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

Vol. 39, N 5, 2005, pp. 863–882 DOI: 10.1051/m2an:2005038

ROBUST A PRIORI ERROR ANALYSIS FOR THE APPROXIMATION OF DEGREE-ONE GINZBURG-LANDAU VORTICES

S¨ oren Bartels

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 param- eter 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.

Mathematics Subject Classification. 35K59, 35Q99, 53A10.

Received: April 12, 2004. Revised: May 24, 2005.

1. Introduction

A mathematical model due to Ginzburg and Landau [18] for the description of certain superconducting materials involves a minimization of the energy functional

Gκ(u, A) =

Ω|∇u−iAu|²dx+κ 2

Ω(|u|²−1)²dx+

Ω|curlA−hext|²dx

where Ω⊆Rⁿ, n= 2,3, is a bounded Lipschitz domain and represents the region occupied by the supercon- ductor. The complex-valued functionuis the condensate wave function,Ais the magnetic potential associated to the induced magnetic field, hext represents an external magnetic field, and κ >0 is the Ginzburg-Landau parameter. The quantity|u|² gives the density of cooper pairs of electrons that produce the superconductivity and zeros of|u|² are calledvortices. 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 onκ. The caseκ= 1/√

2 has been studied by Jaffe and Taubes [21] and it has been shown that vortices can be located anywhere in Ω and do not tend to interact. Ifκ <1/√

2 then vortices tend to attract each other and concentrate in one place. The caseκ >1/√ 2 is called the repulsive case since for such values ofκvortices of same sign tend to repulse each other. Materials for whichκ <1/√

2 orκ >1/√

2 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, 8, 13, 21, 24–27, 33, 34]

and references therein.

Keywords and phrases.Ginzburg-Landau equations, numerical approximation, error analysis, spectral estimate, finite element method.

1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA.[email protected]

c EDP Sciences, SMAI 2005

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2005038

The parameterκalso influences the stability of numerical approximation schemes for superconducting ma- terials. In particular, the dynamics of superconducting materials are often modeled by the gradient flow of the energy functional Gκ and numerical results sensitively depend upon κ. It is not difficult to see that standard error estimates depend exponentially uponκwhich can make simulations useless ifκis large. In this article we aim to design and analyze approximation schemes for which the discretization error grows in κonly 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 [15–17] 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 [1, 36]. Interesting numerical studies of stationary and time-dependent Ginzburg-Landau equations which motivated this work can be found in [7, 11, 12, 20, 32]. An a posteriori error analysis based on ideas of this work can be found in [2].

For ease of presentation we will restrict the analysis to a simplified version ofGκ, in whichA≡0,ε²= 1/(2κ), andn= 2,i.e., Ω⊆R²,

Jε:H¹(Ω;C)→R, v→ 1 2

Ω|∇v|²dx+ε⁻² 4

Ω(|v|²−1)²dx.

In this model, Dirichlet type boundary conditions replace an applied magnetic field. The dynamics of a su- perconducting material can then – in a greatly simplified manner – be described by the gradient flow of Jε. In order to define the time-dependent problem, we assume that we are given a parameter T >0, initial data u^ε₀∈H¹(Ω;C), and boundary datag^ε=u^ε₀|∂Ω∈H^1/2(∂Ω;C). We then aim to solve the following problem:

(P)















Findu^ε∈H¹(0, T;H⁻¹(Ω;C))∩L²(0, T;H¹(Ω;C))such that for almost allt∈(0, T)and all v∈H₀¹(Ω;C)there holds

u^εt;v+ (∇u^ε;∇v) +ε⁻²(f(u^ε);v) = 0, u^ε|∂Ω =g^ε,

u^ε(0) =u^ε₀.

Here,f(a) = (|a|²−1)afora∈C, (·;·) denotes the scalar product inL²(Ω;C), ·;·denotes the duality pairing ofH¹(Ω;C) andH⁻¹(Ω;C),u^ε_t denotes the time derivative ofu^ε, and we use the notationa·b:= (ab+ab)/2 fora, b∈C. Existence and uniqueness of a solutionu^εfollow from standard techniques. Throughout this work we abbreviate

u=u^ε, u₀=u^ε₀ and g=g^ε

but stress that the dependence of the error of numerical approximation schemes uponεis the main contribution of this work.

