A QUASI-VARIATIONAL INEQUALITY PROBLEM ... - Semantic Scholar

Mathematical Models and Methods in Applied Sciences Vol. 20, No. 5 (2010) 679706 # .c World Scienti¯c Publishing Company DOI: 10.1142/S0218202510004404

A QUASI-VARIATIONAL INEQUALITY PROBLEM IN SUPERCONDUCTIVITY

JOHN W. BARRETT Department of Mathematics, Imperial College London, London SW7 2AZ, UK [email protected] LEONID PRIGOZHIN Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990 Israel [email protected] Received 19 September 2008 Revised 8 May 2009 Communicated by J. Ball We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical ¯nite element method based on the lowest order RaviartThomas element, which yields approximations to both the primal and dual variables (the magnetic and electric ¯elds). We prove the subsequence convergence of this approximation, and hence prove the existence of a solution to both the dual and primal formulations, for strictly star-shaped domains. The e®ectiveness of the approximation is illustrated by numerical examples with and without this domain restriction. Keywords: Superconductivity; Kim model; quasi-variational inequalities; critical-state problems; mixed methods; ¯nite elements, existence; convergence analysis. AMS Subject Classi¯cation: 35D05, 35K85, 49J40, 49M29, 65M12, 65M60, 82C27

1. Introduction 1.1. Primal problem Macroscopically, magnetisation of type-II superconductors can be regarded as an eddy current problem described by the Faraday and Ampere laws @ t b þ r e ¼ 0; 679

r b ¼ j;

ð1:1Þ

680

J. W. Barrett & L. Prigozhin

with a nonlinear and, often, multi-valued current-voltage relation characterising the superconducting material. The magnetic permeability of superconductors is assumed equal to that of a vacuum and scaled to unity; hence, we will not distinguish between the magnetic ¯eld and the magnetic °ux density b. Typically, the current-voltage relation represents the electric ¯eld inside the superconductor, e, as the subgradient of a convex functional, e 2 @ðjÞ;

ð1:2Þ

where j is the current density, see Bossavit.9 To model the hysteretic response of superconductors to variations of the external magnetic ¯elds and transport currents, it is convenient to formulate these problems as evolutionary variational or quasivariational inequalities, see Bossavit9 and Prigozhin.22 Of much interest, for technological applications and in physical experiments, are the energy loss estimates. Hence, the simultaneous determination of the current density and the electric ¯eld in a superconductor is often necessary. Unfortunately, the variational inequality formulations,9,22 which we will call \primal", allow one to compute only the current density; and determining the electric ¯eld remains di±cult, e.g. if the relation (1.2) is multi-valued. It has been shown recently, see Barrett and Prigozhin,6 that a dual variational inequality formulation, based on an equivalent representation of the current-voltage relation, j 2 @ ðeÞ;

ð1:3Þ

where is the convex conjugate of , can be the basis for an e±cient method for determining both of the variables in the Bean critical-state model,8 which is the basic model for the magnetisation of type-II superconductors. Bean's model postulates that (in an isotropic superconductor) the current density cannot exceed some critical value, jjðx; tÞj Jc , the electric ¯eld is parallel to the current density, and is zero wherever jjðx; tÞj < Jc . In this case, is the characteristic function of the set of admissible currents K 0 ; that is, 0 if v 2 K 0 ; ðvÞ ¼ K 0 ðvÞ :¼ ð1:4Þ þ1 if v 62 K 0 : In this paper we will study a similar dual formulation for the model in which the critical current density, Jc , depends on the magnetic ¯eld and the inequality becomes quasi-variational, see Prigozhin.22 Such a modi¯cation of the Bean model, where Jc ¼ Jc ðjbjÞ is a monotonically decreasing function of the magnetic ¯eld, has been proposed by Kim et al.21 to account for the decrease of the magnetic moments in strong external ¯elds, which is typical of most type-II superconductors. There are also materials demonstrating a secondary peak in their magnetisation hysteresis loops. The latter phenomenon, often called \the ¯shtail e®ect", can be described by the eddy current model with a non-monotonic Jc ðjbjÞ dependence; see, e.g. Johansen et al.19 We mention here also the primal variational formulation for a generalised double critical-state model, see Badía and Lopez2 and Kashima,20 in which is the

A Quasi-Variational Inequality Problem in Superconductivity

681

characteristic function of the set of admissible currents satisfying jðx; tÞ 2 ðbðx; tÞÞ and ðbÞ R 3 being a given family of closed convex sets. Below we consider a simple geometric con¯guration of an in¯nite superconducting cylinder having a cross section R 2 and placed into a parallel non-stationary uniform external magnetic ¯eld be ðtÞ. In this case the variational inequality for Bean's model, and the quasi-variational inequality for Kim's model, are most easily written in terms of the magnetic ¯eld which has only one non-zero component and can be regarded as a scalar function. Let be a bounded connected domain with a Lipschitz boundary @; if is not simply connected we allow it to have a ¯nite number of \holes" i ; i ¼ 1 ! I; and SS i Þ: In this geometry, the induced magnetic ¯eld wðx; tÞ ¼ set ¼ ð i¼1!I bðx; tÞ be ðtÞ is zero on @ , the outer boundary of , and depends only on time in each of the holes. We adopt the standard notation for curls in two dimensions: r vðxÞ ¼ @ x1 v2 ðxÞ @ x2 v1 ðxÞ and r vðxÞ ¼ ½@ x2 vðxÞ; @ x1 vðxÞ T . To allow for at least some kind of spatial inhomogeneity on we shall assume throughout the majority of this paper that Jc ðx; bÞ ¼ kðxÞMðbÞ;

ð1:5Þ

with kðxÞ k0 > 0 for all x 2 , and M : R ! ½M0 ; M1 R, with where k 2 CðÞ, M0 > 0, are given functions. We do not assume that Jc is monotonically decreasing with respect to jbj, so we can deal with \the ¯shtail e®ect" mentioned above. In addition, we do not require M to be continuous for our results on the dual formulation (Q), (1.15a), (1.15b); whereas we do require M to be continuous for our results on the primal formulation (P), (1.8). Using the laws (1.1) and the constitutive relation (1.2), we obtain for any 2 H 01 ð Þ that ð@ t b; wÞ ¼ ðr e; wÞ ¼ ðe; r ð wÞÞ ð r wÞ ð r Þ; where ð; Þ is the standard inner product on L 2 ð Þ: In the Kim and similar models ¼ K 0 ðbÞ , the characteristic function of the set of admissible current densities K 0 ðbÞ :¼ fv 2 ½L 2 ð Þ 2 : jvðxÞj Jc ðx; bÞ for a:e: x 2 ; v ¼ 0 a:e: in i ; i ¼ 1 ! Ig:

ð1:6Þ

Since, by Ampere's law, j ¼ r b ¼ ½@ x2 b; @ x1 b T we have that jjj ¼ j rbj ¼ j rwj and so one can replace the set of admissible currents in the variational formulation by the set of admissible induced magnetic ¯elds. Hence, for any b 2 H 1 ð Þ, we de¯ne KðbÞ :¼ f 2 H 01 ð Þ : j rðxÞj Jc ðx; bÞ for a:e: x 2 ; r ¼ 0 a:e: in i ; i ¼ 1 ! Ig:

ð1:7Þ

682


We therefore arrive at a primal variational formulation (quasi-variational inequality): T (P) Find w 2 L 2 ð0; T ; Kðw þ be ÞÞ H 1 ð0; T ; L 2 ð ÞÞ such that wð; 0Þ ¼ w 0 ðÞ and Z T ð@ t ðw þ be Þ; wÞ dt 0 8 2 L 2 ð0; T ; Kðw þ be ÞÞ: ð1:8Þ 0

1.2. Analytical solution of primal problem For the special case of Bean's model, MðÞ M0 > 0 in (1.5), the above inequality is variational, and its analytical solution for simply connected cross sections is known in the spatially homogeneous case kðÞ ¼ k0 > 0, see Barrett and Prigozhin.5 We will now generalize this solution to the quasi-variational case under the additional assumption that, initially, the magnetic ¯eld depends only on the distance to the boundary of the domain , so that w 0 ðxÞ ¼ W 0 ðdistðx; @ÞÞ; and then extend it, under a similar condition, to multiply connected domains . In applications, for the geometric con¯guration considered here, the initial magnetic ¯eld is usually uniform, so this condition is trivially satis¯ed. Let , Jc be given by (1.5) with k 1 and M 2 CðR; ½M0 ; M1 Þ, and 0 w 2 Kðb 0 Þ, where b 0 ðxÞ ¼ w 0 ðxÞ þ be ð0Þ. The latter condition implies that W 0 ð0Þ ¼ 0 and jds W 0 ðsÞj MðB 0 ðsÞÞ, where B 0 ðsÞ ¼ W 0 ðsÞ þ be ð0Þ. Let us assume ¯rst that dt be ðtÞ 0 for all t 0. Then we de¯ne uðs; tÞ as a solution to the initial value problem @ s u ¼ MðuÞ;

uð0; tÞ ¼ be ðtÞ:

This yields that uðs; tÞ ¼ F 1 ðF ðbe ðtÞÞ sÞ; F 0 ðsÞ

ð1:9Þ

1

where F ðsÞ is such that ¼ ½MðsÞ and F ð0Þ ¼ 0: For all t 0, on noting that uð0; tÞ ¼ be ðtÞ be ð0Þ ¼ B 0 ð0Þ, we de¯ne the non-negative penetration depth dðtÞ :¼ supfs : uðr; tÞ B 0 ðrÞ 8 r 2 ½0; sg: We then de¯ne

