Finite element approximation of a periodic Ginzburg-Landau ...mgunzburger/files... · [12] which uses a Monte Carlo/simulated annealing approach and represents the best known effort

Numer. Math. 64, 85-114 (1993) Numerische MathemaUk ~ Springer-Verlag 1993

Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors Qiang Du 1, Max Gnnzburger 2, and Janet Peterson 2 1 Department of Mathematics, Michigan State University, East Lansing, MI 48224, USA 2 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA

Received January 21, 1992

Summary. We consider efficient finite element algorithms for the computational simulation of type-II superconductors. The algorithms are based on discretizations of a periodic Ginzburg-Landau model Periodicity is defined with respect to a non-orthogonal lattice that is not necessarily aligned with the coordinate axes; also, the primary dependent variables employed in the model satisfy non-standard "quasi"-periodic boundary conditions. After introducing the model, we define finite element schemes, derive error estimates of optimal order, and present the results of some numerical calculations. For a similar quality of simulation, the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods.

Mathematics Subject Classification (1991): 65N30, 35J60

1. Introduction

Scientists, engineers, and mathematicians who wish to design superconducting devices and/or who wish to study the physics of superconductivity are in need of robust and efficient algorithms for the numerical simulation of superconducting phenomena. Our long-range goal is to develop methods with these properties that are applicable to high-temperature superconductors. A shorter-range goal that is a necessary stepping stone towards meeting the ultimate goal is to develop and implement methods for low-temperature superconductors; in this paper we report on our effects towards meeting the second goal.

For a broad introduction to the subject of superconductivity, the reader may consult [4, 7, 10, 13, 19, 22, and 24]. Superconductors are divided into two types, known as type-I and type-II superconductors. From a technological point of view, type-II superconductors are of greater interest, mainly because, compared to type-I superconductors, they can maintain superconducting properties in the presence of

Correspondence to: M. Gunzburger

86 Q. Duet al.

higher external magnetic fields. Here we are particularly interested in periodic Ginzburg-Landau models for type-II superconductivity. These macroscopic models are among those of greatest interest in the design of superconducting devices such as wires, films, and computer chips. Such models have been used in the past as a setting for analyzing and approximating phenomena in type-II superconductors; see, e.g., [1, 5, 12, 15, 17, 18, 20, and 21]. Most of these deal with some sort of series solution of a periodic Ginzbug-Landau model. One notable exception is [12] which uses a Monte Carlo/simulated annealing approach and represents the best known effort so far for simulating type-II superconductors.

Of relevance to the present work are the observations that electromagnetic phenomena in type-I superconductors may be described by relatively smooth functions that can be well-approximated through the use of standard discretization techniques, e.g., finite element methods, while on the other hand, such phenomena in type-II superconductors exhibit significant variations over length scales of the order of 1000 or fewer Angstroms. Suppose now that one wishes to effect a numerical simulation of electromagnetic phenomena in a sample of superconducting material which is part of some device; typically, the size of this sample would be of the order of a centimeter or more. For a type-I superconductor, this poses little problem since, except for possible edge effects, all variables to be approximated are smooth with respect to the scale of the sample size. Finite element methods for just this situation are considered in [13]. However, for a type-II superconductor, it is not possible, using currently available computers, to simulate phenomena exhibi- ting spatial variations having length scales of the order of 1000 Angstroms. For example, if one were to use a finite element method, then the number of degrees of freedom necessary to well-approximate such functions would be prohibitively large.

The inability to perform simulations for typical material samples of type-II superconductors gives rise to the use of the common practice of neglecting the effects due to the fact that the sample has boundaries. Thus, one assumes that one is far removed from the boundary of the superconducting sample, and that in such regions the physically relevant variables, e.g., the magnetic field, the current, etc., are, in some sense, periodic. The use of a periodic model allows one to focus on a piece of the sample that is of roughly the same size as that of the scale of variations in the interesting phenomena. In this case it is possible to resolve these phenomena on currently available computers.

The particular periodic Ginzburg-Landau model considered here is studied, in detail in [14] and is closely related to the models used in [1, 5, 12, 15, 17, 18, 20, and 21]. From a numerical analysis viewpoint, one of the more interesting features that one must deal with is the periodicity of the model. In the first place, this periodicity is not necessarily with respect to a rectangular lattice, but may involve, e.g., triangular lattices. In fact, we consider general lattices so that all possible topolo- gies for periodicity in the plane can be treated. Perhaps of even greater interest is the fact that the primary variables employed in the model are not themselves periodic. One variable is scalar, complex-valued, and its phase suffers a jump across the lattice defining the periodicity, while another is vector-valued and itself suffers such a jump.

The plan of the rest of the paper is as follows. In the remainder of this section we introduce the notation that is to be used throughout the rest of the paper. In Sect. 2 we describe the periodic Ginzburg-Landau model. In Sect. 3 we develop finite element algorithms for the approximation of solutions of the equations of Sect. 2.

Finite element methods for type-II superconductors 87

We also show how the problem in hand can be put into a framework that allows us to apply some useful known results. We then provide, in Sects. 4 and 5, estimates for the difference between the exact solution and finite element approximations. Finally, in Sect. 6, we illustrate the effectiveness of our algorithms by presenting the results of some computational experiments.

1.1 Preliminaries

Throughout, we only deal with ]R 2. Thus, all vectors will have only two components, say in the x~ and x2 directions, and these, as well as all scalar-valued functions, will depend only on the two variables xl and x2. We adopt the convention that the vector product of the two vectors a = (al ,a2) T and b = (hi, b2) T is the scalar a x b = alb2 - a2bl. Since we are dealing with the plane, it is convenient to introduce two curl operators:

curl A cOA2 ~A1 ~x2 - - ~ X 1 (~X2 and curl ~k = d@ / "

We also have the divergence and gradient operators defined by

OA1 t~A2 ~x1 div A = ~ + c~x~- and grad ~ = | c~r '

\ respectively.

Given two arbitrary vectors tl and t 2 that span ]R 2, we say that a function f ( x ) is periodic with respect to the lattice determined by t~ and t2 if

(1.1) f ( x + tk) =f (x ) for k = 1, 2 and Vx ~. ]R 2 .

Here, f may be scalar or vector-valued and may be real or complex-valued. The vectors tk, k = 1, 2, will be referred to as lattice vectors; without loss of generality, we assume that the counter-clockwise angle between tl and t2 is less than ~. For the sake of brevity, we will often refer to functions satisfying (1.1) merely as being periodic. We will not need to explicitly specify the lattice vectors tl and t2 until we consider, in Sect. 6, computational examples.

Given any point P e ]R 2, a cell o f the lattice with respect to the point P is the open parallelogram f2 e c IR 2 depicted in Fig. 1. The boundary of the cell f2v is denoted by Fe. When P corresponds to the origin, we denote the corresponding cell by 12 and its boundary by F. We denote by If21 the area of the cell f2v; note that If21 is independent of the choice of P.

We will make use of the following function spaces. First, for any bounded set e IR 2, we have, for any nonnegative integer m, the Sobolev space Hm(N) consist-

ing of square integrable functions over N such that all derivatives of order up to and including m are also square integrable. We also have the space ~ff'~(N) of complex-valued scalar functions having real and imaginary parts belonging to

88 Q. D u e t al.

x2

!

x 1

Fig. 1. The cell ~e determined by the lattice vectors t~ and t2 and the point P

H=(~) and the space H " ( ~ ) of real vector-valued functions having components belonging to Hm(~). Norms for functions belonging to H " ( ~ ) , oVf"(D), and / / ~ ( ~ ) will be denoted by [l'][,,,~; when there is no chance of confusion, we will omit the subscript ~ from this notat ion for norms. Then we may define the spaces

H~oc(]R 2) ---- {t~:]R 2 --~ ]R[~ 6 Hra(~) V bounded ~ = jR2},

~Fo~(~ 2) = {@:N 2 ~ CEIl(@), Z(~) s Hm(~) V bounded ~ = IR 2} ,

where ~ ( . ) and Z ( . ) denote the real and imaginary parts, respectively, and

m 2 H1or ) = {A = (A1, A2)T: IR 2 ~ IR2[A1, A2 ~ Hm(~) V bounded ~ c ~2} .

Given the lattice vectors tk, k = 1, 2, we can also define, for m > 1, the space

m 2 Hv,r(R ) = (q~ ~ H~c(R2)[~b(x + tk) = q~(x) for k = 1, 2 and Vx ~ IR 2} ,

a n d the analogous spaces " 2 m 2 ,~per(~X ) and Hper ( ]R ) for complex and vector-valued functions. Details concerning these spaces may be found in, e.g., [2].