The main results of this work prove that under moderate conditions on the initial data u₀ and the domain Ω and if the time-step size parameterk >0 and the mesh-sizeh >0 of an implicit in time, lowest order finite element in space discretization of (P) satisfy (up to logarithmic and constant factors) k≤ε⁸ andh≤ε⁴, then the spatial discretization error inL² is bounded byε⁻⁴(k+h²) for allt∈(0, T). The assumptions involve the condition that vortices ofu₀are well separated and of degree one.

The outline of this article is as follows. We state and discuss theoretically and physically motivated assump- tions on the solution u of (P) in Section 2. 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 robusta priorierror estimates.

2. Assumptions on the solution

In order to derive robusta priorierror estimates for the numerical approximation of (P) we will assume that the solutionuof (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 2.1 [19]. Let f₀: [0,∞)→Rsatisfy f₀(0) = 0,lims→∞f₀(s) = 1,f₀≥0, and

−f₀−s⁻¹f₀ +s⁻²f₀=f₀(1−f₀²).

Given any real numberH˜ ∈Rthe function

U_H˜(x) :=f₀(r(x)/ε)e^iθ(x)e^−iH˜,

where(r(x), θ(x)) denote the polar coordinates ofx∈R², satisfiesU_H˜(0) = 0and

−∆UH˜ +ε⁻²f(U_H˜) = 0 inR².

The following assumption states that in certain neighborhoods of zeros the solutionuof (P) is given by small perturbations of functionsU_H˜ as in the previous lemma. Away from zeros,|u|is assumed to be close to 1.

Assumption I. There exist1 ≥ε₀ >0,δ₀, c₀ >0,c₁≥1, and an integer d≥0 such that for all ε∈(0, ε₀), for allt∈(0, T], and j= 1,2, ..., dthere exist

a^t,ε_j ∈R², p^t,ε_j , q_j^t,ε:Bδ₀(a^t,ε_j )→C, H_j^t,ε:Bδ₀(a^t,ε_j )→R, H˜_j^t,ε∈R such that

u(t, x) =f₀

r(x−a^t,ε_j )/ε

e^{iθ(x−at,εj )}e^{−iHjt,ε(x)}+εp^t,ε_j (x) +ε²q_j^t,ε(x) forx∈Bδ₀(a^t,ε_j ) and |u(t, x)| −1≤c₀δ⁻²₀ ε² for x∈Ω\ ∪^dj=1Bδ₀/2(a^t,ε_j ).

The functions p^t,ε_j , q^t,ε_j , H_j^t,ε,j= 1,2, ..., d, satisfy

|Hj^t,ε(x)−H˜_j^t,ε| ≤c₀ε², f Uj(x)

p^t,ε_j (x)≤c₀ε, |qj^t,ε(x)| ≤c₀ for x∈Bδ₀(a^t,ε_j ), where for each j= 1,2, ..., d,

Uj(x) :=f₀

r(x−a^t,ε_j )/ε

e^{iθ(x−at,εj )}e^{−iH˜jt,ε(x)} forx∈Bδ₀(a^t,ε_j ).

Moreover, for almost all(t, x)∈(0, T)×Ωthere holds|u(t, x)| ≤c₁.

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 datau₀.

(ii) Assumption I implicitly requires that the vortices are well separated, i.e., have a distance δ₀. Since f₀(r) = 1−1/(2r²) +O(1/r⁴) forr→ ∞the assumed expansion ofu(t, x) in a neighborhood of a vortex and the condition|u(t, x)| −1≤c₀δ⁻²₀ ε²away from the vortices are compatible. Note thatf₀(r) =ar−ar³/8 +O(r⁵) forr→0 and a constanta∈R[19].

(iii) Similar assumptions on the solutions of related problems have been made in [13, 28] to study the dynamics of vortices, in [6,9] to study the stability of interfaces in phase field equations, and in [15,16,22] 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 degree-one vortices is specified in the next theorem and will be one key ingredient for oura priorierror analysis.

