Robust a priori error analysis for the approximation of degree-one Ginzburg-Landau vortices

Transcription

1 Robust a priori error analysis for the approximation of degree-one Ginzburg-Landau vortices Sören Bartels Department of Mathematics, University of Maryland, College Park, MD 74, USA Abstract. This article discusses the numerical approximation of time dependent Ginzburg-Landau equations. Optimal error estimates which are robust with respect to a large Ginzburg-Landau parameter are established for a semi-discrete in time and a fully discrete approximation scheme. The proofs rely on an asymptotic expansion of the exact solution and a stability result for degree-one Ginzburg-Landau vortices. The error bounds prove that degree-one vortices can be approximated robustly while unstable higher degree vortices are critical. The theoretical results are illustrated by numerical experiments. Key Words. Ginzburg-Landau equations, numerical approximation, error analysis, spectral estimate, finite element method. AMS Subject Classification: 35K59, 35Q99, 53A.. INTRODUCTION A mathematical model due to Ginzburg and Landau [8] for the description of certain superconducting materials involves a minimization of the energy functional G κ (u, A = u iau dx + κ ( u dx + curl A h ext dx Ω Ω where Ω R n, n =, 3, is a bounded Lipschitz domain and represents the region occupied by the superconductor. The complex-valued function u is the condensate wave function, A is the magnetic potential associated to the induced magnetic field, h ext represents an external magnetic field, and κ > is the Ginzburg-Landau parameter. The quantity u gives the density of cooper pairs of electrons that produce the superconductivity and zeros of u are called vortices. Since a superconducting property is violated in such points it is interesting to predict the number, the location, and the dynamics of vortices. It is known that the properties of vortices sensitively depend on κ. The case κ = / has been studied by Jaffe and Taubes [] and it has been shown that vortices can be located anywhere in Ω and do not tend to interact. If κ < / then vortices tend to attract each other and concentrate in one place. The case κ > / is called the repulsive case since for such values of κ vortices of same sign tend to repulse each other. Materials for which Ω

2 κ < / or κ > / are also known as type-i or type-ii conductors, respectively. For details on the qualitative and theoretical study of Ginzburg-Landau vortices we refer the reader to [4,, 7, 34, 3, 4, 5, 6, 7, 33] and references therein. The parameter κ also influences the stability of numerical approximation schemes for superconducting materials. In particular, the dynamics of superconducting materials are often modeled by the gradient flow of the energy functional G κ and numerical results sensitively depend upon κ. It is not difficult to see that standard error estimates depend exponentially upon κ which can make simulations useless if κ is large. In this article we aim to design and analyze approximation schemes for which the discretization error grows in κ only in a low order polynomial and thereby allow for reliable simulations. We remark that our analysis is inspired by recent work of Feng and Prohl [5, 6, 7] on phase field models and we employ several of their arguments. Owing to a lack of appropriate a priori information for the vectorial problem considered here, their proofs are however not directly transferable. We remark that we also employ several techniques from [, 36]. Interesting numerical studies of stationary and time-dependent Ginzburg-Landau equations which motivated this work can be found in [,, 3, 8, ]. An a posteriori error analysis based on ideas of this work can be found in []. For ease of presentation we will restrict the analysis to a simplified version of G κ, in which A, ε = /(κ, and n =, i.e., Ω R, J ε : H (Ω; C R, v Ω v dx + ε 4 Ω ( v dx. In this model, Dirichlet type boundary conditions replace an applied magnetic field. The dynamics of a superconducting material can then in a greatly simplified manner be described by the gradient flow of J ε. In order to define the time-dependent problem, we assume that we are given a parameter T >, initial data u ε H (Ω; C, and boundary data g ε = u ε Ω H / ( Ω; C. We then aim to solve the following problem: (P Find u ε H (, T ; H (Ω; C L (, T ; H (Ω; C such that for almost all t (, T and all v H (Ω; C there holds u ε t ; v + ( uε ; v + ε (f(u ε ; v =, u ε Ω = g ε, u ε ( = u ε. Here, f(a = ( a a for a C, ( ; denotes the scalar product in L (Ω; C, ; denotes the duality pairing of H (Ω; C and H (Ω; C, u ε t denotes the time derivative of u ε, and we use the notation a b:= (ab + ab/ for a, b C. Existence and uniqueness of a solution u ε follow from standard techniques. Throughout this work we abbreviate u = u ε, u = u ε, and g = gε but stress that the dependence of the error of numerical approximation schemes upon ε is the main contribution of this work. The main results of this work prove that under moderate conditions on the initial data u and the domain Ω and if the time-step size parameter k > and the mesh-size h > of

3 an implicit in time, lowest order finite element in space discretization of (P satisfy (up to logarithmic and constant factors k ε 8 and h ε 4, then the spatial discretization error in L is bounded by ε 4 (k + h for all t (, T. The assumptions involve the condition that vortices of u are well separated and of degree one. The outline of this article is as follows. We state and discuss theoretically and physically motivated assumptions on the solution u of (P in Section. Section 3 gives a priori bounds on various norms of the exact solution. Discrete counterparts of those a priori bounds are proved in Section 4 for a semi-discrete in time approximation of (P. Optimal error estimates in terms of the time discretization parameter which are robust with respect to the small parameter ε are established in Section 5. In Section 6 we discuss a fully discrete approximation scheme and prove robust a priori error estimates. Finally, some numerical experiments illustrate the theoretical results of this contribution in Section ASSUMPTIONS ON THE SOLUTION In order to derive robust a priori error estimates for the numerical approximation of (P we will assume that the solution u of (P admits vortices only of degree one and allows for an asymptotic expansion. The following lemma is essential for the asymptotic expansion and characterizes degree one-vortices. Lemma. ([9]. Let f : [, R satisfy f ( =, lim s f (s =, f, and Given any real number H R the function f s f + s f = f ( f. U H(x:= f (r(x/εe iθ(x i e H, where (r(x, θ(x denote the polar coordinates of x R, satisfies U H( = and U H + ε f(u H = in R. The following assumption states that in certain neighborhoods of zeros the solution u of (P is given by small perturbations of functions U H as in the previous lemma. Away from zeros, u is assumed to be close to. Assumption I. There exist ε >, δ, c >, c, and an integer d such that for all ε (, ε, for all t (, T ], and j =,,..., d there exist such that and a t,ε j R, p t,ε j u(t, x = f ( r(x a t,ε j The functions p t,ε j H t,ε j (x, qt,ε j : B δ (a t,ε j C, Ht,ε j /ε e iθ(x at,ε j e iht,ε j : B δ (a t,ε j R, Ht,ε j R (x + εp t,ε j (x + ε q t,ε j (x for x B δ (a t,ε u(t, x c δ ε for x Ω \ d j=b δ /(a t,ε j., qt,ε j, Ht,ε j H t,ε j c ε,, j =,,..., d, satisfy ( f U j (x p t,ε j (x c ε, q t,ε j (x c for x B δ (a t,ε j, j