We now give a precise definition of gauge invariance. For any X e H2oc(IR2), let the linear t ransformation G x from ~ o o ( I R 2) x Hloc(P, 2) into itself be defined by

Gz( ~, A) = (~, ~) e YF~o~(IR 2) x H~oo (lR 2) V(~b, A) ~ O~ ~o~(]R 2) x Hlo~(R 2)

if 1

= ~Oe ix a n d Q = A + - grad X , K

where •, a parameter appearing in the model, is defined below. Note that if ((, Q) = Gx(~0, A), then (if, A) = G_~((, Q).


Definition. (d/, A) and ((, Q) are said to be 9auoe equivalent if and only if there exists a g E n2oc(lR 2) such that ((, Q) = Gz(~b, A).

2. A periodic Ginzburg-Landau model for superconductivity

We now present the periodic Ginzburg-Landau model that we intend to discretize. Details related to the derivation of the model can be obtained in the references cited in Sect. 1.

The variables employed in Ginzburg-Landau models for superconductivity are the real, vector-valued magnetic potential A and the complex, scalar-valued order parameter ~/. From these one may deduce (appropriately nondimensionalized) physical variables, e.g., the magnetic field h = curl A and the density of superconducting charge carriers Ns = t~/[ 2.

Periodicity assumptions about the physical variables can be expressed in terms of the magnetic potential A and order parameter ~, as

(2.1) curl A, I~bl, and ( 1 grad q~ - A) are periodic ,

where ~b denotes the phase of the order parameter, i.e., ~b = [~b]e i*, and x, the Ginzburg-Landau parameter, is a material constant whose value depends on the temperature. Recall that here and throughout, periodicity is defined with respect to the lattice determined by the vectors tl and t2.

The average magnetic field/~ over a cell ~e is defined by

~ 1 f curl A d a . B = 1 h d(2= ~ [ o e

Note that/3 is independent of the choice of cell, i.e., of the choice of the point P. The relation

xB] (21 = 2nn ,

between the average magnetic field and the area of a lattice cell results from the "fluxoid quantization condition"; here n, the number of fiuxoids associated with the lattice cell, is a positive integer.

Let

and

(2.3) Ok(x) = - ~ (x x t~), k = 1, 2 .

Then, if A and ff = I~,[e 14' satisfy (2.1), it is shown in [14] that (~,, A) is gauge equivalent to ((, Q - Ao) where

(2.4) I~1 and Q are periodic,

90

(2.5) div Q = 0 a.e. ,

and

(2.6) ~o(x "t- tk) - - O)(X) ----- tOOk(X), k = 1, 2 ,

Q. Duet al.

where co denotes the phase of if, i.e., ( = ]~[e i~ Fur thermore , the magnetic field h and the density of superconduct ing charge carriers may be recovered from ( and Q through the relations

(2.7) h = c u r l A = curl Q + / ~ and Ns : [~l z : [~l 2 �9

2.1 Weak and strong forms o f the Ginzburg-Landau equations

The Ginzburg -Landau equat ions are a consequence of the basic postulate of the Ginzburg -Landau theory, namely that a superconduct ing material is in a state such that the Gibbs free energy is minimized. Intermediate between the minimizat ion principle and the strong form of the Ginzburg -Landau equations is a weak form of these equations. It is the latter that forms the basis for defining finite element discretizations. Here we will merely provide these equations; details may be found in, e.g., [14-1.

Denote the four sides of the paral le logram f2e by F+ 1, F_ 1, F+ 2, and F_ 2, using the convent ion of Fig. 2. No te that for k = 1 or 2, F+k is the locus of points y ~ IR 2 such that y = x + tk for x ~ F-k.

We define the function spaces

~ef~(t-2e) = {( ~ ,Yf l((2j,)l((x -b tk) = ( (x)e iKaktx) Vx ~ F-k, k = 1, 2}

and

Hle(f2e) = {Q ~ H~ (I2e)[ Q(x + tk) = Q(x) Vx ~ F-k , k = 1, 2} .

n+ 2

o,

11. a

Fig. 2. Boundary segments and normal and tangential vectors of the cell g2 e


For integers m > 0, we define the spaces

~qua( ]R 2) : (~ ~ ~r -1- t k ) = ( (x )e iKaktx) for k = 1, 2 and u e IR 2} ,

s ~ ( O p ) = {fflaplC ~ X 'qu , (~-2)} , and

DT(f2e) = {Ola /Q ~ H ~ e r ( I R 2 ) } �9

It was shown in [14] that

~ ( O e ) = ~ ( O e ) , and /t~(f2e) = H~,(f2e), and, for m > 2,

#~(Qp) c ~m(f2e) m ~ ( f 2 e ) and /-~e(f2e) c /-/~(f2e) ~ H~(Qe). The weak formulation of the Ginzburg-Landau equations, in terms of the

variables ~ and Q introduced above and with respect to any lattice cell De, is given as follows. Seek (~, Q) ~ i f el (f2e) • H~,(f2e) such that

(2.8, ~effl{(igrad-Ao)~.(-igrad-Ao)~'}df2

+~pgt{:*(igrad-Ao):+((-igrad-Ao):'}'Qd f2 + j" (IQt 2 + Iffl 2 - 1)~t{~*}df2 = 0 V ~ Jf~(g2e)

~e

and

(2.9) (curl Q-curl {~ + div Qdiv Q + I(IZQ. Q)df2 f2e

+, ++o 0 + + + + + , f2p

are satisfied. In case ~ and Q are sufficiently smooth, (2.8) and (2.9) may be integrated by

parts to yield the strong form of the Ginzburg-Landau equations, as well as the associated natural boundary conditions. First, one obtains the differential equations

(2.10) (igrad- Ao).(igrad- Ao)~ + (,Q[2 + [~]2-1)~

. + 2 Q . ( i g r a d - A o ) ~ = 0 inf2e,

Q + [ ~ ] 2 Q + 9 t ~ . ( i g r a d _ A o ' ) ~ = 0 inf2e ,

k ~

curl curl ( \ x / )

(2.11)

and

(2.12) d i v Q = 0 inf2 e.

92

The essential boundary and side conditions are given by

((x + tk) = ((x)e i~gktx~ Yx e F-k, k = 1, 2 , (2.13)

and

(2.14) Q(X + tk) = Q(x) Vx e F_k, k = l, 2 .

The natural boundary conditions associated with (2.9)-(2.10), are

(2.15, e - ' ~ g k ' x ' ( ( i g r a d - Ao) ( + Q( ) x+, "n+k

+ ( ( i g r a d - A ~ ~+ Q~) x ' n - k = O

and

(2.16)

It can

Q. Duet al.

for x ~ F _ k , k = 1,2

curl(Q - Ao)lx + t~ = curl(Q - Ao)lx Vx e F-k, k = 1, 2 .

be shown that (2.13)-(2.16) imply that the physical variables, i.e., the magnetic field, the current, and the density of superconducting charge carriers are periodic across the lattice cell I2e.

Let ((, Q) s ~'~(t2e) x H~,(f2e) denote a solution of (2.8)-(2.9), i.e., a weak solution of the Ginzburg-Landau system (2.10)-(2.16). Then, it is shown in [14] that

and Q can be extended, through the repeated application of the relations

( ( x + t k ) = ( ( x ) e i~gk and Q(X+tk )=Q(x ) y x e R 2 , k = 1,2 ,

to IR 2 as infinitely differentiable functions and they satisfy the Ginzburg-Landau ec~uations in the classical sense. In particular, we have for any integer m > 0, ( .Z#~(12p) and Q e/~e(Oe).

3. Finite element approximations

Finite element approximations of solutions of the periodic Ginzburg-Landau model are based on discretizations of (2.8) and (2.9). We first define the finite element spaces that we will employ to effect such discretizations. Throughout, C will denote a positive, bounded constant whose meaning and value changes with context.

3.1 Finite element spaces

Given any two positive integers NI and N2, let J-h denote a subdivision of the parallelogram f2e into 2NIN2 triangles in the manner depicted in Fig. 3. The triangles in 37-, are denoted by t2 i, j = 1 . . . . . 2N~N2. Let h denote the largest diameter of any of the triangles in ~-,. (There should be no confusion arising from


Fig. 3. A subdivision of O e into 2NIN2 = 24 triangles; N: = 3, N2 = 4

the use of the symbol h to denote both the magnetic field and the grid size.) Denote the space of functions, defined on f2~, consisting of polynomials of degree less than or equal to r by P,(f2j).

We consider the two most practical choices for the underlying finite element spaces, namely spaces of piecewise linear and piecewise quadratic polynomials. Specifically, we define

V h " = {vEC~149 for r = 1 or 2 ,

,r {Vl~R(V),TZ(V)E V h''} for r = 1 or 2 ,

and vh'r= {V=(vl,v2)TlVl,I)2 ~. 1 flh'r} f o r r = 1 o r 2 .