~ tÞ ¼ Bðdistðx; @Þ; tÞ; bðx;

where Bðs; tÞ :¼

uðs; tÞ if s 2 ½0; dðtÞ; B 0 ðsÞ if s > dðtÞ:

ð1:10Þ

~ tÞ be ðtÞ solves the quasi-variational inequality (P). ~ tÞ ¼ bðx; We claim that wðx; ~ @ ¼ 0. If dð0Þ ¼ 0 then Bðs; 0Þ B 0 ðsÞ and wðx; ~ 0Þ ¼ w 0 ðxÞ. In Obviously, wj 0 0 0 addition, since ds B ðsÞ MðB ðsÞÞ and B ð0Þ ¼ be ð0Þ; we obtain that F ðB 0 ðsÞÞ F ðbe ð0ÞÞ s ¼ F ðuðs; 0ÞÞ. Due to the monotonicity of F , this yields that B 0 ðsÞ uðs; 0Þ. Hence, if dð0Þ > 0 the equality uðs; 0Þ ¼ B 0 ðsÞ holds for 0 s dð0Þ and so ~ 0Þ ¼ w 0 ðxÞ. Next, on setting T :¼ ð0; T Þ, once again Bðs; 0Þ B 0 ðsÞ and wðx; we de¯ne þ T :¼ fðx; tÞ 2 T : distðx; @Þ dðtÞg:


683

~ tÞ ¼ @ s uðs; tÞjs¼distðx;@Þ r½distðx; @Þ, and so Then we have a.e. on þ T that r wðx; þ ~ tÞÞ. Whereas on ~ tÞj ¼ j@ s uðs; tÞjs¼distðx;@Þ ¼ Mðbðx; j r wðx; T :¼ T n T , we have 0 ~ tÞ ¼ W ðdistðx; @ÞÞ þ be ð0Þ be ðtÞ, and so it is easily deduced that that wðx; ~ tÞÞ. Therefore w ~ As dt be ðtÞ 0 for all t 0, it follows ~ tÞj Mðbðx; ~ 2 KðbÞ. j r wðx; from (1.9) and (1.10) that @ t uðs; tÞ 0, dt dðtÞ 0, and so @ t b~ 0 a.e. on T . Furthermore, since @ t b~ ¼ 0 on T , to prove the inequality (1.8) we need to show that R ~ We note that functions from the convex set ~ ~ dt 0 for all 2 KðbÞ. dx þ @ t bð wÞ T ~ vanish on @, and w ~ decreases with distance from @ on þ at the ~ 2 KðbÞ KðbÞ T ~ for a function in KðbÞ. ~ Hence we conclude that w ~ maximal possible rate, MðbÞ, ~ and so the inequality (1.8) is satis¯ed. 0 in þ for any 2 KðbÞ, T

If dt be ðtÞ 0 for all t 0, an analytical solution can be found in a similar way. Now uðs; tÞ is a solution to @ s u ¼ MðuÞ with uð0; tÞ ¼ be ðtÞ, and the penetration depth determined as dðtÞ :¼ supfs : uðr; tÞ B 0 ðrÞ 8 r 2 ½0; sg: Finally, since at each moment in time these solutions depend only on the distance to the domain boundary, we can construct an analytical solution to (P) for arbitrary be 2 H 1 ð0; T Þ by combining the solutions described above. Solution for a multiply connected domain can be constructed in a similar way by i , as free Dirichlet sets (see Buttazzo and regarding the hole closures, 11 Stepanov ) and, correspondingly, de¯ning the distance from a point x 2 to the domain boundary as distD ðx; @ Þ Z 1 0;1 :¼ inf kðsðtÞÞjdt sðtÞj dt : s 2 C ð½0; 1; Þ; sð0Þ ¼ x; sð1Þ 2 @ ; 0

ð1:11Þ where

8 1 if x 2 ; > < I [ kðxÞ :¼ i: > : 0 if x 2 i¼1

ðx; @ Þ

Obviously, distD is the same for all points in each hole and, for x 2 , the distance is the length of the shortest path to the outer boundary of assuming the parts of a path inside holes are not counted. Assuming, for a multiply connected cross section, that the initial magnetic ¯eld is a function of distD ðx; @ Þ and taking into account that j rdistD ðx; @ Þj ¼ 1 a.e. in , one can now build an analytical solution to (P) exactly as above but using the distance function (1.11). 1.3. Dual problem The primal formulation allows one to calculate the magnetic ¯eld and, by Ampere's law, also the current density in a superconductor. Nevertheless, determining the electric ¯eld remains di±cult and, to solve this problem for the Bean model, several

684


approaches have been proposed.10,3,6,14 Here we derive a dual (mixed) formulation for models with critical current density depending on the magnetic ¯eld. Returning to possibly spatially inhomogeneous Jc de¯ned by (1.5), and recalling the de¯nition of F in (1.9), F 0 ðsÞ ¼ ½MðsÞ 1 and F ð0Þ ¼ 0, we have that the condition j ¼ r b 2 K 0 ðbÞ is equivalent to ½MðbÞ 1 j ¼ r F ðbÞ 2 K :¼ fv 2 ½L 2 ð Þ 2 : jvðxÞj kðxÞ for a:e: x 2 ; v ¼ 0 a:e: in i ; i ¼ 1 ! Ig: Note that e 2 @K 0 ðbÞ ðjÞ means that j 2 K 0 ðbÞ and ðe; i jÞ 0 for any i 2 K 0 ðbÞ. The latter is possible if and only if e ði jÞ 0 a.e. in for any i 2 K 0 ðbÞ. Hence ðe; v ½MðbÞ 1 jÞ 0 for any v 2 K and the current-voltage relation (1.2), for the choice of ¼ K 0 ðbÞ , can be rewritten as e 2 @K ð r F ðbÞÞ with its dual form as r F ðbÞ 2 @ K ðeÞ; Z kjej dx: where K ðeÞ :¼ sup fðe; vÞ K ðvÞg ¼ v2½L 2 ð Þ 2

ð1:12Þ

The dual form (1.12) yields for any test ¯eld u that Z ð r F ðbÞ; u eÞ kðjuj jejÞ dx:

We set u ¼ Rv and e ¼ Rq; where R½p1 ; p2 T :¼ ½p2 ; p1 T . Since ð r F ðbÞ; Rðv qÞÞ ¼ ð r½F ðbÞ; v qÞ and b be ðtÞ 2 H 01 ð Þ, we obtain for all test ¯elds v that Z kðjvj jqjÞ dx þ ðF ðbÞ F ðbe Þ; r ðv qÞÞ 0: ð1:13Þ

Similarly, Faraday's law takes the form @ t b r q ¼ 0. We introduce the Banach space V

M

2 : r v 2 L 2 ðDÞg; ðDÞ :¼ fv 2 ½MðDÞ

ð1:14Þ

is the Banach space of bounded Radon measures; i.e. MðDÞ ½CðDÞ , where MðDÞ the dual of CðDÞ. Our mixed formulation of (P), (1.8), is then: (Q) Find q 2 L 2 ð0; T ; V M ð ÞÞ and b 2 H 1 ð0; T ; L 2 ð ÞÞ such that bð; 0Þ ¼ b 0 ðÞ and Z T ð@ t b r q; Þ dt ¼ 0 8 2 L 2 ð T Þ; ð1:15aÞ 0

Z T Z 0

kðjvj jqjÞ þ ðF ðbÞ F ðbe Þ; r ðv qÞÞ dt 0 8 v 2 L 2 ð0; T ; V

where b 0 ðÞ ¼ w 0 ðÞ þ be ð0Þ and T :¼ ð0; T Þ.

M

ð ÞÞ;

ð1:15bÞ


685

Clearly, if the pair fq; bg satis¯es (Q), it follows from (1.15b) that rðF ðbÞ F ðbe ÞÞ ¼ 0 in each hole i and so rb ¼ 0 there as well. In addition, fq þ u; bg is also S a solution of (Q) if suppðuÞ i¼1!I i and r u ¼ 0. This corresponds to the wellknown fact that the eddy current problem does not determine the electric ¯eld in nonconducting media in a unique way. Furthermore, it follows immediately from (1.15a), (1.15b) that a solution fq; bg of (Q) is such that q 2 K 1 ð@ t bÞ :¼ fv 2 L 2 ð0; T ; V and