4 4 where for each j =,,..., d, U j (x:= f ( r(x a t,ε j /ε e iθ(x at,ε j t,ε i H e j (x for x B δ (a t,ε j. Moreover, for almost all (t, x (, T Ω there holds u(t, x c. Remarks. (i Assumption I is motivated by the fact that in the repulsive case, i.e., for small values of ε, vortices tend to repulse each other and that higher degree vortices are unstable [3], i.e., split into degree-one vortices immediately. Therefore, Assumption I is merely an assumption on the initial data u. (ii Assumption I implicitly assumes that the vortices are well separated, i.e., have a distance δ. Since f (r = /(r + O(/r 4 for r the assumed expansion of u(t, x in a neighborhood of a vortex and the condition u(t, x c δ ε away from the vortices are compatible. Note that f (r = ar ar 3 /8 + O(r 5 for r and a constant a R [9]. (iii Similar assumptions on the solutions of related problems have been made in [3, 9] to study the dynamics of vortices, in [9, 6] to study the stability of interfaces in phase field equations, and in [5, 6, ] to derive robust error estimates for the approximation of Allen- Cahn and Cahn-Hilliard equations. A spectral estimate for the linearized Ginzburg-Landau operator about symmetric degreeone vortices is specified in the next theorem and will be one key ingredient for our a priori error analysis. Theorem. ([3, 3, 8, 3]. Given H R let U H(x := f (r(x/εe iθ(x e i H. Let L ε, H : H (B δ (; C L (B δ (; C be defined by L ε, Hv := v + ε f (U Hv. Then, the principal eigenvalue of L ε, H is non-negative, i.e., for all v H (B δ (; C there holds (L ε, Hv; v. Proof. Proofs of the theorem can be found in [3, 3, 8, 3] for H =. If H one can check that given any v H (B δ (; C there holds with w := e i Hv (L ε, Hv; v = (L ε, w; w which proves the theorem. The theorem can be generalized to small perturbations of degree one vortices as follows. Proposition.3. Suppose that Assumption I holds. Then there exists an ε-independent constant λ such that for all ε (, ε, for j =,,..., d, for all t (, T ], and for all v H(B δ (a t,ε ; C there holds j ( v; v + ε (f (uv; v λ v L (B δ (a j. Proof. Let j {,,..., d} and t (, T ]. Throughout this proof we abbreviate a:= a t,ε j, p:= pt,ε j, q := qt,ε j, H := Ht,ε j, H := Ht,ε j,

5 5 where a t,ε j There holds, pt,ε j, qt,ε j, Ht,ε j, and H t,ε j are as in Assumption I. For x B δ (a define ( U H(x:= f r(x a/ε e iθ(x a i e H, U H (x:= f ( r(x a/ε e iθ(x a e ih(x. f (u = f (U H + ( f (u f (U H + ( f (U H f (U H and owing to the assumed asymptotic expansion of u in B δ (a, a Taylor expansion of the quadratic function f shows in B δ (a f (u f (U H f (U H (u U H + f (U H u u H ε f (U H p + ε f (U H q + ε f (U H p + εq Cε. Since e ih e i H C H H Cε in B δ (a we have f (U H f (U H f (U H U H U H + f (U H U H U H Cε. Using these estimates we infer with Theorem. for all v H (B δ (a; C that ( v; v + ε (f (uv; v ( v; v + ε (f (U Hv; v C v L (B δ (a This proves the proposition. C v L (B δ (a. We conclude this section by a discussion of Assumption I. The first example specifies (P, allows for a solution without vortices, and yields Assumption I. Example.4 ([]. Suppose that Ω is smooth and g is smooth, independent of ε, and satisfies g(s = for all s Ω. Assume that u = u ε is smooth, independent of ε, and satisfies u (x = for all x Ω. Then, there exist ε > and an ε-independent constant C > such that for all ε (, ε, for almost all (t, x (, T Ω there holds u(t, x Cε. In particular, Assumption I holds with δ =, ε = ε, c = C, and d =. Assumption I can (formally be motivated as follows. Example.5. Given t (, T ] we introduce y := x/ε and make the ansatz u(t, x = U(y + εp(y + ε q(y which leads to the three leading order equations (. (. (.3 ε ( U + ( U U =, ε ( p + ( U p + (U pu p =, ε ( q + ( U q + p U + (U qu q = u t (t,.

6 6 Equation (. admits a solution U subject to boundary conditions U Ω = g. Provided that U(x/ε is minimal for J ε it can be shown [34] that there exists an integer d, a harmonic function ψ : Ω R, and a ε j Ω, j =,,..., d, such that U( /ε e iψ d ( f r( a ε j /ε e iθ( aε j L for ε. (Ω j= We choose p as a solution for (. and let q be the solution of the linear equation (.3 subject to q Ω =. The assertions of the previous example can be made more precise. We only focus on the dominant term in the asymptotic expansion, i.e., on equation (. in the following example. It gives a precise characterization of the solution of (. in neighborhoods of vortices and shows that u is close to away from vortices. Example.6 ([3, 4]. (i Suppose U = U ε satisfies U( =, U L (B δ /ε( C, solves (. in B δ /ε(, and U( /ε is minimal for J ε with Ω = B δ /ε(. Assuming that ε is small enough we may assume that U solves (. in R. It can then be shown that there exists θ R such that for all y R there holds U(y = f (r(ye i(θ(y+θ. (ii Suppose that G : B δ /ε( C is smooth, satisfies G(s = for all s B δ /ε(, and deg(g, B δ /ε( =. Let U satisfy U Bδ /ε( = G, solve (. in B δ /ε, and suppose that U( /ε is minimal for J ε with Ω = B δ (. Then there exists an ε-independent constant C > such that for all y B δ /ε( there holds U(y Cε. 3. A PRIORI BOUNDS The following a priori bounds on a solution u of (P are independent of Assumption I. They solely assume that the modulus of the solution u is uniformly bounded in (, T Ω. Proposition 3.. Suppose that there exists an ε-independent constant c > such that for almost all (t, x (, T Ω there holds u(t, x c. Then, there exists an ε-independent constant c > such that (i ess sup s (,T u L (Ω + (ii (iii T T ess sup s (,T u t L (Ω + u t L (Ω ds T u t (s L (Ω ds J ε(u, c ( ε J ε (u + u ε f(u L (Ω, u tt H (Ω;C ds c ( ε J ε (u + u ε f(u L (Ω + ε 4 J ε (u. Sketch of proof. We formally derive the estimates to verify the dependence on ε and note that the assertions can be made precise by employing techniques from, e.g., [4]. (i This follows from choosing v = u t (note u t Ω in (P u t L (Ω + d ( dt u L (Ω + ε 4 ( u L (Ω =