Theorem 2.2 [3, 23, 29, 30]. Given H˜ ∈ R let U_H˜(x) := f₀(r(x)/ε)e^iθ(x)e^−iH˜. Let L_ε,H˜ : H₀¹(Bδ₀(0);C) → L²(Bδ₀(0);C)be defined by

L_ε,H˜v:=−∆v+ε⁻²f(U_H˜)v.

Then, the principal eigenvalue ofL_ε,H˜ is non-negative, i.e., for all v∈H₀¹(Bδ₀(0);C) there holds (L_ε,H˜v;v)≥0.

Proof. Proofs of the theorem can be found in [3, 23, 29, 30] for ˜H = 0. If ˜H = 0 one can check that given any v∈H₀¹(Bδ₀(0);C) there holds withw:= e^iH˜v

(L_ε,H˜v;v) = (Lε,0w;w)≥0

which proves the theorem.

The theorem can be generalized to small perturbations of degree one vortices as follows.

Proposition 2.3. Suppose that Assumption I holds. Then there exists an ε-independent constant λ₀≥0 such that for all ε∈(0, ε₀), forj = 1,2, ..., d, for all t∈(0, T], and for allv∈H₀¹(Bδ₀(a^t,ε_j );C) there holds

(∇v;∇v) +ε⁻²(f(u)v;v)≥ −λ₀||v||²L²(B_δ0(a_j)). Proof. Letj∈ {1,2, ..., d}andt∈(0, T]. Throughout this proof we abbreviate

a:=a^t,ε_j , p:=p^t,ε_j , q:=q^t,ε_j , H:=H_j^t,ε, H˜:= ˜H_j^t,ε, wherea^t,ε_j ,p^t,ε_j ,q^t,ε_j ,H_j^t,ε, and ˜H_j^t,εare as in Assumption I. Forx∈Bδ₀(a) define

U_H˜(x) :=f₀

r(x−a)/ε

e^iθ(x−a)e^−iH˜, UH(x) :=f₀

r(x−a)/ε

e^iθ(x−a)e^−iH(x). There holds

f(u) =f(U_H˜) +

f(u)−f(UH) +

f(UH)−f(U_H˜)

and owing to the assumed asymptotic expansion ofuin Bδ₀(a), a Taylor expansion of the quadratic function f shows in Bδ₀(a)

|f(u)−f(UH)| ≤ |f(UH)(u−UH)|+1

2|f(UH)||u−uH|²

≤ε|f(UH)p|+ε²|f(UH)||q|+ε²

2|f(UH)||p+εq|²

≤Cε².

Since|e^−iH−e^−iH˜| ≤C|H−H| ≤˜ Cε² inBδ₀(a) we have

|f(UH)−f(U_H˜)| ≤ |f(U_H˜)||UH−U_H˜|+1

2|f(UH)||UH−U_H˜|²≤Cε².

Using these estimates we infer with Theorem 2.2 for allv∈H₀¹(Bδ₀(a);C) that

(∇v;∇v) +ε⁻²(f(u)v;v)≥(∇v;∇v) +ε⁻²(f(U_H˜)v;v)−C||v||²L²(B_δ0(a))

≥ −C||v||²L²(B_δ0(a)).

This proves the proposition.

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 2.4[10]. Suppose that Ω is smooth andg is smooth, independent ofε, and satisfies|g(s)|= 1 for all s∈∂Ω. Assume thatu₀=u^ε₀ is smooth, independent ofε, and satisfies|u₀(x)|= 1 for allx∈Ω. Then, there exist ˜ε₀>0 and anε-independent constantC >0 such that for allε∈(0,ε˜₀), for almost all (t, x)∈(0, T)×Ω there holds |u(t, x)| −1≤Cε². In particular, Assumption I holds withδ₀= 1,ε₀= ˜ε₀,c₀=C, andd= 0.

Assumption I can be motivated as follows.