We have that V h'" ~ Hl(~2e), V h'" ~ ~l(s '2e), and V h'r c H~(Op). Let Xh denote the set of nodes of the triangulation Y-h for the space V h''. Thus,

for V a' 1, x e Xh is a vertex of a triangle t2j ~ Y-r, while for V h' z, x �9 ~ h is a vertex or a midside of such a triangle. Denote by H, h the usual interpolation operator from C(~p) into V h'', i.e., for v �9 C(~e), Flh, v is the unique element of V h' ' such that IIh, v(x) = v(x) for all x e Xh. We assume that the triangulation Y'h is regular so that the following interpolation estimate holds: for r = 1 or 2 there exists a constant C > 0, independent of h and v, such that

(3.1) I lv- I lh , vlls<Ch~+l-Sllvll,+ x V v � 9 0 < s < l ,

where throughout [I" I[~ = qL" lit, ~e, i.e., tacitly, all norms are defined with respect to the lattice cell t2p. See [3] or [8] for details concerning (3.1). We will also denote the interpolation operators into V h'" and r by/-/h. Then, (3.1) obviously also holds for the components of functions belonging to V h'" and for the real and imaginary parts of functions belonging to uh, r.

In the approximate problem, in order to account for the "periodicity" con- straints imposed on functions belonging to 9g'~(f2e) and H~,(I2e), we introduce the spaces

~ , " = {v �9 ~ h " [v ( x + tk) = ei~~ Vx e JV~ c3 r -k , k = 1, 2}

f o r r = l o r 2 ,

94 Q. Du et aL

and

v ~ r = { v E vh 'r [ V(x + tk) = V(x) V x e ~A#h ~ F_k, k = 1,2}

f o r r = l o r 2 .

Functions belonging to ~ are required to satisfy the constraint (2.13) only at the nodes on the boundary of f2e. Since such functions do not necessarily satisfy this constraint at other points on the boundary, we have that ~ r r J(c~el(t2e). However, one can still obtain the following approximation results.

Lemma 3.1. L e t v e )~'~,(12e); then, f o r r = 1 or 2,

(3.2) inf [ ] v - v h I l l ~ O a s h ~ O . V h E 'Y/',~"

L e t v e ~ e + i (f2e); then, f o r r = 1 or 2,

(3.3) inf I I v - v n l l s < h ' + i - ~ l t v % + l 0 _ < s < l . V h e 1r r

Proof. Although ~e'h'" ~: ~el(f2e), we do have that H h v e ~ h , r for any v e �9 ~ , + ~ (f2e). Then, (3.3) follows by applying (3,1) to the real and imaginary parts ofv. To obtain (3.2), we can employ a standard density argument. Let ~ec~ ~ (Op) ~ ~el(f2e), where the space c~(~e ) has an obvious definition. Then, by the triangle inequaIity,

I I v - nh'711x _--< I I v - ~ltl + I1~- //~11~ �9

Through an appropriate choice of ~, the first term on the right can be made as small as one wishes due to the density of cg~176 c~ ~ei(f2e) in ~ ( O p ) ; as h ~ 0, (3.1) implies that the second term on the right also vanishes. Thus, (3.2) holds. []

For the space V h'r w e do have that Vhe '" c Hl,(f2e) so that one easily obtains the following result by applying (3.1) to the components of V.

Lemma 3.2. Le t V e H~(f2e); then, f o r r = 1 or 2,

(3.4) inf I t V - rI11--*0 ash- -*0 . V h ~ ~V'~ r

L e t V e H~,+l(I2e); then, f o r r = i or 2,

(3.5) inf I[ V - rl[~ _-< h'+l-~ll vii,+1 0 < s _< 1.

We also assume that the space V h'' satisfies the standard inverse assumption so that the following inverse inequality holds: for any triangle side Fh

(3.6) llvhll~,rh =< Chq-Sllvhllq, rh Vvhe V h'r and 0 __< q _< s _< r .

For details concerning (3.6), one may again consult 1-3] and I-8]. Obviously, these inequalities also hold for the components of functions belonging to V h'" and for the real and imaginary parts of functions belonging to ~h, , .


3.2 The approximate problem

Once the approximating spaces V~' r and Vhe '" are chosen, finite element approximations are defined in the usual manner, starting with the weak formulation (2.8) and (2.9). Thus, we seek (h e ~h , r and Qh e V h'" such that

(3.7, ~ a e 9 1 { ( i g r a d - A o ) ( h . ( - i g r a d - - A o ) ( h . ) d f 2

+ J" (IQhl z + ICht = - a)91{~h~h*}dQ = 0 V~he3 U'h'r f2e

and

(3.8) /

5 (curl Qh-curl 0 h + div Qh.div Q. h + I(hfZQh. O a t2p \ {(i )}) +9t (h* grad--Ao ~n .0h dO=0 u r.

For the computational examples reported on in Sect. 6, we use piecewise quadratic polynomials, i.e., r = 2.

(H1)

(n2)

and

3.3 Quotation of results concerning the approximation of a class of nonlinear problems

The error estimates to be derived in Sects. 4 and 5 make use of results of [6] and [9] (see also [16]) concerning the approximation of a class of nonlinear problems. Here, for the sake of completeness, we will state the relevant results, specialized to our needs.

The nonlinear problem considered is of the type

(3.9) F(K, u) -- u + TG(~c, u) = 0.

Let X and Y denote two Banach spaces and let A denote a compact interval of IR. Let ~(Y; X) denote the space of bounded linear operators from Y into X. Assume that

T ~ ~'(Y; X ) ,

G is a C 2 mapping from A x X into Y,

(H3) all second Frech6t derivatives of G are

bounded on all bounded sets of A x X .

We say that {(x, u(•)): x e A} is a branch of solutions of (3.9) if x ~ u(~) is a continuous function from A into X such that F(x, u(x)) = 0. The branch is called a regular branch if we also have that DuF(X, u(x)) is an isomorphism from X into X for all r e A. (Here, Du denotes the Frechet derivative with respect to u.)

96 Q. Du et al.

Approximations are defined by introducing a subspace X h c X and an approximating operator

(H4) T h ~ .~(Y; X h ) .

Then, we seek u h ~ X h such that

(3.10) Fh(x, u h) -- u h + ThG(~c, u h) = 0 .

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

(H5) D~G(K, u) E ~(X; Z) VK ~ A and u ~ X .

Concerning the operator Z h, we assume the approximation properties

lim I](T h - T ) v l l x = 0 V v ~ Y h ~ 0

(H6)

and

(n7) lim tI(T h - T)ll~tz~x~ = O . h- ,O

Note that (H7) follows from (H6) whenever the imbedding Z c Y is compact. We now may state the first result of 1.6] or 1.9] that will be used in the sequel.

Theorem 3.3. Let X and Y be Banach spaces and A a compact subset of lR. Let Z be a Banach space contained in Y, with continuous imbedding. Assume that (H1)-(H7) hold and that {(K, U(K)); K ~ A} is a branch of regular solutions of(3.9). Then there exists a neighborhood 0 o f the origin in X and, for h sufficiently small, a unique C 2 function ~c ~ uh(r.) ~ X h, such that {(x, uhffc)); K ~ A} is a branch of regular solutions of(3.10) and Uh(K) -- U(tC) ~ O f of all ~: e A. Moreover, there exists a constant C > O, independent o f h and K, such that

(3.11) ][u(K)- -uh(r ) l lx<C][(Tn-- T)G(K,u(x))lIx V x ~ Z .

For the second result, we have to introduce two other Banach spaces H and W, such that W c X c H, with continuous imbeddings, and assume that

(H8) for all u ~ W, the operator DuG(x, u) may be extended as a linear operator of ~ (H : Y), the mapping u ~ DuG(x, u) being continuous from W onto ~ (H : Y).

We also suppose that

(H9) lim [I T h -- Tl l~v;m = O, h ~ 0

(H10) for each x E A, u(x) e W and the function x --, u(K) is continuous from A into IV,

and

(H11) for each r ~ A, DuF(K, u(K)) is an isomorphism of H .

Then we may state the following additional result of 16] or [9-1.