7 7 and integrating this equation over (, s for any s T. (ii We formally differentiate the first equation in (P in time, (3. u tt u t + ε f (uu t =, and test this equation with u t, i.e., d dt u t L (Ω = u t L (Ω ε (f (uu t ; u t. Using u L ((,T Ω c and integrating over (, s for any s T we infer T T u t (s L (Ω + u t L (Ω ds Cε u t L (Ω ds + u t( L (Ω Cε J ε (u + u t ( L (Ω. We then use (i and u t ( = u ε f(u to verify the assertion. (iii Using the estimate f (u L (Ω C we verify with (3. u tt H (Ω;C u t L (Ω + ε (f (uu t ; ϕ sup ϕ H (Ω;C ϕ H (Ω u t L (Ω + Cε u t L (Ω. Assertion (iii is then deduced from squaring this estimate, integrating the resulting equation over (, T, and employing previous estimates. In order to exploit the estimates of the proposition we need to assume bounds on J ε (u = J ε (u ε and u ε f(u L (Ω. In general, it is impossible to bound these quantities independently of ε in the considered two-dimensional setting. Assumption II. There exists a positive function γ : R R such that γ(ε γ(ε, J ε (u γ ε := γ(ε, u H (Ω + u ε f(u L (Ω γ εε 4 for all ε (, ε. For an optimal choice of u, which extends g from Ω to Ω, we may assume that γ grows at most logarithmically in ε for ε. Example 3. ([4]. Assume that Ω is smooth and simply connected and suppose that g is smooth, independent of ε, and satisfies g(s = for all s Ω. Then there exist ε >, an ε-independent constant C >, and g H (Ω; C with g Ω = g such that for all ε (, ε there holds J ε ( g π d log(ε + C where d = deg(g, Ω. The estimate is sharp in the sense that if Ω is starshaped and d >, e.g., Ω = B ( and g(s = s for s Ω, then there holds lim ε { minv Ω =g J ε (v π d log(ε } = C for an ε-independent constant C >.

8 8 4. SEMI-DISCRETE IN TIME APPROXIMATION SCHEME We analyze the following semi-discrete in time approximation (P k of (P which is defined through a time step size parameter k >. Find (U m : m M H (Ω; C such that for all m M and all V H(Ω; C there holds (P k (d t U m ; V + ( U m ; V + ε ( f(u m ; V =, U m Ω = g, U = u. Here, M is the largest integer for which Mk T and given any sequence (a m : m M the discrete time derivative d t a m is for m M defined by d t a m := (a m a m /k. We set t m := mk for m M and define e m := u(t m U m. The following a priori bound for a solution (U m : m M is the discrete counterpart of assertion (i in Proposition 3.. Proposition 4.. Suppose that k ε. There holds ( max m M U m L (Ω + ε Um 4 L (Ω M ( + k d tu m L (Ω + kε dt ( U m J 4 L (Ω ε (u. m= Proof. The choice V = d t U m for m M in (P k yields d t U m L (Ω + ( Um L k (Ω U m L (Ω ( + ε f(u m ; d t U m d t U m L (Ω + ( U m ; d t U ( m + ε f(u m ; d t U m =. We write f(u m = ( Um ( U m + U m + kd t U m ( and use d t U m (U m + U m = d t Um = to verify ( f(um ; d t U m = ( Um ( ; d t ( U m + k ( ( Um ; d t U m ( ( Um ; d t ( U m k d tu m L (Ω. Employing the identity a (a b = a b + ( a b for a, b C we deduce ( ( Um ; d t ( U m = ( Um ( U m k L (Ω + k( Um Um L (Ω L (Ω = k dt ( U m + L (Ω d t Um. L (Ω

9 9 We may therefore estimate ( ε k dt U m L (Ω + k U m L (Ω k U m L (Ω + kε 4 dt ( U m + ε L (Ω 4 d t Um. L (Ω Multiplying this estimate with k and taking sum over m =,..., l for any l M we deduce U l L (Ω + ε Ul 4 L (Ω l ( ( kε + k dt U m L (Ω + kε d t ( U m 4 L (Ω m= U L (Ω + ε 4 U L (Ω = J ε(u. Since U = u and kε / / we deduce the proposition. 5. ANALYSIS OF THE SEMI-DISCRETE IN TIME APPROXIMATION SCHEME In this section we aim to derive a robust error estimate for the approximation scheme (P k. The following definition introduces for each time step t m > a partition of unity which is adapted to the neighborhoods of vortices in Assumption I. Definition 5.. Given an integer m M let ( ψ m,j : j =,,..., L m C (Ω; R be a partition of unity with finite overlap α > such that for an ε-independent constant c 3 > and for j =,,..., L m, there holds ψ m,j, L m j= ψ m,j =, and ( ψ / m,j c 3 δ in Ω, and, for j =,,..., d, ψ m,j = in B δ /(a tm,ε j and supp ψ m,j B δ (a tm,ε j. The nonlinearity in the error equation defined by (P and (P k requires a different treatment in neighborhoods of vortices and in the remaining parts of Ω. Lemma 5.. Suppose that Assumption I holds. For all ε (, ε, m M, and j = d +, d +,..., L m there holds ε (f(u(t m f(u m ; ψ m,j e m c (c + δ ψ / m,j e m L (Ω ε /4 ψ 8 m,j e m 4. L 4 (Ω Proof. Given m M we abbreviate u = u(t m, U = U m, e = e m, and ψ j = ψ m,j for j = d +, d +,..., L. Assumption I guarantees u c δ ε in supp ψ j and we may