Example 2.5. Given t∈(0, T] we introducey:=x/εand make the ansatz u(t, x) =U(y) +εp(y) +ε²q(y) which leads to the three leading order equations

ε⁻²

−∆U+ (|U|²−1)U

= 0, (2.1)

ε⁻¹

−∆p+|U|²p+ 2(U·p)U−p

= 0, (2.2)

ε⁰

−∆q+|U|²q+|p|²U+ 2(U·q)U−q

= ut(t,·). (2.3)

Equation (2.1) admits a solutionU subject to boundary conditionsU|∂Ω=g. Provided thatU(x/ε) is minimal for Jε it can be shown [34] that there exists an integer d≥0, a harmonic function ψ : Ω→ R, and a^ε_j ∈ Ω, j= 1,2, ..., d, such that

U(·/ε)−e^iψ d j=1

f₀

r(· −a^ε_j)/ε

e^{iθ(·−aεj)}

L^∞(Ω)→0 forε→0.

We choosep≡0 as a solution for (2.2) and letqbe the solution of the linear equation (2.3) subject toq|∂Ω= 0.

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 (2.1) in the following example. It gives a precise characterization of the solution of (2.1) in neighborhoods of vortices and shows that|u|is close to 1 away from vortices.

Example 2.6 [4, 31].

(i) Suppose U = U^ε satisfies U(0) = 0, 1− |U|²

L²(B_δ0/ε(0)) ≤ C, solves (2.1) in Bδ₀/ε(0), and U(·/ε) is minimal for Jε with Ω = Bδ₀/ε(0). Assuming thatε is small enough we may assume that U solves (2.1) in R². It can then be shown that there existsθ₀∈Rsuch that for ally∈R²there holdsU(y) =f₀(r(y))e^i(θ(y)+θ0). (ii) Suppose that G : ∂Bδ₀/ε(0) → C is smooth, satisfies |G(s)| = 1 for all s ∈ ∂Bδ₀/ε(0), and deg(G, ∂Bδ₀/ε(0)) = 0. Let U satisfy U|∂B_δ

0/ε(0) = G, solve (2.1) in Bδ₀/ε, and suppose that U(·/ε) is minimal for Jε with Ω = Bδ(0). Then there exists an ε-independent constant C > 0 such that for all y∈Bδ₀/ε(0) there holds |U(y)| −1≤Cε².

3. A

bounds

The following a prioribounds on a solution uof (P) are independent of Assumption I. They solely assume that the modulus of the solutionuis uniformly bounded in (0, T)×Ω.

Proposition 3.1. Suppose that there exists an ε-independent constant c₁ > 0 such that for almost all (t, x)∈(0, T)×Ωthere holds |u(t, x)| ≤c₁. Then, there exists an ε-independent constantc₂>0such that

(i) ess sup_s∈(0,T)1

2||∇u||²L2(Ω)+ T

0 ||ut(s)||²L2(Ω)ds≤2Jε(u₀), (ii) ess sup_s∈(0,T)||ut||²L²(Ω)+

T

0 ||∇ut||²L²(Ω)ds

≤c₂

ε⁻²Jε(u₀) +||∆u₀−ε⁻²f(u₀)||²L²(Ω)

,

(iii) T

0 ||utt||²H⁻¹(Ω;C)ds≤c₂

ε⁻²Jε(u₀) +||∆u₀−ε⁻²f(u₀)||²L²(Ω)+ε⁻⁴Jε(u₀) .

Sketch of proof. We formally derive the estimates to verify the dependence on ε and note that the assertions can be proved rigorously by employing techniques from,e.g., [14].

(i) This follows from choosingv=ut (noteut|∂Ω≡0) in (P)

||ut||²L²(Ω)+ d dt

1

2||∇u||²L²(Ω)+ε⁻²1

4||(|u|²−1)²||L¹(Ω)

= 0 and integrating this equation over (0, s) for any 0≤s≤T.

(ii) We formally differentiate the first equation in (P) in time,

utt−∆ut+ε⁻²f(u)ut= 0, (3.1)

and test this equation with ut, i.e., 1 2

d

dt||ut||²L²(Ω)=−||∇ut||²L²(Ω)−ε⁻²(f(u)ut;ut).

Using||u||L^∞((0,T)×Ω)≤c₁ and integrating over (0, s) for any 0≤s≤T we infer

||ut(s)||²L²(Ω)+ T

0 ||∇ut||²L²(Ω)ds≤Cε⁻² T

0 ||ut||²L²(Ω)ds+||ut(0)||²L²(Ω)

≤Cε⁻²Jε(u₀) +||ut(0)||²L²(Ω). We then use (i) andut(0) = ∆u₀−ε⁻²f(u₀) to verify the assertion.