Theorem 3.4. Assume the hypotheses of Theorem 3.3 and also assume that (H8)- ( H l l ) hold. Then, for h sufficiently small, there exists a constant C, independent of h and ~, such that

(3.12) !luh(~c) - - u(~c)lI~ ~ C i l ( T ~ - - T)G(x,u(x))I[u + Ilu~(~:) - u(~c)l i~ W c ~ A

3.4 The periodic Ginzburg-Landau model in the framework of Sect. 3.3.

We now show that the weak formulat ion (2.8)-(2.9) and its discretization (3.7)- (3.8) fit into the f ramework presented in Sect. 3.3. To begin with, we let

X = J/~l (f2e) x He~(Oe), Y = ( ~ i ( O e ) ) ' x (H~,(f2e))',

X h = ~/ 'h ,r X ~p,r C: X , Z = ~3/2(~C~p)X L 3 / 2 ( O p ) ,

H = ~La2(Q/,) x L2(QI,), a n d W = ~ 2 ( Q e ) • H2(Qp) ,

where ( .) ' denotes the dual space. For (~, Q) e x or H, we have the norms

ll(~, Q)llx = ll~l[~ + [IQIti and [l(~, Q)Iln = [l~llo + IlQJ[o,

respectively. Note that Z c Y and W c X c H, and that the imbeddings Z c Y and X c H are compact .

We define the opera tor T : Y - + X as follows. For any ( p , F ) e Y , (rl, R) = T(p, F) e X if and only if

(3.13) ~ 9~ {(igrad - A o ) r / - ( - i g r a d - -4o)~'* + ~/~*} dr2 f2e

f2e

r/(x + tk) = r/(x)e iak(x) Vx e F-k, k = 1, 2 , (3.14)

and

(3.15)

where

(curl R . curl 0 + div R- div 0 + R . {~)df2 = ~ F . 0 d[2 V {~ e m ( O e ) , Qp I2e

,n(x2) Jo = ~Ao = -xl = I-5] -x,

and K/~ 7~n

~k(X) = Xgk(X) = -- ~ iX X tk) = -- ~I (X X tk), k = I, 2.

Note that the parameter tc does not explicitly appear in the definitions of ,4o or ~k(X). Also, note that Tmaps Yinto J~ei(f2e) x H~,(f2e) c X. In fact, the domain of Tmay be taken to be (~ei(t2p)) ' x (H~,(f2p))' ~ Y; we have restricted the domain of T to Y in order to account for the fact that "//'he" r Jc6ei(t2e).

One easily sees that (3.13)-(3.14) is a weak formulation of the problem

(3.16) (igrad - ,~o) -(igrad - Jo)~/+ r/= p in f2e,

(3.17) r/(x + tk) = r/(x)e i~k(x) Vx e F-k, k = 1, 2 ,

98 Q. D u e t al.

and

(3 .18) e i~ tx) ( ( ig r a d - - .,4o)r/)jx + tk " n + k

+ ((igrad - , ~ o ) r l ) l x ' n - k = 0 for x e F - k , k = 1, 2 ,

and (3.15) is a weak formulation of the problem

(3.19) - A R + R = c u r l c u r l R - grad d i v R + R = F in [2e,

(3.20) R(x + tk) = R(x) VX ~ F-k, k = 1, 2 ,

and

(3.21) curlRlx + tk = curlRIx Vx e F-k, k = 1, 2 .

Thus, the operator T is the solution opera tor of the system of second-order elliptic partial differential equations and boundary conditions (3.16)-(3.21). Note that (3.13)-{3.14) and (3.15) are uncoupled, as are (3.16)-(3.18) and (3.19)-(3.21). Also, note that T is independent of x.

We define the operator Th: Y--- ,X h as follows. For any ( F , o ) e Y, (rl h, R h) = T(p, F) ~ X h if and only if

(3.22)

and

I !R {(igrad-.4o)~/a- ( - igrad - A o ) ( % + ~/h~'h,} d~2 ~p

= I {o&}da ,r Oe

(3.23) j" (curl R h- curl O h + div R h. div O h + R h" O h) dQ Oe

= f r . 02da v u h , " .

~e

Note that (3.22) and (3.23), which are uncoupled, are discretizations of (3.13)- (3.14) and (3.15), respectively.

Let A denote a compact interval of R +. Then, we define the nonlinear operator G: A x X ~ Y as follows. For any ~c e A and ((, Q) e X, (p, F) = G(x, ((, Q)) ~ Y if and only if

(3.24)

and

~ { p ( * } d I 2 = - t r j" 2 1 ~ { ( ' * ( i g r a d - . 4 o ) ( } - Qdf2 Qe ~e

+ ~ [1 - x2( le[ 2 + [(I 2 - 1)]!R {(('*} d~2 ~e

(3.25) F. OdY2 = j ' (1 - - 1~12)Q-Qd! f2 Q,e Qe

_ 1 !R I { [* ( igrad - A o ) ( } " Qdf2 K ~e

vO


One easily sees that (3.24) and (3.25) are weak formulations of

p : ( - - K2(IQ[ 2 4 - I ( [ 2 -- 1)( -- 2xQ.( igrad - ,4o)( in f2e and

F = Q - I ( I2Q - - L r {(*(igrad - -4o)(} in t2e, K

respectively. The combination of (3.13)-(3.15) and (3.24)-(3.25) implies that (2.8)-(2.9) may be

expressed in the form (3.9) where u = ((, Q). Likewise, the combination of (3.22)- (3.23) and (3.24)-(3.25) implies that (3.7)-(3.8) may be expressed in the form (3.10) where u h = ((h, Qh). In order to apply Theorems 3.3 and 3.4, we need to verify that the hypotheses (H1)-(Hll) hold in the present setting. In the next section, we consider the hypotheses associated with the linear operators T and T h, defined by (3.13)-(3.15) and (3.24)-(3.25), respectively.

4. Convergence and error estimates for the associated linear problem

The problems (3.13)-(3.14) and (3.15) uncouple, as do the approximate problems (3.22) and (3.23). Therefore, we consider (3.13)-{3.14) and (3.22) separately from (3.15) and (3.23), starting with the former pair.

4.1 The linear problem f o r the order parameter

(3.13)-(3.14) may be expressed in the following form: for p E(a~l(t2p)) ', seek q e ~ ( f 2 e ) such that

(4.1) a(r/. C') = 91{(p, (')} V~'e ~r

where the bilinear form a(-,-):~ff1(t2e)x ~ t ( f 2 e ) ~ / R is defined by

a(q, (') = S 9t {(igrad r / - .ior/). ( - - igrad ('* - -Jo('*) + r/~} dr2 Vr/, ( e ~ 1 (i2e) Qe

and where (p, (') = ~oe P~'* dr2. In the notation of (4.1), (3.22) may be expressed in the following form: for

p e (,~ffl(f2e))' and for r = 1 or 2, seek r/k e "t ~h'" such that

(4.2) a(r/h, ~'h) = 91{(p, ~'h)} v(h e -/F~,'.

Standard variational arguments may then be used to obtain the following result. (Note that, on ~le(t2e) or on uh, r, r = 1 or 2, ~ q) defines a norm equivalent to the standard 1-norm. Also, I[" [[, denotes the norm on the dual space (~e 1 (~) ) , . )

Proposition 4.1. Let r = 1 or 2. Then, for p e (~'~l(t2e))', the problems (4.1) and (4.2) have unique solutions rle ~ ( f 2 p ) and qhe ~ , r , respectively. Moreover, IIr/lt~ 5 Cl lp l l . and llr/hll~ < C l lp l l . .

100 Q. Duet al.

With regard to the convergence of the solutions of (4.2) to solutions of (4.1), we have the following result.

Proposition 4.2. Le t r = 1 or 2. For p e (~vfl(Qp)),, let r~ e ,g~(Y2v) and ~I h e ~Vhe '" denote solut ions of(4.1) and (4.2), respectively. Then

(4.3) lim II r/ - - r/h It1 = 0 . h ~ O

Proof. From Proposition 4.1 we see that {r/h} forms a uniformly bounded sequence in ~l( f2p) . Thus, it has a weak limit 4. For any ~'~ ~ffle(f2e), we have, using (4.2),

a(r ~') - ~R {(p, ~')} = a(r - r/h, ~') + a(r/h, ~ ' - ~'h) + a(r/h, ~h)

- ( ' ) } + m { ( p , C h - C)}

= a(r -- r/h, ~') + a(r/h, ~ ' - ~'h) + ~{(p , ~'h _ ~')},

where ~'h e ~//'~," is arbitrary. Then, as a result of (3.2) and the weak convergence of r/h to 4, all the terms on the right-hand side of the last equation vanish in the limit h --* 0. Thus, ~ is the weak solution of (4.1), i.e., r = r/. This proves the weak convergence. As a result of (4.1) and (4.2), we have that

a(r/, r/) -- a(r/h, r/h) = 0t{(p, r/ -- r/h)} ,

SO that the weak convergence implies convergence in norm and the strong convergence (4.3) follows. []

We now turn to error estimates in case the solution r/of (4.1) is smoother. Note that, since ~V'~'" ~: ~el(f2e), we cannot choose, in (4.1), ~ to be an element of ~ " . T h e r e f o r e , we cannot obtain the usual finite element "orthogonali ty" relation. Instead, we proceed as follows. First, it is easily shown that if r/e ~2(f2e) is a solution of (4.1), then

(4.4) a(r/, ~') + b(r/, ~') = ~N{(p, ~')} V~'e ~Vfl(f2e),

where the bilinear form b( . , . ): ~'~2(t2p)x ~'~l(f2e)~ ~ is defined by

b(r/, ~') = S ~.R{i(igradr/- .4o~/)-n~'*} dF Vr/e ~2([2e) and ~'e ~{'~t(Oe). s

Note that unlike the case of(4.1), in (4.4) the test functions do not need to satisfy the "quasi"-periodicity condition (3.14). Also,

(4.5) b(r/, ~) = 0 Vq e , ,~ , ( f2e) , ( e .g~,(f2e) .