10 therefore deduce (f(u f(u; ψ j e = ( u u; ψ j e ( ( U U; ψ j e = ( ( u e; ψ j e + ({ ( u ( U } U; ψ j e = ( ( u ( u + e; ψ j e + ({ ( u ( U } U; ψ j e c δ ε (c + ψ / j e L (Ω + ( ( u U U; ψ j e. We manipulate the second term in the right-hand side as follows, ( ( u U U; ψ j e = ({ (u U (u + U } U; ψ j e = ({ e (e + U } U; ψ j e = ( e U; ψ j e + ( (e UU; ψ j e = ( e U; ψ j e + ψ / j (e U L (Ω / ψ j e 8 / L (Ω ψ j (e U + L (Ω ψ/ j (e U L (Ω = / ψ j e 8 = /4 L (Ω ψ j e 4 8. L 4 (Ω The lemma follows from a combination of the two estimates. Lemma 5.3. Suppose that Assumption I holds. For all ε (, ε, m M, and j =,,..., d there holds ε ( f(u(t m f(u m ; ψ m,j e m λ ( ε ψ / m,j e m L (Ω ( ε (ψ / m,j e m L (Ω 3c ε ψ /3 m,j e m 3 L 3 (Ω ψ/ m,j e m L (Ω. Proof. As in the previous proof we abbreviate u = u(t m, U = U m, e = e m, and ψ j = ψ m,j for j =,,..., d. Given any a, b C there holds and (f(a f(b (a b f (a(a b (a b 3 a a b 3 f (a(a b (a b a b. We thus have for any θ [, ], ( f(u f(u; ψj e ( f (ue, ψ j e 3 ( u e 3 ; ψ j = ( θ ( f (ue, ψ j e 3c ψ /3 j e 3 L 3 (Ω + θ( f (ue; ψ j e ( θ ( f (ue, ψ j e 3c ψ /3 j e 3 L 3 (Ω θ ψ/ j e L (Ω. Applying the spectral estimate of Proposition.3 to v := ψ / j e we infer ( f (ue; ψ j e = ( f (uv; v λ ε v L (Ω ε v L (Ω. Choosing θ = ε we verify the assertion of the lemma. The following lemma gives bounds on the higher norms of the error that arose in the previous estimates.

11 Lemma 5.4. Suppose that there exists a positive function ρ : R R such that ρ ρ and e m L (Ω ρ ε := ρ(ε for all m M and all ε (, ε. Then there exists an ε-independent constant c 4 > such that for all ε (, ε and all m M there holds e m 3 L 3 (Ω + e m 4 L 4 (Ω c 4ρ ε k d t e m L (Ω + c 4ρ ε e m L (Ω e m L (Ω. Proof. Employing the estimate v L 4 (Ω C v L (Ω v L (Ω for v H (Ω; C with v Ω = and the bounds C e m L (Ω e m L (Ω ρ ε we infer for m M 8 e m 4 L 4 (Ω e m e m 4 L 4 (Ω + e m 4 L 4 (Ω = k 4 d t e m 4 L 4 (Ω + e m 4 L 4 (Ω Ck 4 d t e m L (Ω d t e m L (Ω + e m 4 L 4 (Ω = Ck d t e m L (Ω (e m e m L (Ω + e m 4 L 4 (Ω Cρ εk d t e m L (Ω + e m 4 L 4 (Ω Cρ ε k d t e m L (Ω + C e m L (Ω e m L (Ω. Similarly, using v 3 L 3 (Ω C v L 4 (Ω v L (Ω and v L 4 (Ω C v L (Ω, we derive 4 e m 3 L 3 (Ω e m e m 3 L 3 (Ω + e m 3 L 3 (Ω = k 3 d t e m 3 L 3 (Ω + e m 3 L 3 (Ω Ck 3 d t e m L 4 (Ω d te m L (Ω + e m 3 L 3 (Ω Ck 3 d t e m L (Ω d t e m L (Ω + e m 3 L 3 (Ω Cρ ε k d t e m L (Ω + e m 3 L 3 (Ω Cρ ε k d t e m L (Ω + C e m L (Ω e m L (Ω. This proves the lemma. The next lemma gives a bound on the time discretization residual. Lemma 5.5. For m M let R(u tt ; m:= u t (t m d t u(t m. There holds M k R(u tt ; m H (Ω;C T 3 k u tt H (Ω;C ds. m= Proof. Repeated integration by parts shows R(u tt ; m = u t (t m d t u(t m = k tm t m (s t m u tt (s ds.

12 Therefore, we have k m= m= tm M M R(u tt ; m H (Ω;C = k (s t m u tt (s ds k m= t m M ( t m ( t m (s t m ds u tt (s H k (Ω;C ds t m t m = 3 k M This proves the lemma. tm m= t m u tt (s H (Ω;C ds 3 k T H (Ω;C u tt (s H (Ω;C ds. We are now in position to prove the main result of this section. Theorem 5.6. Suppose that Assumptions I and II hold and assume that { k min 96 ε 7 γ c c / ε exp(c δ T 3/,, ε γε, ε }, c 4 4C δ 9c c 4 where C δ holds (5. = (λ + c 3 αδ + + c (c + δ. For all ε (, ε and l M there max e m L m=,,...,l (Ω + k l ( k d te m L (Ω + ε e m L (Ω m= c k ε 6 γ ε exp(c δ t l. Proof. Step : Verification of the assumptions of the employed lemmas. A combination of Propositions 3. and 4. with Assumption II yields e m L (Ω u(t m L (Ω + U m L (Ω 6J ε(u 6γ ε so that we may choose ρ ε = 4γε / Lemma 5.5 imply (since ε ε (5. k in Lemma 5.4. Assumption II, Proposition 3., and M R(u tt ; m H (Ω;C 3 k c (ε γ ε + ε 4 γ ε c k ε 4 γ ε. m= Step : Derivation of an error equation. Choosing v = V = e m for m M and subtracting the first equation in (P k from the first equation in (P we deduce (d t e m ; e m + e m L (Ω + ε ( f(u(t m f(u m ; e m = ( R(utt ; m; e m. We use the identity the estimate (5.3 ( R(u tt ; m; e m ε (d t e m ; e m = d t e m L (Ω + k d te m L (Ω, e m L (Ω + ε R(u tt; m H (Ω;C,

