n 0 onF.

4.1. Quotation of some results concerning the approximation of a class of nonlin-ear problems. The error estimates to be derived in 4.2 make use of results of [7] and[12] (see also [20]) concerning the approximation of a class of nonlinear problems. Sim-ilar results may be also found in [17] and [18]. Here, for the sake of completeness, wewill state the relevant results, specialized to our needs. The nonlinear problems to beconsidered are of the type

F(a, u) =_ u + TG(g, u) O,

where T B(Y; X), G is a C2 mapping from A x X into Y, X and Y are Banachspaces, A is a compact interval of IR, and 13(Y; X) denotes the space of bounded linearoperators from Y into X. We say that {(n, u(n)) : A} is a branch of solutions of(4.7) if n u(n) is a continuous function from A into X such that F(, u()) 0. Thebranch is called a regular branch if we also have that DF(e;, u(e;)) is an isomorphismfrom X into X for all A. Here, D, denotes the Frech6t derivative with respectto u.

Approximations are defined by introducing a subspace Xh c X and an approximat-ing operator Th B(Y; xh). Then, we seek uh Xh such that

(4.8) Fh(e;, uh) uh + ThG(n, uh) O.

We will assume that there exists another Banach space Z, contained in Y, with continu-ous imbedding, such that

(4.9) DG(,, u) B(X; Z) V ff A and u X.

Concerning the operator Th, we assume the approximation properties

(4.10) ’---II(Th T)v]lx 0 V v elim Yh--+0

and

(4.11) lim II(Th T)ll(z;x O.h--,0

Note that (4.11) follows from (4.10) whenever the imbedding Z c Y is compact.We now may state the results that will be used in the sequel.THEOREM 4.1. Let X and Y be Banach spaces and A a compact subset of ]IL As-

sume that G is a C2 mappingfrom A x X into Y and that all second Frecht derivativesof G are bounded on all bounded sets of A x X. Assume that (4.9)-(4.11) hold and that{(n, u()); A} is a branch ofregularsolutions 0]’(4.7). Then there exists a neighborhood

GINZBURG-I.ANDAU MODEL OF SUPERCONDUCTIVITY 73

(9 ofthe origin in X and, for h sufficiently small, a unique C2 function e; uh(e;) E Xh,such that {(, uh()); n E A} is a branch ofregularsolutions 0]’(4.8) and uh(n) (e;) 0for all e; A. Moreover, there exists a constant C > O, independent of h and n, such that

(4.12) Ilu(e;)- uh(e;)llx < CII(Th -T)G(n, u())llx v e A.

4.2. Error estimates. We begin by recasting the weak formulation (3.1), (3.2) andits discretization (4.5), (4.6) into a form that fits into the framework of 4.1. Let

X--- -l(a) x H(f) Y= (7-/1(f)) (I-In(f)) ’,Z :/3/2() x L3/2() and xh= Sh x Vh,

where (.)’ denotes the dual space. Note that Z c Y with a compact imbedding.tthe operator T B(Y; X) be defined in the following manner: T(, P) (0, Q)

for (, P) Y and (0, Q) X, if and only if

(4.13) (VO. V* + V. VO* + O* + O*)dO (f* + f*)dfl V E 1()

and

(4.14) fa (div Qdiv + curl Q. curl K)da fa P"It is easily seen that (4.13) and (4.14) are weak formulations.ofo uncoupled Poisson-pe equations for 0 and Q, and that T is the solution operator of these equations. In-deed, (4.13) is a weak formulation of

(4.15) -A+O= in and V0.n=0 onF

and (4.14) is a weak formulation of

(4.16) -AQ=P in and Q.n=0 and curlQxn=O onF.

tthe operator Th B(Y; Xh) be defined in the following manner: T(, P) (0h, Qh)for (, P) e Y and (0h, Qh) Xh, if and only if

[v0 + v(0 ) +(4.17)

+ V

and

(4.18) (div Qhdiv Xh + curl Qh. curl.h)df p. khdf g Xh Vh

Now, (4.17) and (4.18) consist of two discrete Poisson-type problems that are discretiza-tions of (4.13) and (4.14), respectively; also, Th is the solution operator for these discreteproblems.

Concerning the operators T and Th, we have the following result.LEMMA 4.2 The operator T is well defined by (4.13), (4.14). Let the finite element

subspaces satisfy the inclusions Sh c 7-l (f) and Vh C I’Iln (’). Then, the operator Th is


also well defined by (4.17), (4.18). Let the finite element spaces satisfy (4.1), (4.2). Then,ashO,

(4.19) II(T- Th)(,P)[Ix -, O.