Instead of the usual "orthogonali ty" relation, one obtains from (4.2) and (4.4) that

(4.6) a(q - q*, ~'h) = _ b(r/, ~'h) V~'h e ~/'hp, r .

The key to obtaining optimal error estimates is to show that the right-hand side of (4.6), although not zero, is of sufficiently high order in h. To this end, we have the

Finite element methods for type-II superconductors t0t

following result whose proof may be found in the Appendix. Here, Fh denotes a side of a finite element, and ~ r , ~ re denotes the sum over element sides that have a nontrivial intersection with the boundary Fe of f2e.

Lemma 4.3. There ex is t s a constant C > O, with a value independent o f h and r/, such that

(4.7)

and

tb(r/,(h)l <- Ch21}r/tl2it~hllLr, Vr/e~ez(f2p), . ~ h e ~ ' t

)1/2 (4.8) Ib(r/,(h)l < Ch411nll3 ~ h 2 ~ , 2 lift l[2,re Vr/e ~g(g2e), (hE �9

F h c Fp

Consequently, we have the following result.

Corollary 4.4. For r = 1 or 2, there ex is ts a constant C > O, with a value independent o f h and ~l, such that

(4.9) Ib(r/ ,(h)] < C h ' + l / : l l r / I I , + l l l C h l l l V r / e Y ? 7 1 ( Q p ) , ( h e ~ g '" .

Proof. The result follows from an application of the inverse inequality (3.6) and trace theorems. []

As a result of (3.1), (4.6), and (4.9), we have the following error estimate.

Proposition 4.5. Let r = 1 or 2, and let the solution o f (4.1) satisfy r~ e ~ ' e + 1 (f2e). Le t r/h ~ 3 ~ , r denote the solution of(4.2). Then, there ex is ts a constant C, with a value independent o f h and r/, such that

(4.10) IIn - nhlll <_- Ch'llr/I1,+,.

r / h . ~ r + 1 Eel ~ __. 3g'h, r P r o o f Let ~,,.~,~ e t-oeJ denote the standard interpolation operator; recall that II,kr/~ 3wg,". Then, since on ~ " , a x / ~ , v h) defines a norm equivalent to the standard 1-norm,

C ll l~hr/ _ r/hn~ < a(Hhr/ _ r/h, liar~ -- rl h)

= a ( l - l h r / _ r/, l-lhrt __ r/h) + a(n - - r/h, i l h r / _ r /h) .

Obviously,

la(Hh, r / - - r/, II)r/ -- r/h)l < cIIn)r/- r/Ill It n,"r/- r/hll~ < Ch'll r/II,+ 111/-/,hrt - r/hll ~ ,

where we have used (3.1). Also, using (4.6) and (4.9), we have that

l a ( r / - r/h, n h r / - - nh)l = I b (r / , / - / ) r / - r/h)l < Ch "+ 1/2 II n I1,+1 II n , h r / - r/h ]11 �9

The combinat ion of the last three results, along with (3.1) and the triangle inequality, yields (4.10). []

102 Q. Duet al.

To obtain an ~2(12e)-norm estimate for the error, we will use a duality argument. To this end, we consider the following adjoint problem: seek r e ~ffe~ (f2e) such that

(4.11) a(~, 4) = 9 t { ( r / - r/h, ~')} V~e ovf~(12e).

Using regularity results obtained in 1-14], we have that

(4.12) [[~[1,+1 --< C [ [ r / - r/h[lr_l for r = 1 or 2 .

Also, consider the approximate problem corresponding to (4.11): for r - - 1 or 2, seek ~h ~ ~F'~'" such that

(4.13) a(~h, ~h) = ~l{(tl _ ~h, ~h)} V~'h e ~he,,

Then, using the result for ~ analogous to (4.10) and also (4.12), we have that

(4.14) l[~ - ~hIlx < Ch'll~[lr+1 < Ch'l[rl - ~/hllr-1 for r = 1 or 2 .

Analogous to the form b( . , - ) , we define the bilinear form / ; ( . , . ) : J f2 ( f2 r )x ~i(Oe) -~ C by

6(~, if") = ~ Ot{i(igrad~ + .4o~).n~*dF) V~ ~ ~/f2(f2e) and ~'~ ~ ( f 2 e ) . Fe

Then, for/~( . , . ) we have results analogous to (4.4)-(4.8) for the form b ( . , . ); we will denote these analogous results by (4.4')-(4.8'), e.g., corresponding to (4.4) we have that if ~ e ~e2(f2~,) is a solution of (4.11), then

(4.4') a(~, 4) + 6(~, ~-) = 9 l { ( r / - r/h,C')} V~e ~ ( ~ e ) .

We will need the following stability results.

Lemma 4.6. For r = 1 or 2, let tl ~ ~'e+ ~(f2e) denote the solution of(4.1) and let tlh e ~1/'~'" denote solution of(4.2). Then, for h sufficiently small,

II~hll~,r,_- < C l l~ lh / f r = 1 (4.15)

and

/1/2 (4.16) ~ Ilr/hll~,rh < Cllr/]13,a / f r = 2 .

F h c Fp

The analogous results hold for the solutions ~ and ~h of(4.11) and (4.13).

Proof First, consider the case r = 1. Then,

I] ~h Ill, r,,,

<

<

<

C {h- 1/2 H ~/h _ H~ r/[11 + 1[ r/1[ 1, rp }

C{h-X/211~ - n ~ l l x + h-X/~ll~ h - rlllx + Ilrlll2}

C{h~/21l~211 + 11~112} =< Cl l~[h �9


The first of the above inequalities follows from the triangle inequality, the second from the inverse inequality (3.6), trace theorems, and the stability of the interpolation operator, the third from the triangle inequality and trace theorems, the fourth from (3.1) and (4.10), and the last by readjusting the constant for sufficiently small h. For details concerning trace theorems and the stability of the interpolation operator, one may consult [2] and [8], respectively.

For r = 2, we proceed in a similar manner, i.e.,

( Y {( Y Z IIti h < c Z Ilrl h //~'I112 II~,rh = - 2,rh Fn ~ Fp Fh ~ Fp

( Y} + ~ [I/7~t/]]zZ, r . F h c Fp

< C ( h - 1 I1 r - r t~t i II1, r , + 11 ~ IT 2, r , )

<= C ( h - s/2 [1 r/h - II~t1111 + [I ti [1 s.ae)

< C(h-S/21lrl - / / ~ r / l l l + h-S/2[lr/h - till1 + lltllls, o~)

< C(h I/2 II ti I I s ,~ + I1~ Ils, QA < e l l ~ 113,o~ �9

We have one additional inequality, the second, which follows from the inverse inequality (3.6). []

We may now obtain the desired 2~z(f2e)-norm estimates.

Proposition 4.7. L e t r = 1 or 2. L e t ~I �9 Y7% + l (Oe) denote the solut ion r and let tlh �9 3V~ '~ deno te the solut ion of(4.2). Then, f o r h suf f iciently small,

(4.17) IIti - '711o < Ch "§ tl ~ I1,+~ �9

P r o o f We have that

(4.18) [[tl -- rlhl]2 = a ( t l -- rlh, r + l)(~,tl - rl h)

= a(t / -- r/h, ~ -- ch) + a(q -- tt h, ch) + t;(r r /-- t/h)

= a(tl -- tl n, ~ - ~") -- b(ti, ~h) _ ~(r tln),

where the first equality follows from (4.11) and (4.4'), the second is trivial, and the third from (4.5) and (4.6').