13 3 and employ the partition of unity (ψ m,j : j =,,..., L m to verify d t e m L (Ω + k d te m L (Ω + ( ε em L (Ω + L m j= ε ( f(u(t m f(u m ; ψ m,j e m ε R(u tt; m H (Ω;C. Employing Lemma 5. and 5.3, (ψ / m,j L (Ω c 3δ, and the finite overlap α of (ψ m,j : j =,,..., L m we verify with L m j= ψ m,j = almost everywhere in Ω, L m j= Estimating ε ( f(u(t m f(u m ; ψ m,j e m ( λ ( ε + + c (c + δ 3c ε e m 3 L 3 (Ω ε em 4 8 ( L 4 (Ω ε = ( λ ( ε + + c (c + δ L m j= (ψ / m,j e m L (Ω em L (Ω 3c ε e m 3 L 3 (Ω ε em 4 8 ( L 4 (Ω ε e m L (Ω ( ε ( (λ + c 3 αδ ( ε + + c (c + δ ε e m 4 8 ( L 4 (Ω ε e m L (Ω. we deduce from the previous estimates L m j= (λ + c 3 αδ ( ε + + c (c + δ C δ, em L (Ω e m (ψ / m,j L (Ω em L (Ω 3c ε e m 3 L 3 (Ω d t e m L (Ω + k d te m L (Ω + ε e m L (Ω ε R(u tt ; m H (Ω;C + C δ e m L (Ω + ε max{6c, /4} ( e m 3 L 3 (Ω + e m 4 L 4 (Ω. Since c we have max{6c, /4} = 6c. Lemma 5.4 leads to d t e m L (Ω + k d te m L (Ω + ε e m L (Ω ε R(u tt ; m H (Ω;C + C δ e m L (Ω + 6c c 4 ρ ε k ε d t e m L (Ω + 6c c 4 ρ ε ε e m L (Ω e m L (Ω The assumption 6c c 4 ρ ε kε / (where we substituted ρ ε = 6γ ε allows to absorb (/k d t e m L (Ω on the right-hand side of the previous estimate and implies d t e m L (Ω + k d te m L (Ω + ε e m L (Ω ε R(u tt ; m H (Ω;C + C δ e m L (Ω + 6c c 4 ρ ε ε e m L (Ω e m L (Ω.

14 4 Multiplying the last estimate with k and summing over m =,,..., l for any l M and abbreviating C := 6c c 4 yields with 5. and e = e l L (Ω + k l ( k d te m L (Ω + ε e m L (Ω m= c γ ε k ε 6 + C δ k l e m L (Ω + C ρ ε ε k m= l e m L (Ω e m L (Ω. The assumption C δ k / allows to absorb e l L (Ω/ on the right-hand side. Employing once more e = we verify (5.4 m= m= l ( e l L (Ω + k k d te m L (Ω + ε e m L (Ω c γ ε k ε 6 + C δ k l m= e m L (Ω + ( C ρ ε ε 4 max e l m L m=,,...,l (Ω k ε e m L (Ω. Step 3: Proof of (5. by induction over l. With l = estimate (5.4 reads ( e L (Ω + k k d te L (Ω + ε e L (Ω c γ ε k ε 6 and proves (5.. Suppose that (5. holds with l replaced by l. Using that estimate to bound the last term in the right-hand side of (5.4 we verify that l ( e l L (Ω + k k d te m L (Ω + ε e m L (Ω m= c γ ε k ε 6 + C δ k l m= m= e m L (Ω + C ρ ε ε 4( c k ε 6 γ ε exp(c δ t l 3/. Since the assumptions on k imply (5.5 C ρ ε ε 4( c k ε 6 γ ε exp(c δ t l 3/ c γ ε k ε 6 we have l ( e l L (Ω + k k d te m L (Ω + ε e m L (Ω m= A discrete Gronwall inequality proves (5.. c γ ε k ε 6 + C δ k l m= e m L (Ω. Remark. Employing only Assumption II and using a local Lipschitz estimate to bound the nonlinear term (provided that u(t m L (Ω + U m L (Ω C, ε ( f(u(t m f(u m ; e m C f ε e m L (Ω,

15 one can prove the error estimate (5.6 max e m L m=,,...,l (Ω + k l ( k d te m L (Ω + e m L (Ω m= Ck ε 4 exp(c f ε t l for l M. This estimate is, owing to its exponential dependence on ε, useful only if ε is large or if t l = O(ε. The estimate may be used to impose Assumption I only for t [Cε, T ]. This is of interest if the initial data involve higher degree vortices which split into well separated degree-one vortices within a time O(ε DISCUSSION OF A FULLY DISCRETE APPROXIMATION SCHEME For a time and space discretization of (P we assume that Ω is polygonal and let T be a quasi-uniform regular triangulation of Ω with maximal mesh-size h. We let S (T ; C denote the lowest order finite element space which consists of all T -elementwise affine, globally continuous functions. The subset S(T ; C is defined as S (T ; C := {v h S (T ; C : v h Ω = }. We let u h, be the nodal interpolant of u, and denote by g h the restriction of u h, to Ω. As for the semi-discrete problem (P k we assume that we are given a time step size k >. With the notation of Section 4, the fully discrete problem reads: (P k,h Find (U h,m : m M S (T ; C such that for all m M and all V h S (T ; C there holds (d t U h,m ; V h + ( U h,m ; V h + ε ( f(u h,m ; V h =, U h,m Ω = g h, U h, = u h,. Existence of a unique solution (U h,m holds if k ε. For m M we define E m := u(t m U h,m. The following a priori bound for the solution of (P k,h is proved as Proposition 4.. Proposition 6.. Suppose that k ε. There holds ( max m M U h,m L (Ω + ε Uh,m 4 L (Ω M ( + k d tu h,m L (Ω + kε dt ( U h,m J 4 L (Ω ε (u h,. m= Additional regularity of u is needed for the a priori error analysis of (P k,h. The proofs need further assumptions. Assumption III. The domain Ω is convex, there holds g(s = for all s Ω, and there exist an ε-independent constant c > and an integer σ such that (6. lim s + u t(s L (Ω c ε σ for all ε (, ε.