Z

T 0

M

ð H ÞÞ : ð r v; Þ HT ¼ ð@ t b; Þ HT 8 2 L 2 ð HT Þg

Z

Z

kjqj ¼

T

Z

min

v2K 1 ð@ t bÞ

kjvj: 0

ð1:16Þ

One can exploit this formulation to obtain q if @ t b is known. As was shown above, if k 1 in and the initial magnetic ¯eld is a function of distD ðx; @ Þ, an analytical solution is known for the solution b of the primal formulation (P); that is, bðx; tÞ ¼ BðdistD ðx; @ Þ; tÞ. One could use this b in (1.16) to recover q. Problems, similar to (1.16), and their dual formulations have been considered, for simply connected domains, in several works.24,17,18,12,14 We only mention here that the ridge (cut locus) of the domain plays an important role in the analysis in these works and that an integral representation of the solution to (1.16) with k 1 has been derived for being a polygonal domain18 and for domains with a smooth boundary.12,14 In our analysis of problem (Q) presented below we will also restrict ourselves to the case when and, moreover, assume that the domain is strictly starshaped. The later assumption is to ensure that certain density results hold; see (1.21) and (1.22a)(1.22c) below. However, our numerical method applies to the case of a multiply connected cross section and we will present a numerical example. In addition, to avoid perturbation of domain errors in our ¯nite element approximation, we will assume that is polygonal for ease of exposition. The outline of this paper is as follows. In the next section we introduce a regularised version, (Qr ), of our mixed formulation, (Q), by smoothing the nonR di®erentiable functional jvj by replacing jvj with 1r jvj r with r > 1. We then consider T the ¯nite element approximation, (Q h; r Þ, of (Qr ) using RaviartThomas elements of the lowest order with vertex sampling on the nonlinear term. We then establish stability bounds on this approximation, independent of the mesh parameter h, time step parameter and the regularisation parameter r. In Sec. 3, under the assumption (1.5), we prove subsequence convergence of this approximation, as the parameters, h and , go to zero and r goes to one, and establish existence of a solution fq; bg to (Q). Moreover, on further assuming that M 2 CðR; ½M0 ; M1 Þ, we show that w ¼ b be is a solution of (P). Finally in Sec. 4, we present some numerical experiments based on the discretisation (Q h; r ). This paper extends the dual formulation in Barrett and Prigozhin6 from variational to quasi-variational inequalities. We introduce, and prove the convergence of, a fully practical numerical scheme based on the dual formulation, (Q), which enables

686


one to approximate simultaneously both the electric and magnetic ¯elds. Hence, approximating the local rate of energy dissipation, j e ¼ Jc ðx; bÞjej, becomes straightforward. In proving the existence of a solution to the dual formulation (Q), we also show the existence of a solution to the primal quasi-variational inequality (P) involving a gradient constraint. We are not aware of any previous existence results for (P) in the literature, apart from the analytical solution above which holds for k 1 and special initial data w 0 . However, existence results are available in the case k 1 and M 2 CðR; ½M0 ; M1 Þ for a modi¯ed problem (Pp ) which includes a p-Laplacian term, and this term plays a crucial role in the analysis.23,1 Finally, we note that there are existence results20 for a primal quasi-variational 3d formulation of the double critical-state model where two di®erent constants, Jcjj and Jc? , limit the magnitudes, respectively, of the parallel and orthogonal (to the magnetic ¯eld) components of the current density. 1.4. Notation We end this section with a few remarks about the notation employed in this paper. Above and throughout we adopt the standard notation for Sobolev spaces on a bounded domain D with a Lipschitz boundary, denoting the norm of W ‘;p ðDÞ (‘ 2 N, p 2 ½1; 1) by jj jj‘;p;D and the semi-norm by j j‘;p;D . Of course, we have that j j0;p;D jj jj0;p;D . We extend these norms and semi-norms in the natural way to the corresponding spaces of vector functions. For p ¼ 2, W ‘;2 ðDÞ will be denoted by H ‘ ðDÞ with the associated norm and semi-norm written as, respectively, jj jj‘;D and j j‘;D . We set W 01;p ðDÞ :¼ f 2 W 1;p ðDÞ : ¼ 0 on @Dg, and H 01 ðDÞ W 01;2 ðDÞ. jDj will denote the measure of D and ð; ÞD the standard inner product on L 2 ðDÞ. When D , for ease of notation we write ð; Þ for ð; Þ . denote the space of continuous functions on D. As one can identify Let CðDÞ L 1 ðDÞ as a closed subspace of the Banach space of bounded Radon measures, ½CðDÞ H ; it is convenient to adopt the notation MðDÞ Z jj jjjjMðDÞ sup h; iCðDÞ ð1:17Þ :¼ < 1; D

Þ 2CðD

jj0;1;D 1

H CðDÞ: The condition r where h; iCðDÞ denotes the duality pairing on ½CðDÞ 2 2 v 2 L ðDÞ in (1.14) means that there exists u 2 L ðDÞ such that hv; riCðDÞ ¼ ðu; ÞD for any 2 C 01 ðDÞ. then there exist a We note that if fn gn0 is a bounded sequence in MðDÞ, subsequence fnj gnj 0 and a 2 MðDÞ such that as nj ! 1 i:e: hn ; iCðDÞ ð1:18Þ n ! vaguely in MðDÞ; ! 0 8 2 CðDÞ: j

j

Moreover, we have that

Z

Z lim inf nj !1

see e.g. p. 223 in Folland.16

D

jnj j

jj; D

ð1:19Þ


As well as the Banach space V the Banach space

M ðDÞ,

687

recall (1.14), we require for a given s 2 ½1; 1

V s ðDÞ :¼ fv 2 ½L s ðDÞ 2 : r v 2 L 2 ðDÞg:

ð1:20Þ

We note that V M ðDÞ and V s ðDÞ for s 2 ½1; 2Þ are not of local type; that is, v 2 does not imply that v 2 V M ðDÞ ½V s ðDÞ, see e.g. V M ðDÞ ½V s ðDÞ and 2 C 1 ðDÞ 25 p. 22 in Temam. Therefore, one has to avoid cuto® functions in proving any required density results. If is strictly star-shaped, which we shall assume for the analysis in this paper, one can show, using the standard techniques of change of variable and molli¯cation, that 2 ½C 1 ðÞ Moreover, for any v 2 V

is dense in V s ðÞ

M ðÞ,

if s 2 ð1; 1Þ:

ð1:21Þ

2 such that there exist fv j gj1 2 ½C 1 ðÞ 2 as j ! 1; vaguely in ½MðÞ

ð1:22aÞ

weakly in L 2 ðÞ as j ! 1; Z Z lim sup jv j j dx ¼ jvj

ð1:22bÞ

vj ! v r vj ! r v

j!1

ð1:22cÞ

We brie°y outline the proofs of (1.21) and (1.22a) for any positive 2 CðÞ. (1.22c). Without loss of generality, one can assume that is strictly star-shaped with respect to the origin. Then for v de¯ned on and > 1, we have that v ðxÞ ¼ vð 1 xÞ is de¯ned on :¼ . Applying standard Friedrich's molli¯ers J" to v , and a diagonal subsequence argument yield, for ! 1 and " ! 0 as j ! 1, the desired sequences fvgj1 demonstrating (1.21) if v 2 V s ðÞ and satisfying (1.22a) (1.22c) if V M ðÞ; see e.g. Lemma 2.4 in Barrett and Prigozhin,7 where such techniques are used to prove similar density results. Finally, throughout C denotes a generic positive constant independent of the regularisation parameter, r 2 ð1; 1Þ, the mesh parameter h and the time step parameter . Whereas, CðsÞ denotes a positive constant dependent on the parameter s. 2. Numerical Approximation of (Q) First, we gather together our assumptions on the data. (A1) Let R 2 be a strictly star-shaped domain with boundary @, so that with in (P), (1.8), and (Q), (1.15a), (1.15b). Let be 2 H 1 ð0; T Þ, k 2 CðÞ k1 kðxÞ k0 > 0 for all x 2 , and M : R ! ½M0 ; M1 R with M0 > 0. We shall assume that b 0 ðÞ ¼ w 0 ðÞ þ be ð0Þ with w 0 2 Kðb 0 Þ; i.e. b 0 ðÞ be ð0Þ 2 H 01 ðÞ and j rb 0 j kMðb 0 Þ a.e. in . We note that the assumptions (A1) do allow for an MðsÞ that is strictly positive on any bounded interval of R, but goes to zero as jsj ! 1. This follows since a solution of (P) is such that w ¼ b be ¼ 0 on @ ð0; T Þ and j rwj k1 M1 a.e. in T . Hence it follows that jbj maxt2½0;T jbe ðtÞj þ k1 M1 diamðÞ a.e. on T . Therefore an MðsÞ

688


that is strictly positive on any bounded interval, but goes to zero as jsj ! 1 can always be replaced by ( MðsÞ if jsj L; ð2:1Þ ML ðsÞ :¼ MðLÞ > 0 if jsj L k M diamðÞ þ max jb ðtÞj 1 1 e t2½0;T

without changing the problem (P). In order to prove the existence of, and approximate, solutions to (Q), (1.15a), (1.15b), we introduce F 2 CðRÞ, G 2 C 1 ðRÞ such that F ð0Þ ¼ Gð0Þ ¼ 0;

F 0 ðsÞ ¼ ½MðsÞ 1

and

G 0 ðsÞ ¼ F ðsÞ

8 s 2 R:

ð2:2Þ

Hence it follows from (A1) that G is strictly convex, i.e. GðaÞ GðcÞ < F ðaÞða cÞ 8 a; c 2 R;

a 6¼ c:

ð2:3Þ

We note also for later purposes that ðF ðcÞ F ðaÞÞc GðcÞ F ðaÞc c2 jajjcj c2 M a2 12 M0 2M1 4M1 M0

8 a; c 2 R:

ð2:4Þ

Next, for any r > 1, we regularise the non-di®erentiable nonlinearity j j by the strictly convex functional 1r j jr . We note for all a, c 2 R 2 that 1 @jaj r 1 ¼ jaj r2 ai ) jaj r2 a ða cÞ ½jaj r jcj r ; r @ai r ðjaj r2 a jcj r2 cÞ a

r1 ½jaj r jcj r : r

ð2:5aÞ ð2:5bÞ

In addition, for r 2 ð1; 2 we have that 1

jjaj r2 a jcj r2 cj ð1 þ 2 2ð2rÞ Þ 2 ja cj 8 a; c 2 R 2 ;