First consider the case r = 1. The continuity of the form a( . ,- ), (4.10), and (4.t4) yield that

(4.19) [ a f t / - r ~ - ~h)t < C[]rl -- r II~ - ~hllx ----< Ch2llr/[12l[q - qh]to �9

Also, by (4.7) and (4.15'), and (4.12),

(4.20) lb(q, ~h)l < Ch2 []q{Izll ~hIll,rp < Ch2ll~llI2ll~llz <= Ch21lr/][2[lr/- Cl io

104 Q. Duet aL

and by (4.7), (4.15), and (4.12),

(4.21) Ib(~, r/n)l < Ch21l~ll2ll~hllx,rp< Ch2ll~ll2tlr/[12 < Ch2 II~/-r/hllo il ~/II2.

The combination of (4.18)-(4.21) then yields (4.17) with r = 1. For the case r = 2, we have, from the continuity of the form a(- ,. ), (4.10), and

(4.14), that

(4.22) [ a ( r / - r/h,~ - Ch)[ < Cll~ - r/hllx [l r -- ~h][1 < Ch2l]rl - ~hll2 < Ch6l[~/I[ 2 �9

Also, (4.8), (4.16), (4.16'), and (4.14) yield that

( Y Ib(~, ~h)l < Ch411~[13 ~ I[~hll~,r~ < Ch411~113t1~II3 < Ch611~ll~ F h c Fp

(4.23)

and

(4.24)

The combination of (4.18) and (4.22)-(4.24) then yields (4.17) with r = 2.

Ib(~, ~/h)l ~ Ch411~113( F~ c F e

2 ~1/2 IIr/hll2,r,j < Ch41l~1{311~/113 < Ch6H~I 2 .

[]

4.2 The linear problem for the magnetic potential

Results entirely analogous to those of Sect. 4.1 can be obtained for the linear problem associated with the magnetic potential, i.e., for (3.15) and its discretization, (3.23). The situation is somewhat easier here since we have that Vhe '" c H~,(t2e), while in Sect. 4.1 we had that ~h , r r ~le(g2e). For this reason, here we give the results without proofs.

Proposit ion 4.8. Let r = 1 or 2. Then, for F e (~r ', the problems (3.15) and (3.23) have unique solutions R e tt~,(Oe) and Rh e vhe '', respectively. Moreover, I[Rllx _-< CIIFt[. , Ilgn[lx ~ C[IF[I., and

(4.25) lim [1R - R h I[i = 0 . h~0

I f R e ~l'e + 1 (f2e), then there exists a constant C, with a value independent ofh and R, such that

(4.26) I[R - Rhl[1 =< Ch~I[R[[,+~

and, for h sufficiently small,

(4.27) IIR- Rhllo ~ Ch "+l t]RIl,+l. []

4.3 Convergence and estimates for the operator T and Z h

The results of Sects. 4.1 and 4.2, along with the definitions of the operators T and T h given in Sect. 3.4, may be combined to show that the hypotheses of Sect. 3.3 on these operators hold in the current setting.


Proposit ion 4.9. Le t the operators T and T h be defined by (3.13)-(3.15) and (3.22)- (3.23), respectively. Then, these operators satisfy the hypotheses (H1), (H4), (H6), (H7), and (H9) o f Sect. 3.3. Moreover ,

(4.28) II(r/, R) - - ( r / h , Rh)llx < Ch'(ltr/II,+~ 4- IIRL+0

and

(4.29) II (r/, R) - O1 h, R h) II H < C h" + 1 ( [I r~ tl r + 1 + II R II , + 1) .

P r o o f (H1) and (H4) follow from Propositions 4.1 and 4.8. (H6) follows from (4.3) and (4.25). Since Z c Y and X ~ H with compact imbeddings, (H7) and (H9) follow from (H6). (4.28) follows from (4.10) and (4.26); (4.29) follows from (4.17) and (4.27). []

5. Error es t imates for the periodic Ginzburg-Landau model

We now turn to the derivation of error estimates for the nonlinear periodic Ginzburg-Landau model and its approximate solution. The key ingredient in this derivation is the estimates obtained in Sect. 4 for associated linear problems.

Let A denote a compact interval of IR § Then, given any ((, Q) e X, we have that (~, F) ~ Y satisfies

(/0, F) = Dg, o)GOc, (~, Q))(r/, P)

for (r/, P) e X if and only if

y ~R{:('*} df2 = S 9t{r/('*} d f ~ - x y 2!R{(*(igrad- do)r/}.QdO I2 e ~2p lap

- x f 2iR{('*(igrad - Ao)(} . P d O Oe

_ ~2 ~ (IQI2 + i~12 _ 1)ffi{r/~'*} d~2 Qe

and

-- x2 f (2Q. e + 2!R {q{*})9t {{~'*} dE2 ~e

VCE ~ 1 (~p)

(5.2) S p" Qdf2 = - 1 9t S {~*(igrad - .,~o)r/+ q*(igrad - Ao)~)" QdO Oe K ~e

+ I P . Q d O - f ( l~12P+29~{r /~*}Q) .Q_df2 ~p ~e

One easily sees that (5.1) and (5.2) are weak formulations of

(5.3) /~ = r / - x2(IQ] 2 + I([ 2 - 1) r / - x2(2Q.P + 2~R{q(*})(

- 2KQ-( igrad - A o ) r / - 2 x P . (igrad - A o ) ( in me

106 Q. Duet al.

and

(5.4) F = P - IClup - 291{~C*} 0

_ t_ 9t {(*(igrad - A0)~/+ r/*(igrad - Ao)~} in f2e, x

respectively. Given any (if, Q) e X and x e A, one may use the definitions (3.13)-(3.15) and

(5.1)-(5.2) for the operators T: Y--* X and D~.o)G(x, (~, Q)): X ~ Y, respectively, to define the operator D~r162 (~, Q)):X --, X, i.e.,

D~r (~, Q.))(rl, P) = (I + TD~r (~, Q)))(t/, P) V(~/, P) ~ X .

A branch of solutions { [x, (~(r), Q(~:))]; x ~ A} is called a regular branch if, for each x e A, the problem

DIr (x, (~, Q) )(rl, P) = (~, F)

has a unique solution for each (p, F) ~ Y. The first estimate we obtain is for the error measured in H~(f2e)-norms.

T h e o r e m 5.1. Let r = 1 or 2. Let A denote a compact interval of IR +. Assume that { Ix, (~(x), Q(x))]; x ~ A} is a branch of regular solutions of(2.8)-(2.9) such that, for each x, (((r), Q(x)) e X = ~ l ( f 2 e ) x H~,(f2e). Then, there exists a neighborhood 0 of the origin in X and for h sufficiently small, a unique C 2 function

~ ((h(r.), Qh(r)) ~ X h = 3r "" x re", such that {(x, ((h(K), Qh(~:))); x ~ A} is a branch of regular solutions of (3.7)-(3.8) and (~h(~C), Qh(K)) -- (((x), Q(x)) ~ O for all

~ A. Moreover, there exists a constant C > O, independent of h and ~c, such that

(5.5) II~(x) - (h(x)llx + HQ(x) - ~(K)llx ~ Ch'(]](ll,+1 + t[QI[,+I) �9

Proof. The results follow from Theorem 3.3 once the hypotheses (H1)-(H7) of Sect. 3.3 are verified in the current setting. As a result of Proposition 4.9, (H1), (H4), (H6), and (H7) have already been verified.

From its definition (3.24)-(3.25), one easily sees that the operator G is a polynomial mapping in ( and Q;, its simple dependence on x is also evident. A straightforward, but tedious, calculation, can be effected to compute the second derivatives of G; from these it is easily shown that hypotheses (HI) and (H2) hold.

Now let us examine the operator D~.Q)G(x,((, Q)) for given ~c~A and ((, Q)~ X. It is simplest to examine its strong form (5.3)-(5.4). Note that, by the Sobolev imbedding theorem and for (~, Q), (r/, P ) ~ x = Jt~l(f2p)• Het(f2p), we have that (, r/~ .~6(~r~p), Q, P c L6(Op), 69(/dxj, 6~rl/~3X j E ~2(~C~p), and aQ/3xj, dP/~xj e L2(f2e) forj = 1 or 2. As a result, one easily concludes that the right-hand sides of (5.3) and (5.4)-belong to ~3/2(f2e) so that for each x e A and ((, Q) e x , D~r ((, Q)) ~ :~(X; z) with Z = -L~3/2(f2e) x ~3/2(f2e). Thus, (H5) is verified.

-It was shown in 1-14] that if ((, Q)s X is a solution of (2.8)--(2.9), i.e., if ( ( , Q ) = - TG(x(~,Q)), then certainly ~e~C~,+l(g2p) and Q~H'e+l(f2e). As a


result, we have, from (4.28), that

l l ( T - Tn)G(K, (~, Q))[lx < Chr(II~]l,+l + I[QII,+I) �9

Combining with (3.tl) then yields (5.5). []

We can also obtain estimates for the error measured in LZ(~2e)-norms.