Also, if the finite element spaces satisfy (4.3), (4.4) and if (0, Q) T(,P) satisfies(0, Q) E 7"/"+1(9t) x Hm+l(fl) N H()), then

(4.20) I](T- Th)(,P)I]x < Ch’(llll,+ + IlAllm + ).

Proof. Bywell defined we mean, for example, that T does indeed belong to B(Y; X).The left-hand side of (4.13) defines a Hermitian, positive definite sesquilinear form on7-/(2) x 7-/(fl) and the left-hand side of (4.14) defines a symmetric, positive definitebilinear form onH(fl) H(2). Moreover, whenever (, P) E Y, the right-hand sidesof (4.13) and (4.14) define bounded linear functionals on 7-/ (fl) andH(fl), respectively.Thus, by the Lax-Milgram theorem, both (4.13) and (4.14) have unique solutions andthe solution operator is bounded, i.e., the operator T is well defined. Similarly, it can beshown that the operator Th is also well defined.

Standard finite element arguments applied to the pairs (4.13) and (4.17) and (4.14)and (4.18) imply that

I10-- oh]ll_

inf II0- hll(bh8h

and

Aevrespectively. Then, (4.19) follows from (4.1) and (4.2), and, if 0 7-/m+l(f) and QHm+l(f), (4.20) follows from (4.3) and (4.4). [:]

Let A be a compact subset of IR+. Next, we defined the nonlinear mapping G.A X Y as follows: G(n, (0, Q)) ((, P) for n. 6 A, (0, Q) 6 x, and (, P) 6 Y, ifand only if

(4.21)

and

n b*da fn [ ;2(1012 1) + Iql 1] O(p*da

*vo]

P. Adft IOl Q. A +(4.22)

fn(nH" curlh)dn V

It is easily seen, with the association u (, hA), that the system (3.1), (3.2) is equivalentto

(4.23) (, aA) + TG(a, (, aA)) 0

and that the discrete system (4.5), (4.6) is equivalent to

(4.24) (h, aAh) + ThG(a, (h, aAh)) 0.


Thus, we have cast our problem into the form of 4.1.A solution ((), A(t)) of (3.1), (3.2), or equivalently, of (4.23), is called regular if

the linear problem

(4.25) T-((P,e.) + D2G(n, (, A))(, n/i,) (,)

has a unique solution (, fit) E X for every (, ) E Y. An analogous definition holds forregular solutions of the discrete system (4.5), (4.6), or equivalently, of (4.24). In (4.25),DuG(n, (.)) denotes the Frech6t derivative of G with respect to the second argument.For given (0, Q) X,,a direct computation yields that (, P) Y satisfies

((, P) D2G(s, (0, Q))(, ()

for (, () x, if and only if

(4.26)

and

(4.27)

Remark. We will assume throughout that the system (3.1), (3.2), or equivalently (4.23),has a branch of regular solutions for n belonging to a compact interval of IR+. It can beshown, using techniques similar to those employed for the Navier-Stokes equations (see[37] and the references cited therein) that for almost all values of n and for almost alldata H, the system (3.1), (3.2), or equivalently, (4.23), is regular, i.e., is locally unique.See [31] for some other results in this direction.

Concerning the operator G(., .), we have the following result.LEMMA 4.3. Let A be a compact subset oflR+, and let G A X -- Y be defined

by (4.21), (4.22). Then, G is a C mappingfrom A X bto Y. Let D2G(., .), defined by(4.26), (4.27), denote the Frechdt derivative ofG with respect to the second argument. Then,for any (0, Q) X,

(4.28) D2G(n, (0, Q)) e B(Z; X)

Moreover, all second Frecht derivatives ofG are bounded on bounded sets ofA X.Proof. Clearly, G(n, (0, Q)) is a polynomial map in n, the components of Q, and the

real and imaginary parts of 0, and thus it can be shown that, in fact, G is a C mappingfrom A x X into Y.

Through an examination of (4.26), (4.27) we see that

D2G(t, (0, Q))(, () Cx + C2, C3 + Ca ),


where

Since 0 and belong to 7-( (f), we have that 0 and belong to E6 (f) and, for i 1,..., d,O0/Oz and OO/Oz belong to/22 (f). Likewise, since Q and belong to H1 (f), we havethat Q and ( belong to L6(f) and, for i, j 1,..., d, Oqj/Oz and 0tj/0z belong toL2(Vt). From these we may conclude that C1 E E(f), Cz E a/(f), Ca Lg(gt), and(34 La/2(f). Then, C + C Ea/(f) and Ca + (34 E La/Z(ft) so that (4.28) holds.