ð2:6Þ

see e.g. Lemma 1 in Chow.13 For a given r > 1, we then consider the following regularisation of (Q): (Qr ) Find q r 2 L 2 ð0; T ; V r ðÞÞ and br 2 H 1 ð0; T ; L 2 ðÞÞ such that br ð; 0Þ ¼ b 0 ðÞ and Z T ð@ t br r q r ; Þ dt ¼ 0 8 2 L 2 ðT Þ; ð2:7aÞ 0

Z

T 0

½ðkjq r j r2 q r ; vÞ þ ðF ðbr Þ F ðbe Þ; r vÞ dt ¼ 0 8 v 2 L 2 ð0; T ; V r ðÞÞ: ð2:7bÞ

For ease of exposition, we shall assume that is a polygonal domain to avoid perturbation of domain errors in the ¯nite element approximation. We make the following assumption


689

(A2) is polygonal. Let fT h gh>0 be a regular family of partitionings of into disjoint open triangles with h :¼ diamðÞ and h :¼ max2T h h , so that ¼ [2T h . Let @ be the outward unit normal to @, the boundary of . We then introduce the lowest order RaviartThomas ¯nite element spaces V

h

:¼ fv h 2 ½L 1 ðÞ 2 : v h j ¼ a þ c x; a 2 R 2 ; c 2 R 8 2 T h g V

1

ðÞ; ð2:8aÞ

S h :¼ f h 2 L 1 ðÞ : h j ¼ y 2 R 8 2 T h g: Here the constraint V , 0 2 T h that

h

V

1 ðÞ

ð2:8bÞ

yields for all v 2 V

v h j @ þ v h j 0 @ 0 ¼ 0

h

and for all adjacent triangles

on @ \ @ 0 :

ð2:9Þ

Let P h : L 1 ðÞ ! S h be such that ððI P h Þz; h Þ ¼ 0

8 h 2 S h:

ð2:10Þ

It follows for all s 2 ½1; 1, that if z 2 L s ðÞ, then jP h zj0;s; jzj0;s;

and

lim jðI P h Þzj0;s; ¼ 0:

h!0

ð2:11Þ

In addition, let 0 ¼ t0 < t1 < < tN1 < tN ¼ T be a partitioning of ½0; T into possibly variable time steps n :¼ tn tn1 , n ¼ 1 ! N. We set :¼ maxn¼1!N n and b ne :¼ be ðtn Þ

n ¼ 0 ! N:

ð2:12Þ

Our fully practical approximation of (Qr ) by V h is then: n h and B n 2 S h such that (Q h;¿ r ) For n 1, ¯nd Q r 2 V r h h h ðB nr n r Q nr ; h Þ ¼ ðB n1 r ; Þ 8 2 S ;

ðkjQ nr j r2 Q nr ; v h Þ h þ ðF ðB nr Þ F ðb ne Þ; r v h Þ ¼ 0 8 v h 2 V h ;

ð2:13aÞ ð2:13bÞ

where B 0r ¼ F 1 ðP h F ðb 0 ÞÞ. P Here ðv; zÞ h :¼ 2T h ðv; zÞ h , and 3 1 X ðv; zÞ h :¼ jj vðP j Þ zðP j Þ 8 v; z 2 ½Cð Þ 2 ; 3 j¼1

8 2 T h;

ð2:14Þ

where fP j g 3j¼1 are the vertices of . Hence ðv; zÞ h averages the integrand v z over each triangle at its vertices and is exact if v z is piecewise linear over the partitioning T h . For any r 1 and for any v h 2 V h , we have from the equivalence of

690


norms for v h and the convexity of j j r that Z 3 1 X jv h ðP j Þj r Cr ðjv h j r ; 1Þ h jv h j r dx ðjv h j r ; 1Þ h jj 3 j¼1

8 2 T h:

ð2:15Þ

It follows from (2.15) that X ðkjv h j r ; 1Þ h ¼ ½ð½P h kjv h j r ; 1Þ h þ ððI P h Þkjv h j r ; 1Þ h

2T

h

2T

h

X

ð½P h k jv h j r ; 1Þ jðI P h Þkj0;1; ðjv h j r ; 1Þ h

ðkjv h j r ; 1Þ CjðI P h Þkj0;1; jv h j r0;r;

8 v h 2 V h:

ð2:16Þ

We note from the de¯nition of B 0r , (2.2), (2.11) and (A1) that for all s 2 ½1; 1 jB 0r j0;s; ¼ jF 1 ðP h F ðb 0 ÞÞj0;s; M1 jP h F ðb 0 Þj0;s; M1 jF ðb 0 Þj0;s; M 1 jb 0 j0;s; C: M0

ð2:17Þ

Theorem 2.1. Let the Assumptions (A1) and (A2) hold. Then for all r > 1, for all regular partitionings T h of , and for all n > 0, there exists a unique solution, Q nr 2 V h and B nr 2 S h , to the nth step of (Q h; r ). Moreover, we have that max jB nr j0; þ max jF ðB nr Þj0; þ

n¼0!N

n¼0!N

N X

n ðk jQ nr j r ; 1Þ h CðT Þ;

ð2:18aÞ

n¼1

n 2 N N X X B r B n1 r1 r n r h þ max ðkjQ r j ; 1Þ þ n n j r Q nr j 20; C: r n¼1!N n 0; n¼1 n¼1 ð2:18bÞ Proof. It follows from (2.13a) that þ n r Q nr : B nr ¼ B n1 r

ð2:19Þ

On noting this, (2.5a) and (2.3), we see that (2.13b), on noting (2.13a) is the EulerLagrange equation for the strictly convex minimisation problem min E nr ðv h ; B n1 þ n r v h Þ; r

v h 2V

ð2:20aÞ

h

where E nr ðv h ; h Þ :¼

n ðkjv h j r ; 1Þ h n F ðb ne Þð r v h ; 1Þ þ ðGð h Þ; 1Þ r 8 v h 2 V h;

8 h 2 S h:

ð2:20bÞ


691

Hence the desired existence and uniqueness results for Q nr and B nr immediately follow from this and (2.19). Choosing h F ðB nr Þ F ðb ne Þ in (2.13a) and v h Q nr in (2.13b), and combining yields that n ðkjQ nr j r ; 1Þ h þ ðF ðB nr Þ F ðb ne Þ; B nr B n1 r Þ ¼ 0:

ð2:21Þ

We have from (2.21), (2.3), (2.2), (A1), (2.12) and (2.4) that ðGðB nr Þ F ðb ne Þ B nr ; 1Þ þ n ðkjQ nr j r ; 1Þ h n1 n1 n n1 n1 ðGðB n1 r Þ F ðb e Þ B r ; 1Þ ðF ðb e Þ F ðb e Þ; B r Þ n b e b n1 e n1 n1 1 jB n1 ðGðB n1 r Þ F ðb e Þ B r ; 1Þ þ M 0 n r j0; n 0; Z tn n M1 jj 2 n1 n1 ðGðB n1 jB n1 jd b j 2 dt r Þ F ðb e Þ B r ; 1Þ þ r j 0; þ 4M1 M 02 tn1 t e Z M1 jj tn n1 2 n1 n1 ½jb e j þ jdt be j 2 dt: ð1 þ n ÞðGðB n1 r Þ F ðb e Þ B r ; 1Þ þ M 02 tn1

ð2:22Þ It follows immediately from (2.22) that for n ¼ 1 ! N Z ðGðB nr Þ F ðb ne Þ B nr ; 1Þ e tn ðGðB 0r Þ F ðb 0e Þ B 0r ; 1Þ þ C

tn

½jbe j 2 þ jdt be j 2 dt :

0

ð2:23Þ In addition, we have from (2.4), (2.10), (A1) and (2.17) that ðGðB 0r Þ F ðb 0e Þ B 0r ; 1Þ ðF ðB 0r Þ F ðb 0e Þ; B 0r Þ ¼ ðF ðb 0 Þ F ðb 0e Þ; B 0r Þ ½jF ðb 0 Þj0;1; þ jF ðb 0e ÞjjB 0r j0;1; C:

ð2:24Þ

The ¯rst two bounds in the desired result (2.18a) then follow from (2.23), (2.24), (2.4), (2.2) and (A1). Summing (2.22) from n ¼ 1 to N yields, on noting (2.23), (2.24) and (A1), that N N ðGðB N r Þ F ðb e ÞB r ; 1Þ þ

N X

n ðkjQ nr j r ; 1Þ h CðT Þ:

ð2:25Þ

n¼1

Hence the third bound in (2.18a) follows from (2.25), (2.4) and (A1). ðb e ÞF ðb e Þ Choosing h ½F ðB r ÞF ðB r Þ½F in (2.13a), and noting (2.13b) and n (2.5b), yields for n ¼ 2 ! N that n n n1 B r B n1 ½F ðB nr Þ F ðB n1 r r Þ ½F ðb e Þ F ðb e Þ n ; n n n n n1 ¼ ð r Q r ; F ðB r Þ F ðB r Þ ½F ðb ne Þ F ðb n1 e ÞÞ n

n1

n

n1

r2 n1 ¼ ðk½jQ nr j r2 Q nr jQ n1 Q r ; Q nr Þ h r j r1 r h ðk½jQ nr j r jQ n1 r j ; 1Þ r