Theorem 5.2. Assume the hypothesis o f Theorem 5.1. Then, for sufficiently small h, there exists a constant C > O, independent o f h and •, such that

(5.6) II ~(~c) - ~h(x)ilo + l1Q(x)- Qh(K)[Io < Chr+l~(ll~ll,+a, IlQIb+l) �9

where ~ ( . , . ) : IR 2 ~ IR, is a quadratic polynomial.

Proof The result follows from (3.12), (4.29), and (5.5) once the hypotheses (H8)- (Hl l ) of Sect. 3.3 are verified in the current setting. (H9) has been verified in Proposition 4.9. The remaining hypotheses follow from straightforward, but tedious, calculations; the main ingredients needed are the Sobolev imbedding theorem and the regularity results ( ~ 9r + 1 (f2e) and Q ~/I~,+ 1 (f2e) of [14]. []

Remark. The estimates (5.5) and (5.6) are optimal with respect to the rate of convergence. Indeed, we have that

II~(x) - ~h(x)ll, + I1Q(~c) - Qh(~c)lls = O(h r+l-s) for s = 0 or 1 ,

where r = 1 for piecewise linear finite elements and r = 2 for piecewise quadratic finite elements. It seems possible to obtain similar optimal error estimates for higher-order finite element spaces. The key is to deduce results corresponding to those of Lemma 4.3; these should be derivable, perhaps through a judicious choice for the interpolation nodes.

Remark. In Theorems 5.1 and 5.2 we assumed that x ~ A determines a branch of regular solutions of the Ginzburg-Landau equations. That these equations possess such branches can be shown using techniques similar to those employed for the Navier-Stokes equations; see [23] and the references cited therein. However, just as in the Navier-Stokes case, it is impossible to predict exactly at what values of the parameters of the model singularities appear.

6. Numerical example

We begin by listing the input data necessary to specify the periodic Ginzburg- Landau model, and also some of the quantities of interest that one would like to obtain as output data from the model. For a complete discussion of both the input and output data, see [14].

6.1 Input data

The following is a list of the parameters that uniquely specify the periodic Ginzburg-Landau model.

108 Q. Duet al.

1. The Ginzburg-Landau parameter x, a material constant which, for type-II superconductors, satisfies ~ ~ (l/x/2, ~).

2. The positive integer n, the number of fluxoids associated with a lattice cell. 3. The position of the point P which determines the position of the lattice cell with

respect to the origin; the usual choice for P is the origin itself. 4. The average magnetic field/~; for a given value of x,/~ may be chosen in the

interval (0, x]. 5. The directions and relative magnitudes of the lattice vectors tl and t2. Without

loss of generality, one may choose tl to be aligned with the xl-axis. One also chooses 0 * 0, the angle between tl and t2, and 7 > 0, the ratio of the magnitudes of t2 and tx. The absolute magnitudes of tl and tz are then determined with the help of the fluxoid quantization condition, i.e., ]O] = 2nn/xB.

Thus, the parameters that determine the model are P, x, n, 7, ~9, and/~.

r Output data

The reduced magnetic potential Q and order parameter ( are the primary dependent variables employed in the periodic Ginzburg-Landau model. However, of more interest are the magnetic field h, the current j, and the density of superconducting charge carriers Ns. These may be determined from ~ and Q through the relations

(6.1) h = curl Q +/~, Ns = Ir = I(] 2, and

, O+ o) where co denotes the phase of (.

For specified values for the input parameters, one would like to know the corresponding external magnetic field He. The best method for determining He is developed in 1-11], where it is shown that

(6.2) He - BIOI o, 2 grad + Q - Ao ( + Icur l (Q- Ao)l 2 dO.

Thus, once ((, Q) has been obtained from a given set of input parameters, He can be determined by evaluating integrals. Another quantity of interest is the magnetization (or magnetic moment per unit volume) M defined by

(6.3) - 4r~M = He - / ~ .

Remark. If the finite element approximations (h and Qh are substituted into (6.1)-(6.3), then one can deduce that the approximations to the magnetic field, current, external field, and magnetization so obtained are O(h') and that for the density of superconducting charge carriers is O(h "§ x), where, as always, r = 1 for piecewise linear elements and r = 2 for piecewise quadratic elements.


6.3 Computational results

We will report, in detail, on the results of our extensive computational experiments and their physical interpretation elsewhere. Here, for the sake of completeness, we provide a small sample of these results.

We consider the most interesting periodicity structure, namely that corresponding to an equilateral triangular lattice having one fluxoid associated with each lattice cell. It is well-known (see any of the references listed at the beginning of Sect. 1) that such a lattice yields the smallest value for the Gibbs free energy. For this case we have that n = 1, ~ = 1, and ,9 -- n/3. Then the area It2[ of any lattice cell is given by IOI = x/~ltl12/2 = 2n/kB so that

v/ 4 S �9 t 1 - ~ - - 11 / 4rt ( l i l x//3" t2 +T'U

We make the customary choice of the origin for the point P. The lattice cell f2 is depicted in Fig. 4. The only remaining inputs to be chosen are the Ginzburg- Landau parameter ~c and the average magnetic field/3.

The discretized equations are a nonlinear system of algebraic equations. These are solved by a continuation method coupled with Newton's method. The code is configured so that one chooses a fixed value for x, and then varies/~. For each pair (x,/~), Newton's method is used to solve the nonlinear equations. The initial guess for Newton's method is determined by continuing from the solution determined for a previous value of/3 and the same value of x. The particular continuation method used is a tangent line approximation to the solution at the previous value of/~. We start with a value of B close to the upper limit x for which Newton's method seems to have a large attraction ball; we then continue by successively reducing the value of /3 towards its lower limit 0.

Computational results were obtained, on a Macintosh II, using piecewise quadratic element functions, i.e., r = 2. A uniform grid spacing was used in each of the tl and t2 directions. We fix x = 5 and vary/~. We then obtain approximations

IM0,0) ID X 1

Fig. 4. The lattice cell f2 for an equilateral triangular lattice

110 Q. Du et al.

for ( and Q which are used to determine Ns f rom (6.1), H e from (6.2), and M from (6.3). The graph of the computed approximat ion of - 4 r t M vs. H e is given in Figs. 5 and 6. In Fig. 5, the magnet izat ion obtained using 3 intervals in each direction, i.e., N1 = N2 = 3, is compared with the corresponding graph for the Monte Car lo /s imula ted annealing approx imat ion obta ined in 1-12]. In Fig. 6, the magnetization obta ined using 3 intervals in each direction is compared with the corresponding graph for 4 intervals in each direction, i.e., for N~ = N2 = 4. The level curves, using Nx = N2 = 3, of Ns are given in Fig. 6. The solution in only a single lattice cell was computed; this solution was extended, using periodicity or "quasi"- periodicity relations to obtain the solution outside the computa t iona l cell.

0.2

0.18

Monte C~rlo/SimuJmed am~tlin 8 [12] 0.16 t

0.14

o.12

"~ 0.I F Lite

"$ ~' 0.08 �9

0.06

0.04

0.02

0 i , , i o 0;5 I 1;5 3 3 ; , ,.5

F.xmratl field

Fig. 5. Magnetization ( -4~M) vs. external field (He) for x = 5

0.18

0.16

0 .14

10121 .~ o.11

~ 0.08.

X 0.06 I 0.04 0.02

0

3x3

0.$ 1 1.5 2 2.5 3 3.5 4 4.5

External field

Fig. 6. Magnetization vs. external field for two different grids

Finite element me thods for type-II superconduc tors 111

Fig. 7. Level curves of N~, the densi ty of superconduct ing charge carriers, for ~ = 5, /~ = 10/3, and N~ -- N2 = 3

Appendix

Proo f o f Lemma 4.3 for the case r = 1. It was shown in [14] that for all q e ~ ( ~ 2 v ) , (igrad q - Aoq)lx + t~ "n+k = - - (igrad q - -4o~/)]x" n - k ei#k(x) VX E F-k, k = 1, 2. Then, since n+k = n -k ,

b(~/, (h) = ~ ~B {i(igrad ~ - Ao~/)" n( h* dF } Fe

2

= ~ ~ ~ {i(igrad ~ / - ,4or/)-nbk(~h*)e Ok(x)} d F , k = l Fk

where 6k(~ h) = ( h ( x + t~) - (h(x)ei~k~x~.

Now, let x, and x~+ x denote the two nodes on the boundary on each side of the point x e Fk. Then

x = ;txz + (1 - ~.)xz+l

for some 2 e (0, 1). For the piecewise linear function (h, we also have that

112 Q. Du et al.

Then,

6k((h) = {(1 -- 2)(h(X,+l + tk) + 2(h(X, + tk)) -- {(1 -- 2)(h(X/+l) + 2(h(x~))e i~k(~)

= (I - 2)(h (X, +1) { e i ak tx '+ ' ) - - e ia~(~)} + ~,(h (xl) {ei~ktx,) _ eia~(~)} .