Next, we recall the well-known result, see, e.g., [20], [22], or [36], that wheneveru, v, w, and z belong to n (), then,

(4.29)

and

Ovu--w dfOxi

for some constants C and C. Then, a straightforward (but tedious) calculation showsthat, as a result of (4.29) and (4.30), all second Frech6t derivatives of G are bounded onbounded sets of A X.

Using Theorem 4.1, we are led to the following result.THEOREM 4.4. Assume that A is a compact interval of IR+ and that there exists a

branch {, (,A) t E A} ofregular solutions ofthe system (3.1), (3.2). Assume thatthe finite element spaces Sh and Vh satisfy the conditions (4.1)-(4.4). Then, there existsa neighborhood (.9 of the origin in X 7-l (ft) x H () and, for h sufficiently small, aunique branch {n, (h, Ah)) E A} ofsolutions ofthe discrete system (4.5),(4.6) suchthat (, A) (h, Ah) (.9 for all A. Moreover,

+ IIA(, ) Ah(, )llx --’ 0

as h --. O, uniform@ in tIf, in addition, the solution of the system (3.1), (3.2) satisfies (,A) 7-/’+(f) x

I-Im+ (f), then there exists a constant C, independent of h, such that

(4.39.) Ch( )ll + IIA( ) Ah( )llx <_ +

uniformly inProof. First, as a consequence of Lemma 4.2, we see that (4.10) and (4.11) are sat-

isfied. Then, (4.12) follows since the imbedding Z c Y is compact. The remaininghypotheses of Theorem 4.1 are verified in Lemma 4.2. Then, the present results followfrom Theorem 4.1. In particular, since (T- Th)G(, (, A)) -(,A), (4.31) and(4.32) follow from (4.19) and (4.20), respectively.

Remark. Using some additional results (that maybe found in [7], [12], and [20]) con-ceming the approximation of problems of the type (4.7), we may derive error estimates


for (b, A) in the [z (f) x LZ(f)]-norm. In particular, if solutions of the (formal) adjointof (4.25) can be shown to belong to 7-/z(f) x HZ(f) whenever (,) (f) x L(f),then it can be shown that

Thus, in this case, the error measured in the [9. (f) LZ (9t)]_norm converges at a higherrate (as h 0) than does the error measured in the [(f) H (f)]-norm.

Remark. Any of the many standard iterative methods for the solution of nonlinearsystems of equations may be employed to solve the nonlinear discrete system (4.5), (4.6).For example, using the notation of (4.25), Newton’s method is defined as follows. Givena value for and an initial guess ((),A()) for ()h, Ah), the sequence of Newtoniterates {(b(), A())}> is defined by

T-((s+), A(s+)) + D2G(a, (b(S),A(S)))((s+l),A(s+))G(t, ((), A())) n2G(a, (b(), A()))((), A()) for s 0, 1,...,

It can be shown (using techniques similar to those employed for the Navier-Stokes equa-tions [20]) that if the initial guess (b() A()) is "sufficiently" close to a nonsingular so-lution (ph, Ah) of the discrete system (4.5), (4.6), then the Newton iterates converge to(bh, Ah) with a quadratic rate of convergence. Good initial guesses may be generatedby using, for example, continuation methods wherein the solution at one value of n isused to generate a good initial guess for the solution at another value of n. Modern con-tinuation methods also have the ability to compute the solution, as a function of , evenin the presence of bifurcation points.

5. The periodic Ginzburg-Landau model. From a practical viewpoint, the modeland associated finite element method presented and analyzed above are of use mostlyfor type I superconductors. In this case, solutions are relatively simple in structure, withmost of the interesting phenomena occuring near the boundary. Thus, with perhapssome grid refinement near the boundary, the finite element discretization of 4 wouldbe effective. On the other hand, solutions to the GL model for the case of type II su-perconductors exhibit much more complicated structures, e.g., vortex-like phenomena,with variations occuring over the order of 103 ,mgstroms. Thus, in any computation ofpractical utility, the grid size necessary to resolve these structures would be prohibitivelysmall. An alternate model, using quasi-periodic boundary conditions (see, e.g., [1], [15],[16], [28], and [31]), may be used for type II superconductors. Here, we briefly describethis periodic model. Further details and a discussion of finite element methods for thisalternate model will be the subject of a future paper.