16 6 Proposition 6.. Suppose that there exists an ε-independent constant c > such that for almost all (t, x (, T Ω there holds u(t, x c and suppose that Assumption III holds. There exists an ε-independent constant c > such that (i (ii T T u tt L (Ω ds c ( ε 4 J ε (u + ε σ, u t H (Ω ds c ( ε 4 J ε (u + ε σ, (iii ess sup s (,T u H (Ω c ( ε J ε (u + u ε f(u L (Ω + u H (Ω. Proof. The proof of the proposition is similar to the proof of Proposition in [5]. Some basic properties of approximation operators are summarized in the following proposition. For proofs we refer the reader to [5, 36, 5, 35]. Proposition 6.3. (i Given v H (Ω; C let P h v S (T ; C be defined by ( (Ph v v; w h = for all w h S (T ; C. For m M set φ m := P h E m S (T ; C and θ m := E m φ m. There exists an ε-independent constant c > such that (6. θ m L (Ω + h θ m L (Ω c h 4 u(t m H (Ω k θ m L (Ω c h log h / u(t m H (Ω M T d t θ m L (Ω c h 4 u t H (Ω ds. m= (ii There exists an ε-independent constant c 3 > such that J ε (u h, J ε (u + c 3 h u H (Ω, φ L (Ω + h φ L (Ω c 3h 4 u H (Ω. The following estimates are minor modifications of estimates given in Section 5. Notice that φ m + U m approximates u(t m. Proposition 6.4. Suppose that Assumption I holds, let m M, and suppose that θ m L (Ω ε. There exist ε-independent constants λ, c, c 4, c 4 > and c > such that (i for j =,,..., d and all v H (B δ (a j ; C there holds ( v; v + ε (f (φ m + U h,m v; v λ v L (B δ (a j ; (ii for j = d +, d +,..., L m there holds ε (f(φ m + U h,m f(u h,m ; ψ m,j φ m c ( c + δ ψ/ m,j φ m L (Ω ε 8 (iii for j =,,..., d there holds ψ /4 m,j φ m ε ( f(φ m + U h,m f(u h,m ; ψ m,j φ m λ ( ε ψ / m,j φ m L (Ω 4 ; L 4 (Ω ( ε (ψ / m,j φ m L (Ω 3 c ε ψ /3 m,j φ m 3 L 3 (Ω ψ/ m,j φ m L (Ω ;

17 7 (iv there holds φ m 3 L 3 (Ω + φ m 4 L 4 (Ω c 4 ρ ε k d t φ m L (Ω + c 4 ρ ε φ m L (Ω φ m L (Ω, provided max j=,,...,m φ j L (Ω ρ ε for some ρ : R R with ρ ε := ρ(ε ρ(ε ; (v there holds M k R(u tt ; m L (Ω T 3 k u tt L (Ω ds; (vi there holds m= f(φ m + U h,m f(u(t m L (Ω c 4 θ m L (Ω. Proof. Notice (φ m + U h,m u(t m L (Ω = θ m L (Ω ε. A local Lipschitz estimate for f yields ε ( f (φ m + U h,m v; v ( f (u(t m v; v C v L (Ω and the proof of (i follows from Proposition.3. The proofs of (ii-(v follow the lines of the proofs of Lemma 5., 5.3, 5.4, 5.5, respectively. Assertion (vi follows from uniform bounds for u(t m and φ m + U h,m and local Lipschitz continuity of f. The previous estimates allow to prove a robust a priori error estimate for the approximation scheme (P k,h. Theorem 6.5. Suppose that Assumptions I, II, and III hold and assume that k + h h min{, ε 4 }, { k min ε ρ ε, c c 4 h log h / } C, ε, δ 3c c 3 ε 4 γ / ε, and h 4 ( C C c 3 3 C 4( C / ε4( max{ε σ, ε 8 γ ε } / exp((3 + C δ T 3/, where C = 6 c c 4, ρ ε = (6 + 8c 3 / γε /, and ρ ε γ ε, C δ = ( λ + c 3 αδ + + c ( c + δ, C = (c 3 + c c + 3c c 3 c 4 + c. For all ε (, ε and l M there holds (6.3 max 4 E m L m=,,...,l (Ω + k l ε h E m L (Ω m= 4 C (k + h 4 max{ε σ, ε 8 γ ε } exp((3 + C δ t l + 3c c h 4 ε 4 γ ε ( + ε T. Proof. Step : Verification of the assumptions of the employed lemmas. Propositions 6.3 and 6., Assumption II and the assumption on h imply θ m L (Ω c 3 h log h / 3c ε γ / ε ε.

18 8 H stability of P h, Propositions 3. and 6.3, Assumption II, and the condition h ε 4 prove φ m L (Ω E m L (Ω u(t m L (Ω + U h,m L (Ω 8J ε (u + 8J ε (u h, 6J ε (u + 8c 3 h u H (Ω 6γ ε + 8c 3 γ ε so that we may choose ρ ε = (6 + 8c 3 / γ / ε in Proposition 6.4. Step : Derivation of an error equation. Choosing v = V h = φ m for m M and subtracting the first equation in (P k,h from the first equation in (P we deduce (d t E m ; φ m + ( e m ; φ m + ε ( f(u(t m f(u h,m ; φ m = ( R(utt ; m; φ m. We use the identities (d t E m ; φ m = (d t θ m ; φ m + d t φ m L (Ω + k d tφ m L (Ω and ( (E m φ m ; φ m =, Cauchy inequalities, and (vi of Proposition 6.4 to verify d t φ m L (Ω + k d tφ m L (Ω + φ m L (Ω + ε ( f(φ m + U h,m f(u h,m ; φ m = (d t θ m ; φ m ( R(u tt ; m; φ ( m + ε f(φ m + U h,m f(u(t m ; φ m d tθ m L (Ω + 3 φ m L (Ω + R(u tt; m L (Ω + c ε 4 4 θ m L (Ω. We argue as in the proof of Theorem 5.6 to show with (ii and (iii of Proposition 6.4 ε ( L m f(φ m + U h,m f(u h,m ; φ m = ε ( f(φ m + U h,m f(u h,m ; ψ m,j φ m j= C δ φ m L (Ω 3 c ε φ m 3 L 3 (Ω ε φm 4 8 ( L 4 (Ω ε φ m L (Ω. Estimate (iv of Proposition 6.4 yields 3 c ε φ m 3 L 3 (Ω + ε φm 4 8 L 4 (Ω ε max{3 c, /8} c 4 ( ρ ε k d t φ m L (Ω + ρ ε φ m L (Ω φ m L (Ω A combination of the last three estimates shows d t φ m L (Ω + k d tφ m L (Ω + ε φ m L (Ω d t θ m L (Ω + c 4ε 4 θ m L (Ω + (3 + C δ φ m L (Ω + R(u tt; m L (Ω + ε max{6 c, /4} c 4 ( ρ ε k d t φ m L (Ω + ρ ε φ m L (Ω φ m L (Ω. We note max{6 c, /4} = 6 c, multiply the last estimate with k, and sum over m =,,..., l to verify l ( k φ l L (Ω + k d tφ m L (Ω + ε φ m L (Ω m= φ L (Ω + k + (3 + C δ k l d t θ m L (Ω + c 4ε 4 k m= l θ m L (Ω + k m= l φ m L (Ω + 6 c c 4 ε ρ ε k m= l R(u tt ; m L (Ω m= l φ m L (Ω φ m L (Ω. m=