(iii) Using the estimate||f(u)||L∞(Ω)≤C we verify with (3.1)

||utt||H⁻¹(Ω;C)≤ ||∇ut||L²(Ω)+ε⁻² sup

ϕ∈H₀¹(Ω;C)

(f(u)ut;ϕ)

||ϕ||H¹(Ω)

≤ ||∇ut||L²(Ω)+Cε⁻²||ut||L²(Ω).

Assertion (iii) is then deduced from squaring this estimate, integrating the resulting equation over (0, T), and

employing previous estimates.

In order to exploit the estimates of the proposition we need to assume bounds on Jε(u₀) = Jε(u^ε₀) and

||∆u₀−ε⁻²f(u₀)||L²(Ω). In general, it is impossible to bound these quantities independently ofεin the considered two-dimensional setting.

Assumption II. There exists a positive functionγ:R→Rsuch that γ(ε)≤γ(ε)² and Jε(u₀)≤γε:=γ(ε), ||u0||²H²(Ω)+||∆u0−ε⁻²f(u₀)||²L²(Ω)≤γεε⁻⁴ for allε∈(0, ε₀).

For an optimal choice ofu₀, which extendsg from ∂Ω to Ω, we may assume thatγ grows at most logarith- mically forε→0.

Example 3.2[4]. Assume that Ω is smooth and simply connected and suppose thatg is smooth, independent of ε, and satisfies |g(s)| = 1 for all s∈ ∂Ω. Then there exist ˜ε₀ >0, anε-independent constant C >0, and

˜

g∈H¹(Ω;C) with ˜g|∂Ω=g such that for allε∈(0,ε˜₀) there holds Jε(˜g)≤2πd˜log(ε⁻¹) +C

where ˜d= deg(g, ∂Ω). The estimate is sharp in the sense that if Ω is starshaped and ˜d >0,e.g., Ω =B₁(0) and g(s) =sfors∈∂Ω, then there holds limε→0

minv|∂Ω=gJε(v)−πd|˜log(ε)|

=Cfor anε-independent constant C >0.

4. Semi-discrete in time approximation scheme

We analyze the following semi-discrete in time approximation (Pk) of (P) which is defined through a time step size parameterk >0.

(Pk)















Find(Um: 0≤m≤M)⊆H¹(Ω;C)such that for all 1≤m≤M and allV ∈H₀¹(Ω;C)there holds (dtUm;V) + (∇Um;∇V) +ε⁻²

f(Um);V) = 0, Um|∂Ω =g, U₀ =u₀.

Here,M is the largest integer for whichM k≤T and given any sequence (am: 0≤m≤M) the discrete time derivativedtam is for 1≤m≤M defined by dtam:= (am−am−1)/k. We settm:=mk for 0≤m ≤M and define

em:=u(tm)−Um.

The followinga prioribound for a solution (Um: 0≤m≤M) is the discrete counterpart of assertion (i) in Proposition 3.1.

Proposition 4.1. Suppose that k≤ε². There holds

1≤maxm≤M

1

2||∇Um||²L²(Ω)+ε⁻²

4 |Um|²−1 ²

L²(Ω)

+k M m=1

1

2||dtUm||²L²(Ω)+kε⁻²

4 dt(|Um|²−1) ²

L2(Ω)

≤2Jε(u₀).

Proof. The choiceV =dtUm for 1≤m≤M in (Pk) yields

||dtUm||²L²(Ω)+ 1 2k

||∇Um||²L²(Ω)− ||∇Um−1||²L²(Ω)

+ε⁻²

f(Um);dtUm

≤ ||dtUm||²L²(Ω)+

∇Um;dt∇Um

+ε⁻²

f(Um);dtUm

= 0.