ð2:26aÞ

692


and

1

B 1r B 0r ½F ðB 1r Þ F ðB 0r Þ ½F ðb 1e Þ F ðb 0e Þ ; 1 1 ¼ ð r Q 1r ; ½F ðB 1r Þ F ðB 0r Þ ½F ðb 1e Þ F ðb 0e ÞÞ ð2:26bÞ

¼ ðk jQ 1r j r ; 1Þ h ð r Q 1r ; F ðB 0r Þ F ðb 0e ÞÞ: Next it follows, as W r0 ¼ F 1 ðP h F ðw 0 ÞÞ, r v h 2 S h for all v h 2 V (A1) and (2.15), that

h

and on noting

ð r Q 1r ; F ðB 0r Þ F ðb 0e ÞÞ ¼ ð r Q 1r ; F ðb 0 Þ F ðb 0e ÞÞ ¼ ðQ 1r ; r½F ðb 0 ÞÞ 1 r1 ðk; 1Þ h ðk; jQ 1r jÞ ðkjQ 1r j r ; 1Þ þ r r 1 ðkjQ 1r j r ; 1Þ h þ C: r

ð2:27Þ

Summing (2.26a), including (2.26b) and noting (2.27) yields for n ¼ 1 ! N that ‘ n X r1 B r B ‘1 F ðB ‘r Þ F ðB ‘1 r r Þ ðkjQ nr j r ; 1Þ h þ ‘ ; r ‘ ‘ ‘¼1 n ‘ ‘1 ‘ ‘1 X Br Br F ðb e Þ F ðb e Þ ð2:28Þ Cþ ‘ ; : ‘ ‘ ‘¼1 The ¯rst two bounds in the desired result (2.18b) then follow from (2.28), (2.2), (2.12) and (A1), on using a Young's inequality. Finally the third bound in (2.18b) then follows from the second bound in (2.18b) and (2.19). h, ¿ 3. Convergence of (Q r ) | Existence Theory for (Q)

As the stability bounds (2.18a), (2.18b) do not control spatial derivatives of n N fQ nr ; B nr g N n¼1 , except for f r Q r g n¼1 ; we cannot exploit compactness to get strong n N convergence of fB r g n¼1 , which we require to pass to the limit in F ðB nr Þ in (Q h; r ). h; Hence we prove the subsequence convergence of solutions to (Q r ), (2.13a), (2.13b), to solutions of (Q), (1.15a), (1.15b), as h; ! 0 and r ! 1 in stages. First, we introduce the following intermediate problem, a discrete in time approximation of (Qr ) for r > 1: (Q ¿r ) For n 1, ¯nd q nr 2 V r ðÞ and b nr 2 L 2 ðÞ such that 2 ðb nr n r q nr ; Þ ¼ ðb n1 r ; Þ 8 2 L ðÞ;

ðkjq nr j r2 q nr ; vÞ þ ðF ðb nr Þ F ðb ne Þ; r vÞ ¼ 0

8 v 2 V r ðÞ;

ð3:1aÞ ð3:1bÞ

where b 0r ¼ b 0 . We then show that the unique solution of (Q h; r ) converges to the unique solution of (Q r ) as h ! 0, by exploiting the monotonicity result (2.5a) and the monotonicity


693

of F . In order to achieve this we introduce the generalised interpolation operator I h : V r ðÞ \ ½W 1;r ðÞ n ! V h , where r > 1, satisfying Z ðv I h vÞ @ i ds ¼ 0 i ¼ 1 ! 3; 8 2 T h ; ð3:2Þ @i

where @ follows that

[ 3i¼1 @ i

and @ i are the corresponding outward unit normals on @ i . It ð r ðv I h vÞ; h Þ ¼ 0 8 h 2 S h :

In addition, we have for all 2 T

h

ð3:3Þ

and any s 2 ð1; 1 that

jv I h vj0;s; Cs h jvj1;s;

and jI h vj1;s; Cs jvj1;s; ;

ð3:4Þ

e.g. see Lemma 3.1 in Farhloul15 and the proof given there for s 2 is also valid for any s 2 ð1; 1. Furthermore, it follows from (2.5a), (2.14) and (3.4) for any r > 1 and any 2 T h that Z jI h vj r ðjI h vj r ; 1Þ h Ch jj j½I h v r j1;1; Crh jj jjvjj r1;1;

8 v 2 ½W 1;1 ðÞ 2 :

ð3:5Þ

Hence, similarly to (2.16), it follows from (3.5) and (3.4) that jðkjI h vj r ; 1Þ ðkjI h vj r ; 1Þ h j C½rk1 þ jðI P h Þkj0;1; jjvjj r1;1; 8 v 2 ½W 1;1 ðÞ 2 :

ð3:6Þ

Theorem 3.1. Let the Assumptions (A1) and (A2) hold. For all regular partitionings T h of , and for all time partitions f n g N n¼1 and for all r > 1 the unique h; solution fQ nr ; B nr g N to (Q ) is such that as h ! 0 r n¼1 Q nr ! q nr

weakly in ½L r ðÞ 2 ; n ¼ 1 ! N;

ð3:7aÞ

r Q nr ! r q nr

weakly in L 2 ðÞ; n ¼ 1 ! N;

ð3:7bÞ

B nr ! b nr

strongly in L 2 ðÞ; n ¼ 0 ! N;

ð3:7cÞ

F ðB nr Þ ! F ðb nr Þ

strongly in L 2 ðÞ; n ¼ 0 ! N;

ð3:7dÞ

where fq nr ; b nr g N n¼1 is the unique solution of (Q r ), (3.1a), (3.1b). Moreover, we have that N X r1 max ðkjq nr j r ; 1Þ þ n j r q nr j 20; þ max jb nr j0; þ max jF ðb nr Þj0; n¼0!N n¼0!N r n¼1!N n¼1 N X b n b n1 2 r þ max j rb nr j0;p; CðT Þ; þ n r ð3:8Þ n¼1!N n 0; n¼1

where p ¼

r r1 .

694


Proof. It follows immediately from the bounds (2.18a), (2.18b), (2.15), our n N assumptions on k and (2.5a) that there exist fq nr g N n¼1 and fb r g n¼0 such that n n N n (3.7a), (3.7b) hold for a subsequence ffQ r ; B r g n¼1 ghj >0 of ffQ r ; B nr g N n¼1 gh>0 , the n n N bounds on fq nr g N hold in (3.8) and there exist fb ; g g such that as hj ! 0 r r n¼1 n¼1 B nr ! b nr ;

F ðB nr Þ ! g nr

weakly in L 2 ðÞ; n ¼ 1 ! N:

ð3:9Þ

It follows from (2.11) and (2.2) that as h ! 0 F ðB 0r Þ ! F ðb 0 Þ F ðb 0r Þ;

B 0r ! b 0 b 0r

strongly in L 1 ðÞ:

ð3:10Þ

2 We now show that fq nr ; b nr g N n¼1 is a solution of (Q r ). For any 2 L ðÞ, we choose h h h P 2 S in the hj version of (2.13a). We can now pass to the limit hj ! 0 in this, and obtain, on noting (3.7b), (3.9) and (2.11), the desired result (3.1a) for n ¼ 1 ! N. It follows from (2.13a), (2.13b), (2.5a), the monotonicity of F and (2.16) that for any v h 2 V h and h 2 S h n n n1 1 h þ n r v hÞ n ðF ðB r Þ F ðb e Þ; B r n n n n h h ¼ 1 n ðF ðB r Þ F ðb e Þ; B r n r ðQ r v ÞÞ n n n h ¼ ðkjQ nr j r2 Q nr ; Q nr v h Þ h þ 1 n ðF ðB r Þ F ðb e Þ; B r Þ n n h h r 1 ðk½jQ nr j r jv h j r ; 1Þ h þ 1 n ðF ð Þ F ðb e Þ; B r Þ

ðkjv h j r2 v h ; Q nr v h Þ þ r 1 ½ðkjv h j r ; 1Þ ðkjv h j r ; 1Þ h n n n r h h 1 h þ 1 n ðF ð Þ F ðb e Þ; B r Þ Cr jðI P Þkj0;1; jQ r j 0;r; :

ð3:11Þ

2 and 2 L 2 ðÞ, we choose v h I h v and h P h in the hj For any v 2 ½C 1 ðÞ version of (3.11) with n ¼ ‘, for some integer ‘ 2 ½1; N. Assuming that B ‘1 ! b ‘1 r r 2 strongly in L ðÞ as hj ! 0, we can now pass to the limit hj ! 0 in this, and obtain, on noting (3.1a), (3.7a), (3.9), (2.11), (3.3), (2.6), (3.4) and (3.6) that ‘ ‘ ‘ ‘ 1 ‘ ðg r F ðb e Þ; b r ‘ r ðq r vÞÞ ‘ ‘ ‘1 þ ‘ r vÞ ¼ 1 ‘ ðg r F ðb e Þ; b r ‘ ‘ ðkjvj r2 v; q ‘r vÞ þ 1 ‘ ðF ðÞ F ðb e Þ; b r Þ 2 L 2 ðÞ: 8 fv; g 2 ½C 1 ðÞ

ð3:12Þ

2 is dense in V r ðÞ, recall (1.21), and fq ‘ ; b ‘r g 2 V r ðÞ L 2 ðÞ, it folAs ½C 1 ðÞ r lows that (3.12) holds for all fv; g 2 V r ðÞ L 2 ðÞ. For any ¯xed z 2 V r ðÞ and