19 where we used k6 c c 4 ε ρ ε / to absorb a sum over k d t φ m L (Ω on the right hand side. Using estimates of Propositions 6.3 and 6. and Assumption II to bound the first four terms on the right-hand side and employing (3 + C δ k / to absorb φ l L (Ω on the right-hand side we find l ( k φ l L (Ω + k d tφ m L (Ω + ε φ m L (Ω m= ( c 3 ε 4 γ ε + c c (ε 4 γ ε + ε σ + c 4 ε 4 c 3 3c ε 4 γ ε h 4 + k 3 c (ε 4 γ ε + ε σ + (3 + C δ k l m= C (h 4 + k max{ε σ, ε 8 γ ε } + (3 + C δ k Proposition 6.3 implies l m= φ m L (Ω + 6 c c 4 ε ρ ε k φ m L (Ω + C ε ρ ε k l m= l m= φ m L (Ω φ m L (Ω φ m L (Ω φ m L (Ω. C ε ρ ε k φ L (Ω φ L (Ω C ε ρ ε kh 4 c 3/ 3 ε 6 γ 3/ ε ε 8 γ ε ( C k + ( C ρ ε h8 c 3 3 ( C γ ε C (h 4 + k max{ε σ, ε 8 γ ε }, where we used the assumption on h 4 in the last estimate, so that we may deduce l ( k φ l L (Ω + k d tφ m L (Ω + ε φ m L (Ω m= C (h 4 + k max{ε σ, ε 8 γ ε } + (3 + C δ k m= l m= φ m L (Ω + C ( ε 4 ρ ε max φ l m L m=,,...,l (Ω k ε φ m L (Ω. Step 3: Induction over l. An inductive argumentation with a discrete Gronwall inequality as in Step 3 in the proof of Theorem 5.6 leads to (6.4 l ( k φ l L (Ω + k d tφ m L (Ω + ε φ m L (Ω m= for all l M provided that 4 C (h 4 + k max{ε σ, ε 8 γ ε } exp ( (3 + C δ t l C ε 4 ρ ε (4 C (h 4 + k max{ε σ, ε 8 γ ε } exp ( (3 + C δ T 3/ 9 C (h 4 + k max{ε σ, ε 8 γ ε }

20 which is guaranteed by the assumptions on h + k. Step 4: Proof of (6.3. The estimate follows from a combination of (6.4 together with and the estimates in (6.. 4 E m L (Ω φ m L (Ω + θ m L (Ω, E m L (Ω φ m L (Ω + θ m L (Ω, Remarks. (i As for the semi-discrete in time approximation scheme, imposing Assumptions II and III only, one can derive an error estimate with exponential dependence on ε which may be used to weaken Assumption I (cf. remark below Theorem 5.6. (ii The assumption on h + k can be weakened resulting in a higher power of ε in ( NUMERICAL EXPERIMENTS In this section we illustrate the theoretical results of the previous sections by three numerical experiments. To define the examples and to display the numerical solutions, we identify C and R. In particular, we write xy = ( x y x y, x y + x y for x = (x, x and y = (y, y in R. In all examples we will employ ε = / and note that in this case standard error estimates involve a factor exp(ε 43 while in our error estimates only a factor ε 4 = 4 occurs. The implementation of the scheme (P k,h was done in Matlab with a semi-implicit, linearized treatment of the nonlinearity and a direct solution of linear equations. 7.. Motion of a degree-one vortex. The first example discusses the motion of one degreeone vortex. In this example we expect that U h,m x/ x for (k, h, ε and T. Example 7.. Let Ω:= ( /, /, a:= (/4, /4, ε:= /, and x a ( u (x:= x a + ε for x Ω, / x for x Ω. x Set g := u Ω and T :=. We employed the approximation scheme (P k,h in Example 7. with a uniform triangulation with maximal mesh-size h = /3 and with a time-step size parameter k = 4. Figure displays the numerical solutions U h,m for m =,, 3, 5. We observe that the vortex, initially located at (/4, /4 rapidly moves to the origin and the numerical solution does not vary afterwards. We remark that we obtained similar numerical results for the choice k = ε. 7.. Repulsion of four degree-one vortices. Our second numerical example involves four degree-one vortices which are close to each other at time t =. As discussed in the introduction we expect that they tend to maximize their distance and then attain a stable state.

21 FIGURE. Numerical solutions U h,m in Example 7. for m =,, 3, 5, i.e., for t m =,.,.3, and.5. Example 7.. Let Ω:= ( /, /, a := (, /, a := (, /, a 3 := (, /, and a 4 := (, /. Set ε:= / and define 4 x a j ( u (x:= j= x aj + ε for x Ω, / x 4 for x Ω. x 4 We set g := u Ω and T =. Figure displays the output of the approximation scheme (P k,h for a uniform triangulation with maximal mesh-size h = /3 and with a time-step size parameter k = 4 evaluated at t = mk for m =,,, 5. The vortices repulse each other and the solution becomes stationary once the vortices are close to the corners of Ω. Notice that since the vortices are initially close together, the parameter δ in Assumption I is very small in this example and as it enters the error estimates exponentially we expect a large approximation error in this example. We remark that for other choices of k with k ε 4 we obtained similar results. For larger time step size parameters k > ε we observed a different behavior of the numerical solution Instability of a degree-two vortex. The third example confirms the theoretical result that higher degree vortices are unstable [3], i.e., rapidly split into degree one vortices.