We assume that f c IR2. The periodic model is based on the assumption that awayfrom bounding surfaces, certain physical variables exhibit periodic behavior. Specifically,it is assumed that the magnetic field (h curl A), the current (j ((1/n)Vb- A)II),and the density of superconducting charge carriers ([b[ 9) are periodic with respect to thelattice determined by the fixed vectors t and t2, i.e.,

h(x+tk) h(x), j(x+k)=j(x), and I(x+t)[ I(x)[, x IR2, k 1,2.

The direction and relative lengths of the lattice vectors tl and t2 are determined by thelattice symmetry. For example, for an equilateral triangular lattice, the angle between tand is 60 degrees, and the two lattice vectors are of the same length. (It is well known,


(X2 ’Y2)

(o,o) (X 1’ Y

FIG. 5.1. A cell ofthe lattice determined by the vectors tl and t2.

e.g., see any of the above papers, that the GL free energy is a minimum for a regulartriangular lattice.) The size of a lattice cell, and thus the absolute lengths of the latticevectors, is determined by the fluxoid quantization condition and is given by

where [fl, B, and n, respectively denote the area of the lattice cell, the average magneticfield, and the number of fluxoids carried by the each cell of the lattice.

The order parameter and the magnetic potential A can then be shown to satisfythe "quasi-periodic" boundary conditions

(5.1) (x + x k

and

(5.2) A(x + tk) A(x) + Tgk, x E IR2, k 1, 2,

where gk(x; tk), k 1, 2, is determined from the fluxoid quantization condition and, ifthere is one fluxoid associated with each cell, is given by

l(tk /k3) x,gk --- k 1,2,

where k3 denotes a unit vector perpendicular to the (z, y)-plane.Due to the "periodic" nature of and A, it is customary to focus on a single lattice

cell, such as one with a corner at the origin. Figure 5.1 provides a sketch of such a cell forthe equilateral triangular lattice, where without loss of generality, we can assume thatone of the lattice vectors is aligned with a coordinate axis.

With respect to the cell depicted in Fig. 5.1, the "periodicity" conditions (5.1), (5.2)imply that

A(x + x2, Y2) A(x, 0) + 1/2/}(x2k2 y2kl) for 0 < x < x,

((5.4) A x +--y,y A --y,y + /xk2 for0<y<y2y2 y2

(x + x2, y2) (x, 0)exp{-i1/2e;y2x} for 0 < x < xx,


and

(5.6) Xl -I- Y’Y2 y 22Y’ y exp i-e;Bxy for 0 < y < Y2

where kl and k2 denote the unit vectors in the direction of the x and y axes, respectively.The "periodic" GL model is then to minimize the GL free energy (2.16) over all.

appropriate functions and A, i.e., functions having one square integrable derivative,that also satisfy (5.3)-(5.6), where in (2.16), f now denotes the cell depicted in Fig. 5.1.The application of standard techniques of the calculus ofvariations then yields that min-imizers satisfy the GL equations (2.17), (2.18). Of course, (5.3)-(5.6) appear as essentialboundary conditions. The minimization process also yields natural boundary conditions;it is easy to see that these imply that the current and the magnetic field are periodic withrespect to the lattice. Since (5.5), (5.6) imply that I1 is likewise periodic, we see thatminimizing (2.16) over functions satisfying (5.3)-(5.6) yields all the required physicalperiodicity conditions.

There is one difficulty still to be overcome. The average magnetic field is given by

B - h eft - curl Ada,

where, since we are dealing with a planar problem, h (0, 0, h)T and curl A may beviewed as a scalar-valued function. Obviously, B depends on the solution ofthe problem..On the other hand, the cell f and the lattice vectors t and t2 that are used to define theproblem, depend on B. Thus it seems that not everything, needed to pose the "periodic"problem is known. This difficulty is circumvented as follows. Instead of specifying andthe constant applied field H, we specify t and the average field B. Note that if H is aconstant, it does not explicitly appear in the specification of the problem. Of course, itis also necessary to set the number n of fluxoids carried in each cell. For the triangularlattice determined from the cell depicted in Fig. 5.1, n 1. With n, B, and specified,all the data in the specification of the "periodic" problem are known. The applied fieldcorresponding to the solution obtained is then easily deduced; see, e.g., [14].

Most of the results of 3 and 4 can be extended to the periodic model. As wasindicated above, details will be provided elsewhere.

Acknowledgment. The authors wish to thank Dr. James Gubernatis of the Theo-retical Division of the Los Alamos National Laboratory for many helpful and interestingdiscussions.

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GINZBURG-I_ANDAU MODEL OF SUPERCONDUCTIVITY 81

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