2 R>0 , choosing v q ‘r z and b ‘r in (3.12) yields that ½ðkjq ‘r zj r2 ðq ‘r zÞ; zÞ þ ðg ‘r F ðb ‘e Þ; r zÞ 0:

ð3:13Þ

Sending ! 0, and repeating the above for all z 2 V r ðÞ yields that ðkjq ‘r j r2 q ‘r ; vÞ þ ðg ‘r F ðb ‘e Þ; r vÞ ¼ 0 8 v 2 V r ðÞ:

ð3:14Þ


695

For any ¯xed ’ 2 L 2 ðÞ and 2 R>0 , choosing v q ‘r and b ‘r ’ in (3.12) yields that ½ðF ðb ‘r ’Þ ðg ‘r ; ’Þ 0:

ð3:15Þ

Sending ! 0, and repeating the above for all ’ 2 L 2 ðÞ yields that g ‘r ¼ F ðb ‘r Þ, and hence this and (3.14) yield that (3.1b) holds with n ¼ ‘. It is a simple matter to show that for a given b ‘1 2 L 2 ðÞ the solution fq ‘r ; b ‘r g is r unique. Hence the whole sequence converges in (3.7a), (3.7b) and weakly in (3.7c), (3.7d) for n ¼ ‘. To complete the induction step, we need to show that (3.7c) holds, and hence (3.7d), for n ¼ ‘. From (3.1a) and (2.13a) with n ¼ ‘ and h F ðB ‘r Þ, we have that ‘ ðb ‘r B ‘r ; F ðb ‘r Þ F ðB ‘r ÞÞ ¼ ðb ‘r B ‘r ; F ðb ‘r ÞÞ ðb ‘1 B ‘1 r r ; F ðB r ÞÞ

‘ ð r ðq ‘r Q ‘r Þ; ½F ðB ‘r Þ F ðb ‘e Þ þ F ðb ‘e ÞÞ: ð3:16Þ Next we note from (3.1b) and (2.13b) with n ¼ ‘ and v q ‘r and v h Q ‘r that ð r ðq ‘r Q ‘r Þ; F ðB ‘r Þ F ðb ‘e ÞÞ ¼ ð r q ‘r ; F ðB ‘r Þ F ðb ‘r ÞÞ ð r q ‘r ; F ðb ‘r Þ F ðb ‘e ÞÞ þ ð r Q ‘r ; F ðB ‘r Þ F ðb ‘e ÞÞ ¼ ð r q ‘r ; F ðB ‘r Þ F ðb ‘r ÞÞ þ ðkjq ‘r j r ; 1Þ ðkjQ ‘r j r ; 1Þ h :

ð3:17Þ

In addition, we note from (2.16) and (2.5a) that ðkjq ‘r j r ; 1Þ ðkjQ ‘r j r ; 1Þ h ðk½jq ‘r j r jQ ‘r j r ; 1Þ þ CjðI P h Þkj0;1; jQ ‘r j r0;r; rðkjq ‘r j r2 q ‘r ; q ‘r Q ‘r Þ þ CjðI P h Þkj0;1; jQ ‘r j r0;r; :

ð3:18Þ

The desired result (3.7c), and hence (3.7d), for n ¼ ‘ follow from combining (3.16) (3.18) and noting (2.2), (3.7a) for n ¼ ‘, (3.7c) for n ¼ ‘ 1, the weak convergence versions of (3.7c), (3.7d) for n ¼ ‘ and (2.11). It follows from (3.10) and the above induction step that the desired results (3.7a) (3.7d) hold for the stated range of n; and furthermore, fq nr ; b nr g N n¼1 is the unique solution of (Q r ). The ¯rst ¯ve bounds in (3.8) then follow directly from (2.18a), (2.18b), (3.7a)(3.7d) and (2.5a). Finally, we need to prove the sixth bound in (3.8). First, we note from (3.1b), (A1) and the ¯rst bound in (3.8) that for n ¼ 1 ! N r1 n r jq jðF ðb nr Þ F ðb ne Þ; r vÞj k1 jq nr j r1 jvj j þ C jvj0;r; 0;r; 0;r; r 0;r; r CðT Þjvj0;r;

8 v 2 V r ðÞ:

ð3:19Þ

It follows from (3.19) for any integer n 2 ½1; N, as C 01 ðÞ is dense in L r ðÞ, that the distributional gradient of F ðb nr Þ belongs to the dual of ½L r ðÞ 2 . Hence for n ¼ 1 ! N

696


we have that r½F ðb nr Þ 2 ½L p ðÞ 2 and jð r½F ðb nr Þ; vÞj CðT Þjvj0;r;

8 v 2 ½L r ðÞ 2 :

ð3:20Þ

Choosing vðxÞ ¼ j r½F ðb nr ðxÞÞj p2 r½F ðb nr ðxÞÞ if j r½F ðb nr ðxÞÞj 6¼ 0, and vðxÞ ¼ 0 otherwise, in the above and as F is globally Lipschitz, recall (2.2) and (A1), we obtain the ¯nal bound in (3.8). Let n n n qþ bþ bþ e ðtÞ :¼ b e ; r ð; tÞ :¼ b r ðÞ; r ð; tÞ :¼ q r ðÞ; ðt tn1 Þ n ðt tÞ n1 b r ðÞ þ n b r ðÞ t 2 ðtn1 ; tn ; b cr ð; tÞ :¼ n n

ð3:21Þ n 1:

It follows from (3.21), (3.8), (2.12) and as be 2 H 1 ð0; T Þ that c bþ r br ! 0

strongly in L 2 ðT Þ;

bþ e ! be

strongly in L 1 ð0; T Þ as ! 0:

ð3:22Þ

Adopting the notation (3.21), (Q r ) can be restated as: Z T c @b r r qþ ; dt ¼ 0 8 2 L 2 ðT Þ; r @t 0 Z T þ r2 þ ½ðkjq þ q r ; vÞ þ ðF ðb þ r Þ F ðb e Þ; r vÞ dt ¼ 0 rj

ð3:23aÞ

8 v 2 L 2 ð0; T ; V r ðÞÞ:

ð3:23bÞ

0

Theorem 3.2. Let the Assumptions (A1) and (A2) hold. For all time partitions þ þ f n g N n¼1 and for all r 2 ð1; 2 such that r ! 1 as ! 0; the unique solution fq r ; b r g to þ þ þ þ (Q r ) is such that there exists a subsequence fq r ; b r g j >0 of fq r ; b r g>0 and fq; bg 2 L 2 ð0; T ; V M ðÞÞ ½H 1 ð0; T ; L 2 ðÞÞ \ L 2 ð0; T ; H 1 ðÞÞ such that as j ! 0 qþ r ! q r qþ r ! rq c bþ r ; br ! b

@b cr @b ! @t @t c bþ r ; br ! b

2 Þ; vaguely in L 2 ð0; T ; ½MðÞ

ð3:24aÞ

weakly in L 2 ðT Þ;

ð3:24bÞ

weakly in L 2 ð0; T ; H 1 ðÞÞ;

ð3:24cÞ

weakly in L 2 ðT Þ;

ð3:24dÞ

strongly in L 2 ðT Þ:

ð3:24eÞ

Moreover, fq; bg solves (Q), (1.15a), (1.15b) with .


697

Proof. The bound (3.8) yields immediately, on noting that r 2 ð1; 2, (3.21) and our assumption (A1) on k, the ¯rst four bounds in c2 Z T @b r 2 þ 2 c 2 þ 2 jr qþ j þ jjb jj þ jjb jj þ þ jq j ð3:25Þ r 1; r 1; r 0; r 0;r; dt CðT Þ: @t 0; 0 Next we have from (3.1b), with v q nr , (A1) and (3.8) that for n ¼ 1 ! N k0 jq nr j r0;r; ðF ðb ne Þ F ðb nr Þ; r q nr Þ CðT Þj r q nr j0; :

ð3:26Þ

Therefore the last bound in (3.25) follows immediately from (3.26), (3.21) and the ¯rst bound in (3.25). The subsequence convergence results (3.24a)(3.24d) follow immediately from (3.25) and (3.22). The strong convergence result (3.24e) follows from (3.24c), (3.24d), a standard compactness result and (3.22). As b cr ð; 0Þ ¼ b 0 ðÞ, it follows from the above that bð; 0Þ ¼ b 0 ðÞ. It follows immediately from passing to the limit j ! 0 in (3.23a), on noting (3.24b), (3.24d) and (3.22), that fq; bg satisfy (1.15a) with . 2 Þ, we choose v q þ z in (3.23b) to yield, on Given any z 2 L 2 ð0; T ; ½C 1 ðÞ r noting (2.5a), that Z T Z T þ þ þ r2 þ ðF ðb r Þ F ðb e Þ; r ðq r zÞÞ dt ¼ ðkjq þ qr ;qþ rj r zÞ dt 0 0 Z T r r ð3:27Þ ðk½jq þ r 1 r j jzj ; 1Þ dt: 0

It follows immediately from (3.24b), (3.24e), (2.2) and our assumptions on M that for 2Þ any z 2 L 2 ð0; T ; ½C 1 ðÞ Z T Z T þ þ þ ðF ðb r Þ F ðb e Þ; r ðq r zÞÞ dt ! ðF ðbÞ F ðbe Þ; r ðq zÞÞ dt 0 0 as j ! 0: ð3:28Þ 2Þ Next, we note that for any z 2 L 2 ð0; T ; ½C 1 ðÞ Z T Z T ðkjzj r ; 1Þ dt ! ðkjzj; 1Þ dt r 1 0

as j ! 0:

Finally, it follows from (3.24a), and similarly to (1.19), that Z T Z T Z T Z þ r r ðkjq þ j ; 1Þ dt lim inf ðkjq j; 1Þ dt kjqj lim inf r 1 dt: r r j !0

0

ð3:29Þ

0

j !0

0

0

ð3:30Þ

Combining Eqs. (3.27)(3.30), it follows that fq; bg satisfy (1.15b) for any 2 Þ. The desired result, fq; bg satis¯es (1.15b) with for v 2 L 2 ð0; T ; ½C 1 ðÞ any v 2 L 2 ð0; T ; V M ðÞÞ, and hence fq; bg solves (Q) with , then follows from the density results (1.22a)(1.22c). Theorem 3.3. Let the assumptions of Theorem 3.2 hold. In addition, let M 2 CðR; ½M0 ; M1 Þ. We then have that any solution fq; bg of (Q) with is such that

698


w ¼ b be 2 L 2 ð0; T ; Kðw þ be ÞÞ \ H 1 ð0; T ; L 2 ðÞÞ and wð; 0Þ ¼ w 0 ð0Þ. In addition, we have that Z T Z kMðw þ be Þjqj þ ðw; r qÞ dt ¼ 0: ð3:31Þ 0

Moreover, w solves the quasi-variational inequality (P), (1.8), with . Proof. We adapt the proof of Theorem 3.2 in Barrett and Prigozhin.6 Any solution fq; bg of (Q) yields that w ¼ b be 2 H 1 ð0; T ; L 2 ðÞÞ and wð; 0Þ ¼ w 0 ðÞ. Let Z TZ kjvj þ ðF ðbÞ F ðbe Þ; r vÞ dt: ð3:32Þ ZðvÞ :¼

0

It follows from (1.15b) with and (3.32) that ZðqÞ ¼ Z :¼

min

v2L 2 ð0;T ;V

M ðÞÞ

ZðvÞ Zð0Þ ¼ 0:

ð3:33Þ

If Z < 0 then, for any minimizing sequence fv j g, we obtain that Zð2v j Þ ! 2Z < Z, which is a contradiction. Hence Z ¼ 0, and so we have that ZðvÞ 0 ¼ ZðqÞ for any v 2 L 2 ð0; T ; V M ðÞÞ. Since this is true also for v, we obtain that Z T Z TZ ðF ðbe Þ F ðbÞ; r vÞ dt kjvj dt 0

0

8 v 2 L 2 ð0; T ; V

ðÞÞ

ð3:34aÞ

kjqj dt:

ð3:34bÞ

M

and Z

Z T Z

T

ðF ðbe Þ F ðbÞ; r qÞ dt ¼ 0

0

It follows from (3.34a), as C 01 ð0; T ; ½C 01 ðÞ 2 Þ is dense in L 1 ð0; T ; ½L 1 ðÞ 2 Þ and k 2 CðÞ, that the distributional gradient of F ðbÞ belongs to the dual of 1 1 we have that r½F ðbÞ 2 L ð0; T ; ½L ðÞ 2 Þ. Hence, as kðxÞ k0 > 0 for all x 2 , 2 1 1 L ð0; T ; ½L ðÞ Þ and Z T ðk 1 r½F ðbÞ; vÞ dt jjvjjL 1 ð Þ 8 v 2 L 1 ð0; T ; ½L 1 ðÞ 2 Þ: ð3:35Þ T 0

For any p 2 ½1; 1Þ, choosing vðx; tÞ ¼ j½kðxÞ 1 r½F ðbðx; tÞÞj p2 ½kðxÞ 1 r½F ðbðx; tÞÞ if j r½F ðbðx; tÞÞj 6¼ 0, and vðx; tÞ ¼ 0 otherwise, in the above, and noting the continuity of the p norm for p 2 ½1; 1, we obtain that jjk 1 r½F ðbÞjjL 1 ðT Þ 1: It follows from (3.34a) and (3.36) that Z T Z ½F ðbÞ F ðb Þv n ds dt 2k1 jjvjjL 1 ðT Þ e 0

@

ð3:36Þ

8 v 2 L 2 ð0; T ; V 2 ðÞÞ;

ð3:37Þ


699

where n is the outward unit normal to @. Without loss of generality, we can assume that is strictly star-shaped with respect to the origin and so ½x nðxÞ n0 > 0 for a.e. x 2 @. Choosing v ¼ j ½F ðbÞ F ðbe Þx in (3.37), where j 2 H 1 ðÞ is such that j ¼ 1 on @ and jjj jjL 1 ðÞ ! 0 as j ! 1, yields that F ðbÞ ¼ F ðbe Þ a.e. on @ ð0; T Þ and hence b ¼ be a.e. on @ ð0; T Þ. For example, one can choose j in the following way. For integers j 1, let j 2 H 1 ðÞ be the extension by zero to j :¼ j1 j of the unique solution of Laplace's equation in nj satisfying the Dirichlet boundary conditions j ¼ 1 on @ and j ¼ 0 on @j . The weak maximum principle yields that j ðxÞ 2 ½0; 1 for a.e. x 2 , and so jjj jjL 1 ðÞ ! 0 as j ! 1. Combining b ¼ be a.e. on @ ð0; T Þ with (3.36) yields, on recalling (2.2), that w ¼ b be 2 L 2 ð0; T ; Kðw þ be ÞÞ with . 2 Þ satisfy (1.22a)(1.22c), with v q, for a.a. Let fq j gj1 2 L 2 ð0; T ; ½C 1 ðÞ t 2 ð0; T Þ. It follows as w 2 L 2 ð0; T ; Kðw þ be ÞÞ that Z 0 ½kMðbÞjq j j þ w r q j dx dt Z

T

Z

T

¼ ¼ T

½kMðbÞjq j j rw q j dx dt MðbÞ½kjq j j r½F ðbÞ q j dx dt

Z

M1 Z

T

¼ M1 T

½kjq j j r½F ðbÞ q j dx dt ½kjq j j þ ½F ðbÞ F ðbe Þ r q j dx dt:

ð3:38Þ

Passing to the limit j ! 1 in (3.38), on noting (1.22a)(1.22c), (3.34b) and that M 2 CðR; ½M0 ; M1 Þ, yields the desired result (3.31). Similarly, we have for any 2 Kðw þ be Þ that Z Z Z r q j dx dt ¼ r q j dx dt kMðw þ be Þjq j j dx dt; ð3:39Þ T

T

T

and hence that Z TZ ð; r qÞ dt kMðw þ be Þjqj dt:

Z

T 0

0

ð3:40Þ

Choosing w in (1.15a) with and 2 L 2 ð0; T ; Kðw þ be ÞÞ, and noting (3.31) and (3.40), yields that w solves the primal variational inequality (P), (1.8).

4. Numerical Experiments To compute the unique solution fQ nr ; B nr g of our approximation ðQ h; r Þ at the nth time step we substitute (2.19) into (2.13b). This yields a nonlinear problem for

700


Q :¼ Q nr 2 V h : þ n r QÞ F ðb ne Þ; r v h Þ ¼ 0 ðkjQj r2 Q; v h Þ h þ ðF ðB n1 r

8 v h 2 V h;

ð4:1Þ

which we solved iteratively. At the ðj þ 1Þth iteration, we ¯rst solve the following 1 linear problem for Q jþ 2 2 V h : þ n r Q j Þ; r v h Þ ðF ðB n1 r þ n ð½MðB n1 þ n r Q j Þ 1 r ðQ jþ 2 Q j Þ F ðb ne Þ; r v h Þ r 1

jþ 2 Q j Þ; v h Þ h ¼ 0 þ ðk½jQ j j r2 Q j þ jQ j j r2 " ðQ 1

8 v h 2 V h;

ð4:2Þ

where jvj" :¼ ðjvj 2 þ " 2 Þ 2 with " 1, and we have recalled that F 0 ðsÞ ¼ ½MðsÞ 1 . 1 Clearly, the linear system (4.2) is well-posed. Finally we set Q jþ1 ¼ Q jþ 2 þ ð1 ÞQ j , where 1 is an over-relaxation parameter. In all of our examples below, we choose r ¼ 1 þ 10 7 , " ¼ 10 9 and ¼ 1:2. Let E h be the set of edges of T h , and e ðxÞ be the vector basis function associated with the edge e 2 E h in the RaviartThomas ¯nite element space V h , see e.g. Bahriawati and Carstensen4 for details. Then any v h 2 V h can be represented as P P h h h e2E h v e e , and we de¯ne jjv jjE h :¼ e2E h jv e jjej. Our stopping criterion for the above iterative scheme was 1

jjQ jþ1 Q j jjE h "NL ; jjQ jþ1 jjE h where "NL is a given tolerance. We set "NL ¼ 2 10 4 throughout the examples below. When this stopping criterion was satis¯ed, we set Q nr ¼ Q jþ1 and computed the magnetic ¯eld B nr using (2.19). We used the MATLAB PDE Toolbox for the domain triangulation, and curved domains were approximated by polygons. The ¯nite element meshes in our examples below contained about 7000 triangles. As in Barrett and Prigozhin,6 the parameters in the numerical simulations were chosen on assuming that the dimensionless variables ðx; t; . . .Þ were obtained from the original variables ðx 0 ; t 0 ; . . .Þ as follows: x¼

x0 ; L

t¼

t0 ; t0

j¼

j0 ; j0

b¼

b0 ; Lj0

e¼

e 0 t0 ; L 2 j0

where L is the characteristic cross section size (the maximal horizontal extension in the plots below), j0 is the value of the critical current density Jc for a zero magnetic ¯eld, and the superconductors were homogeneous with kðxÞ 1. In the examples below we assumed that jdt b 0e j is a constant, which was scaled to unity by choosing the time scale t0 appropriately. In our numerical simulations, the time step was uniform with ¼ 0:005. Initially, the magnetic ¯eld was zero, i.e. w 0 be ð0Þ ¼ 0, and we assumed a growing external ¯eld, be ðtÞ ¼ t; except for the last example on hysteresis. As our ¯rst example we compare, see Fig. 1, for a rectangular cross section , the Bean model (Jc ¼ 1 in dimensionless variables) with the Kim model,


701

Fig. 1. Rectangular cross section, be ¼ t ¼ 0:08. The Bean model (top) and the Kim model (bottom). The magnetic ¯eld (a) and the electric ¯eld ((b) jej, (c) the vector ¯eld).