22 FIGURE. Numerical solutions U h,m in Example 7. for m =,,, 5, i.e., for t m =,.,., and FIGURE 3. Numerical solutions U h,m in Example 7.3 for m =,, 5, 4, i.e., for t m =,.,.5, and.4.

23 3 Example 7.3. Let Ω:= ( /, /, set ε:= /, and define x ( for x Ω, x + ε u (x:= / x for x Ω. x We set g := u Ω and T =. The numerical approximations for m =,, 5, 4 in Example 7.3 are shown in Figure 3. As in the previous examples we employed a uniform triangulation with maximal mesh-size h = /3 and a time-step size parameter k = 4. The higher degree singularity splits rapidly into two degree-one vortices which then repulse each other. Notice that this example is not covered by our analysis since the distance between the two vortices is arbitrarily small for small positive t. A large discretization error has to be expected in this example. We observed however, that the numerical solution did not differ significantly for smaller time step sizes. Acknowledgments. S.B. is thankful to A. Prohl for stimulating discussions. This work was supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD. REFERENCES [] H. W. ALT, S. LUCKHAUS: Quasilinear elliptic-parabolic differential equations. Math. Z. 83 (983, [] S. BARTELS: A posteriori error analysis for Ginzburg-Landau type equations. In preparation (4. [3] A. BEAULIEU: Some remarks on the linearized operator about the radial solution for the Ginzburg- Landau equation. Nonlinear Anal. 54 (3, [4] F. BETHUEL, H. BREZIS, F. HÉLEIN: Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA (994. [5] S. C. BRENNER, L. R. SCOTT: The mathematical theory of finite element methods. Texts in Applied Mathematics, Springer-Verlag, New York (. [6] X. CHEN: Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differential Equations 9 (994, [7] X. CHEN, C. M. ELLIOTT, T. QI: Shooting method for vortex solutions of a complex-valued Ginzburg- Landau equation. Proc. Roy. Soc. Edinburgh A 4 (994, [8] Z. CHEN, K.-H. HOFFMANN: Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity. Adv. Math. Sci. Appl. 5 (995, [9] P. DE MOTTONI, M. SCHATZMAN: Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (995, [] S. DING, Z. LIU: Hölder convergence of Ginzburg-Landau approximations to the harmonic map heat flow. Nonlinear Anal. 46 (, [] Q. DU, M. GUNZBURGER, J. PETERSON: Finite element approximation of a periodic Ginzburg-Landau model for type-ii superconductors. Numer. Math. 64 (993, [] Q. DU, M. GUNZBURGER, J. PETERSON: Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (99, 54 8 [3] W. E: Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. Phys. D 77 (994, [4] L. C. EVANS: Partial differential equations. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI (998. [5] X. FENG, A. PROHL: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (3,

24 4 [6] X. FENG, A. PROHL: Numerical analysis of the Cahn-Hilliard equation and approximation of the Hele- Shaw problem. Part I: Error analysis under minimum regularities. SIAM J. Numer. Anal. (to appear. [7] X. FENG, A. PROHL: Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comp. 73 (4, [8] V. GINZBURG, L. LANDAU: On the theory of superconductivity. Zh. Èksper. Teoret. Fiz. (95, In Men of Physics: L.D. Landau, D. TER HAAR, ed. Pergamon, Oxford (965, [9] R.-M. HERVÉ, M. HERVÉ: Étude qualitative des solutions réelles d une équation différentielle liée à l équation de Ginzburg-Landau. Ann. Inst. H. Poincaré Anal. Non Linéaire (994, [] K.-H. HOFFMANN, J. ZOU: Finite element approximations of Landau-Ginzburg s equation model for structural phase transitions in shape memory alloys. RAIRO Modél. Math. Anal. Numér. 9 (995, [] A. JAFFE, C. TAUBES: Vortices and monopoles. Progress in Physics, Birkhäuser Boston, Inc., Boston, MA (994. [] D. KESSLER, R. H. NOCHETTO, A. SCHMIDT: A posteriori error control for the Allen-Cahn problem: circumventing Gronwall s inequality. Preprint (3. [3] E. H. LIEB, M. LOSS: Symmetry of the Ginzburg-Landau minimizer in a disc. Math. Res. Lett. (994, [4] F. H. LIN: Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension- submanifolds. Comm. Pure Appl. Math. 5 (998, [5] F. H. LIN: The dynamical law of Ginzburg-Landau vortices. Proceedings of the Conference on Nonlinear Evolution Equations and Infinite-dimensional Dynamical Systems (Shanghai, 995,, World Sci. Publishing, River Edge, NJ (997. [6] F. H. LIN: Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49 (996, [7] F. H. LIN, Q. DU: Ginzburg-Landau vortices: dynamics, pinning, and hysteresis. SIAM J. Math. Anal. 8 (997, [8] T. C. LIN: Spectrum of the linearized operator for the Ginzburg-Landau equation. Electron. J. Differential Equations 4 (, 5 pp. (electronic. [9] T. C. LIN: The stability of the radial solution to the Ginzburg-Landau equation. Comm. Partial Differential Equations (997, [3] P. MIRONESCU: On the stability of radial solutions of the Ginzburg-Landau equation. J. Funct. Anal. 3 (995, [3] P. MIRONESCU: Les minimiseurs locaux pour l équation de Ginzburg-Landau sont á symétrie radiale. C. R. Acad. Sci. Paris Sér. I Math. 33 (996, [3] M. MU, Y. DENG, C.-C. CHOU: Numerical methods for simulating Ginzburg-Landau vortices. SIAM J. SCI. COMPUT. 9 (998, [33] J. C. NEU: Vortices in complex scalar fields. Phys. D 43, no. -3, [34] F. PACARD, T. RIVÌERE: Linear and nonlinear aspects of vortices. The Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA (. [35] V. THOMÉE: Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, Springer-Verlag, Berlin (997. [36] M. F. WHEELER: A priori L error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. (973,