Wi th h, = [xz+l - xz[, we have that

e i~k(x') - - e i~k(x) = igrad ~k(X)" tkeiOk(x)(~, -- 1)h, - (grad gk(X)" tk)Ze i~k(x) (2 - - 1)2h 2 2

= igrad ~R(X)" tkei~k(~)(2 -- 1)hz + M l h 2 . Similarly,

Therefore,

e ~k(~'+~) - e i~(x) = igrad ~k(X)" tkei~(~)2h~ + M2h 2 �9

6k((h)=igrad~k(X). tkei~k(x)2(2--1)h2((h(xz+l)-h~(h(x ') )

+ MI(1 - 2)h2(h(x,+l) + M2(1 - 2)h2(h(Xt) and

h I]6g(( )}]0,r_~ < Ch2{[ICnll~.r_~ + I[ (h llL~tr_~))

< Ch2[}Ch[ll, r_ k < Ch2H(h[ll,re.

C o m b i n i n g the above results yields

2 [b(r/, (h)] < ~ Iligrad r / - .4orl).ni[o,r_h}lfk((h)llo, r_~

k = l

2 < ~ [1 ( i g r a d q - .'ion ) �9 n [[o,/-_kh 21f (h 111,Fe --<--- Ch2 [I rl I] 2 ]i (h II 1,re,

k = l

i.e., (4.7) holds. [ ]

The p r o o f of L e m m a 4.3 for the case r = 2 will m a k e use of the fol lowing two e lementa ry results, the first of which is we l l -known result a b o u t S impson ' s rule.

L e m m a A.1. Let Ih2 denote the quadratic interpolation operator such that Ih2f = f fo r x = a, b, and (a + b)/2. Then, for any cubic polynomial P3, we have

b b S Ih2(p3) dx = S padx" []

In the next result, Fh denotes the intersect ion of a side of finite e lement with the b o u n d a r y Fp.

L e m m a A.2. There exists a constant C > 0 such that for any 9 E H~ h and f E H } h, c

Fh


Proof. Without loss of generality, we may take Fh = [0, h]. By Taylor's theorem, we have

h h 1 h 1 $ (g -- lh2g)fdx = 0 S ~ g"'(0)[ x3 -- lh2(x3)]f dx + 0 f ~ [g(4)( O)X4 - - lh(gt"(O)x')]f dx 0

for some 0 e (0, h). First using Lemma A. 1,

1 g,,,(0)[X 3 Ih2(x3)]f dx h 1 g,,,(0)[X 3 ' o ~ ~ -- = ! ~ -- l ~ ( x Z ) [ f ( x ) - - f ( O ) ) [ dx

< Ch7/2 II f ( x ) -- f(O)]] o, r, e s s m a x 19'" [ < Ch 4 It f Ill,r, 119114,Fh.

Similarly,

[ j 1 _i~(g(4)(O)x4)]fd x [ g ( 4 ) ( 0 ) X4 ~ Ch9/2 Ilfl[0,r, essmaxlg~*~r

< Ch4tlfllo, rhllgtl5,rh.

Therefore, after a change of variables, we see that the lemma is valid for any F h . []

Proof of Lemma 4.3 for the case r = 2. We have that r/~ ~3(,Qe) and (h e ~F',~" 2 Note that

b(r/,(h)= ~, ( ~ S ( ig radr /_ ,ioq).ne,~k[ih2((he-i~k)_ ~he-,~k])dF " k = i Fh ~ Fk F.

Define, o n Fh, g --= - - (he-i~/k and f--- (igrad r / - .ior/). tie i~k. Then, by Lemma A.2, we have

t S ( igrad r / - - < II f l l 1, r~ II g II 5,r~. 2o~). ~e,~k [I~ ((he i~k) ~he i~k 1 dF Ch 4 Fh

Now, since on Fn, ~h is a quadratic polynomial, there exists a constant C > 0, independent of h, such that

Ilglls, r,<=Cllvhtl2,r, and [Jf]Jl'r"<=C( ~nn 1,r,+ ][~/J[1.r,).

Hence,

Ib(r/,(*)[ < Ch'll~hll2'rh ~nn 1,r, + Ilnlll,r, k = 1 Fh F~

i.e., (4.8) holds.

[ ) < C h4 ~ tl ~h [I 2, r , / [I II 1,r, + H r/II l,r, F, ~ Fp

< Ch4 Z . 2 I[( l[2,r, Hi/H3,o , L F, c Fe

[]

114 Q. Du et al.

Acknowledoements. The authors wish to thank Dr. James Gubernatis of the Theoretical Division of the Los Alamos National Laboratory for many helpful and interesting discussions, and also for providing the data from [12] used in Fig. 5.

References

1. Abrikosov, A. (1957): On the magnetic properties of superconductors of the second type. Zb. Eksperim. i Teor. Fiz. 32, 1442-1452 [-English translation: (1957) Soviet Phys.-JETP 5, 1174-1182]

2. Adams, A. (1975): Sobolev spaces. Academic Press, New York 3. Babugka, I., Aziz, A. (1972): Survey lectures on the mathematical foundations of the finite

element method. In: A. Aziz, ed., The mathematical foundations of the finite element method with application to partial differential equations, pp. 3-359. Academic Press, New York

4. Bardeen, J. (1956): Theory of superconductivity. In: S. Ftfigge, ed., Encyclopedia of Physics XV, pp. 17-369. Springer, Berlin Heidelberg New York

5. Brandt, E. (1972): Ginsburg-Landau theory of the vortex lattice in type-II superconductors for all values of K and B. Phys. Stat. Sol. (b) 51, 345-358

6. Brezzi, F., Rappaz, J., Raviart, P.-A. (1980): Finite-dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math. 36, 1-25

7. Chapman, S., Howison, S., Ockendon, J. (1992): Macroscopic models for superconductvity. SIAM Review (to appear)

8. Ciarlet, P. (1978): The finite element method for elliptic problems. North-Holland, Amsterdam

9. Crouziex, M., Rappaz, J. (1989): On numerical approximation in bifurcation theory. Masson, Paris

10. DeGennes, P. (1966): Superconductivity in metals and alloys. Benjamin, New York 11. Doria, M., Gubernatis, J., Rainer, D. (1989): Virial theorem for Ginzburg-Landau theories

with potential application to numerical studies of type II superconductors. Phys. Rev. B 39, 9573-9575

12. Doria, M., Gubernatis, J., Rainer, D. (1990): Solving the Ginzburg-Landau equations by simulated annealing. Phys. Rev. B 41, 6335-6340

13. Du, Q., Gunzburger, M., Peterson, J. (1992): Analysis and approximation of Ginzburg- Landau models for superconductivity. SIAM Review 34, 54-81

14. Du, Q., Gunzburger, M., Peterson, J. (1992): Modeling and analysis of a periodic Ginzburg- Landau model for type-II superconductors. SIAM J. Appl. Math. (to appear)

15. Eilenberger, G. (1964): Zu Abrikosovs Theorie der periodischen L6sungen der GL-Gleichun- gen ffir Suparaleiter 2. Art. Z. Phys. 180, 32-42

16. Girault, V., Raviart, P.-A. (1986): Finite element methods for Navier-Stokes equations. Springer, Berlin Heidelberg New York

17. Kleiner, W., Roth, L., Autler, S. (1964): Bulk solution of Ginzburg-Landau equations for type II superconductors: upper critical field region. Phys. Rev. 133, A1226-A1227

18. Koppe, H., Willebrand, J. (1970): Approximate calculation of the reversible magnetization curves of type II superconductors. Low Temp. Phys. 2, 499-506

19. Kuper, C. (1968): An introduction of the theory of superconductivity. Clarendon, Oxford 20. Lasher, G. (1965): Series solution of the Ginzburg-Landau equations for the Abrisokov mixed

state. Phys. Rev. A, 140, 523-528 21. Odeh, F. (1967): Existence and bifurcation theorems for the Ginzburg-Landau equations.

J. Math. Phys. 8, 2351-2356 22. St. James, D., Sarma, G., Thomas, E. (1969): Type II superconductivity. Pergamon, Oxford 23. Temam, R. (1983): Navier-Stokes equations and nonlinear functional analysis. SIAM,

Philadelphia 24. Tinkham, M. (1975): Introduction to superconductivity. McGraw-Hill, New York

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Finite element approximation of a periodic Ginzburg-Landau ...mgunzburger/files... · [12] which uses a Monte Carlo/simulated annealing approach and represents the best known effort