702


1 Jc ðsÞ ¼ ð1 þ jsj a Þ ; here and below we set a ¼ 0:02 for this model. Since the critical current density in the Kim model decreases with the growth of magnetic ¯eld, the shielding eddy current is weaker and magnetic ¯eld penetrates further inside the ~ is given by (1.10). To estimate the accuracy of our superconductor; this ¯eld, b ¼ b, ~ ; tn Þ and x ~ n 2 S h , where B ~ n j ¼ bðx numerical solution B nr we compared it with B h is the centroid of for all 2 T . We obtained that

~ n B nr jj0;1; jjB < 0:002: ~ n be ðtn Þjj0;1; jjB The electric ¯eld found using Kim's model is stronger, but qualitatively similar to that in the Bean model.10 It has the same zig-zag shape, is zero in the zero-magnetic¯eld core, it is parallel to level contours of the magnetic ¯eld, and vanishes along the discontinuity lines of the current density. Near concave corners of the electric ¯eld becomes singular, see Fig. 2 where is a circle with a section removed.

(a)

(b)

(c) Fig. 2. Circular cross section with a section removed, be ¼ t ¼ 0:07, and the Kim model. The magnetic ¯eld (a) and the electric ¯eld ((b) jej, (c) the vector ¯eld).


703

Although our analysis in Secs. 2 and 3 holds only for continuous kðxÞ k0 > 0, our numerical method (Q h; r ) extends to piecewise constant k, where the discontinuities are aligned with the mesh. Therefore one can simulate numerically the magnetisation of a superconductor with a multiply connected cross section, see Fig. 3, by ¯lling the hole and setting k ¼ k0 1 there. For this example, we chose k0 ¼ 10 6 and so the eddy current in the hole is negligible; as in the other examples, k 1 in the superconductor. Similarly to the Bean model6 when the penetration zone reaches the hole boundary, the magnetic ¯eld begins to penetrate the hole via an in¯nitesimally thin channel and the electric ¯eld becomes singular. This singularity is evident in Fig. 3, although the channel is slightly smeared by the relatively coarse mesh. For a cross section with only one hole, 1 , we have that distD ðx; @ Þ ¼ minfdistðx; @ Þ; distðx; @1 Þ þ distð@1 ; @ Þg: Calculating this distance is not di±cult and we can substitute it into the derived analytical solution for the magnetic ¯eld, see Sec. 1.2, for multiply connected cross sections. The relative error in the L 1 norm, estimated as

(a)

(b) Fig. 3. Cross section with a hole, be ¼ t ¼ 0:09, and the Kim model. The magnetic ¯eld (a) and the electric ¯eld (level contours of jej and vector ¯eld, (b)).

704


in the ¯rst example, was 0.011 for a mesh with about 7000 elements and 0.007 for a re¯ned mesh with about 15,000 elements (the same time step, ¼ 0:005, was used in both cases). The di®erence between various critical state models is best exhibited by the corresponding magnetisation loops, showing the behaviour of the magnetic momentum of a superconductor when the external ¯eld changes cyclically. For the longitudinal con¯guration considered in R this paper, the magnetic moment of a superconductor per unit of length is m ¼ ðbðx; tÞ be ðtÞÞ dx. The hysteresis loops in Fig. 4 were computed for the superconductor with a rectangular cross section as in the ¯rst example for three di®erent models: the Bean model, the Kim model, and a model with a secondary peak in the Jc ðÞ dependence. In the latter case the critical current density was taken similar to that in Johansen et al.19; that is, 1 þ c ðð jsj c Þ 2 þ c 2 Þ 1 , in dimensionless variables, with a ¼ Jc ðsÞ ¼ ð1 þ jsj 1 a 2 3 a jÞ 0:02 as in the Kim model, c1 ¼ 1, c2 ¼ 8 and c3 ¼ 1.

m

m 0.03 0.05

0

0

-0.05

-0.03

he

he -0.4

0

-0.4

0.4

0

(a)

0.4

(b) m 0.05

0

-0.05 -1

he -0.5

0

0.5

(c) Fig. 4. Hysteresis loops (dimensionless variables): Bean's model (a), Kim's model (b), and the model with a secondary peak in Jc ðbÞ (c).


705

In conclusion, we note that although quasi-variational inequalities, arising in critical-state problems with critical current density depending on the magnetic ¯eld, are much more di±cult mathematical problems than the variational inequalities arising in Bean's model, their numerical solution based on the dual formulation presented in this work is practically as e±cient as the solution of the Bean model problems in Barrett and Prigozhin.6 Acknowledgement We acknowledge support of the Sixth EU Framework Programme Transnational Access implemented as Speci¯c Support Action (Dryland Research SSA). References 1. A. Azevedo and L. Santos, Convergence of convex sets with gradient constraint, J. Convex Anal. 11 (2004) 285301. 2. A. Badía and C. López, The critical state in type II superconductors with cross-°ow e®ects, J. Low Temp. Phys. 130 (2003) 129153. 3. A. Badía-Majós and C. López, Electric ¯eld in hard superconductors with arbitrary cross section and general critical current law, J. Appl. Phys. 95 (2004) 80358040. 4. C. Bahriawati and C. Carstensen, Three Matlab implementations of the lowest-order RaviartThomas MFEM with a posteriori error control, Comput. Meth. Appl. Math. 5 (2005) 333361. 5. J. W. Barrett and L. Prigozhin, Bean's critical-state model as the p ! 1 limit of an evolutionary p-Laplacian, Nonlinear Anal. 42 (2000) 977993. 6. J. W. Barrett and L. Prigozhin, Dual formulations in critical state problems, Interfaces and Free Boundaries 8 (2006) 347368. 7. J. W. Barrett and L. Prigozhin, A mixed formulation of the MongeKantorovich equations, M 2 AN 41 (2007) 10411060. 8. C. P. Bean, Magnetization of high-¯eld superconductors, Rev. Mod. Phys. 36 (1964) 3139. 9. A. Bossavit, Numerical modeling of superconductors in three dimensions A model and a ¯nite-element method, IEEE Trans. Mag. 30 (1994) 33633366. 10. E. H. Brandt, Electric ¯eld in superconductors with rectangular cross section, Phys. Rev. B 52 (1995) 1544215457. 11. G. Buttazzo and E. Stepanov, Transport density in MongeKantorovich problems with Dirichlet conditions, Discr. Cont. Dynam. Syst. 12 (2005) 607628. 12. P. Cannarsa and P. Cardaliaguet, Representation of equilibrium solutions to the table problem for growing sandpiles, J. Eur. Math. Soc. 6 (2004) 435464. 13. S.-S. Chow, Finite element error estimates for non-linear elliptic equations of monotone type, Numer. Math. 54 (1989) 373393. 14. G. Crasta and A. Malusa, A variational approach to the macroscopic electrodynamics of anisotropic hard superconductors, Arch. Rational Mech. Anal. 192 (2009) 87115. 15. M. Farhloul, A mixed ¯nite element method for a nonlinear Dirichlet problem, IMA J. Numer. Anal. 18 (1998) 121132. 16. G. B. Folland, Real Analysis: Modern Techniques and their Applications, 2nd edn. (Wiley-Interscience, 1984). 17. U. Janfalk, Behaviour in the limit, as p ! 1, of minimizers of functionals involving p-Dirichlet integrals, SIAM J. Math. Anal. 27 (1996) 341360.

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18. U. Janfalk, On a minimization problem for vector ¯elds in L 1 , Bull. London Math. Soc. 28 (1996) 165176. 19. T. H. Johansen, M. R. Koblischka, H. Bratsberg and P. O. Hetland, Critical-state model with a secondary high-¯eld peak in Jc ðBÞ, Phys. Rev. B 56 (1997) 1127311278. 20. Y. Kashima, On the double critical-state model for type-II superconductivity in 3D, M 2 AN 42 (2008) 333374. 21. Y. B. Kim, C. F. Hempstead and A. R. Strnad, Critical persistent currents in hard superconductors, Phys. Rev. Lett. 9 (1962) 306309. 22. L. Prigozhin, On the Bean critical-state problem in superconductivity, Euro. J. Appl. Math. 7 (1996) 237247. 23. J. F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. XXX1X (2000) 153169. 24. G. Strang, L 1 and L 1 approximation of vector ¯elds in the plane, Lect. Notes in Num. Appl. Anal. 5 (1982) 273288. 25. R. Temam, Mathematical Methods in Plasticity (Gauthier-Villars, 1985).