We write

f(Um) =1 2

|Um|²−1

Um+Um−1+kdtUm

and usedtUm·(Um+Um−1) =dt

|Um|²−1

and ^k₂(|Um|²;|dtUm|²)≥0 to verify f(Um);dtUm

= 1 2

(|Um|²−1);dt(|Um|²−1) +k

2

(|Um|²−1);|dtUm|²

≥ 1 2

(|Um|²−1);dt(|Um|²−1)

−k

2||dtUm||²L2(Ω). Employing the identitya·(a−b) = ¹₂|a−b|²+¹₂(|a|²− |b|²) fora, b∈Cwe deduce

(|Um|²−1);dt(|Um|²−1)

= 1

2k (|Um|²−1)−(|Um−1|²−1) ²

L²(Ω)

+ 1

2k |Um|²−1 ²

L²(Ω)− |Um−1|²−1 ²

L²(Ω)

= k

2 dt(|Um|²−1) ²

L²(Ω)+1

2dt |Um|²−1 ²

L²(Ω). We may therefore estimate

1−ε⁻²k 2

||dtUm||²L2(Ω)+ 1

2k||∇Um||²L2(Ω)− 1

2k||∇Um−1||²L2(Ω)

+kε⁻²

4 dt(|Um|²−1) ²

L²(Ω)+ε⁻²

4 dt |Um|²−1 ²

L²(Ω)≤0.

Multiplying this estimate withk and taking sum overm= 1, ..., for any 1≤≤M we deduce 1

2||∇U||²L²(Ω)+ε⁻²

4 |U|²−1 ²

L2(Ω)

+k m=1

1−kε⁻² 2

||dtUm||²L²(Ω)+kε⁻²

4 dt(|Um|²−1) ²

L²(Ω)

≤1

2||∇U₀||²L2(Ω)+ε⁻²

4 |||U₀|²−1||²L2(Ω)=Jε(U₀).

SinceU₀=u₀and 1−kε⁻²/2≥1/2 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 (Pk). The following definition introduces for each time steptm>0 a partition of unity which is adapted to the neighborhoods of vortices in Assumption I.

Definition 5.1. Given an integer 1≤m≤M let

ψm,j :j = 1,2, ..., Lm

⊆C^∞(Ω;R) be a partition of unity with finite overlapα >0,i.e., for allx∈Ω there holds

card

j∈ {1,2, ..., Lm}:ψm,j(x)>0

≤α, such that for anε-independent constantc₃>0 and forj= 1,2, ..., Lm, there holds

ψm,j ≥0,

L_m

j=1

ψm,j = 1, and |∇

ψ_m,j^1/2

|²≤c₃δ⁻²₀ in Ω,

and, forj = 1,2, ..., d,

ψm,j = 1 inBδ₀/2(a^t_j^m,ε) and suppψm,j⊆Bδ₀(a^t_j^m,ε).

The nonlinearity in the error equation defined by (P) and (Pk) requires a different treatment in neighborhoods of vortices and in the remaining parts of Ω. Recall thatem=u(tm)−Umfor all 0≤m≤M.

Lemma 5.2. Suppose that Assumption I holds. For all ε∈(0, ε₀), 1≤m≤M, and j =d+ 1, d+ 2, ..., Lm

there holds

ε⁻²(f(u(tm))−f(Um);ψm,jem)≥ −c₀(c₁+ 1)δ₀⁻²ψm,j^1/2em²L²(Ω)−ε⁻²

8 ψ_m,j^1/4em ⁴

L⁴(Ω).

Proof. Given 1≤m≤M we abbreviateu=u(tm),U =Um, e=em, andψj=ψm,j forj=d+ 1, d+ 2, ..., L.

Assumption I guarantees|u| −1≤c₀δ₀⁻²ε²in suppψj and we may therefore deduce (f(u)−f(U);ψje) =

|u|²−1)u;ψje

−

(|U|²−1)U;ψje

=

(|u|²−1)e;ψje

+

(|u|²−1)−(|U|²−1) U;ψje

=

(|u| −1)(|u|+ 1)e;ψje

+

(|u|²−1)−(|U|²−1) U;ψje

≥ −c₀δ⁻²₀ ε²(c₁+ 1)ψ_j^1/2e²L2(Ω)+

(|u|²− |U|²)U;ψje . We manipulate the second term in the right-hand side as follows,

(|u|²− |U|²)U;ψje

=

(u−U)·(u+U) U;ψje

=

e·(e+ 2U) U;ψje

=

|e|²U;ψje + 2

(e·U)U;ψje

=

|e|²U;ψje

+ 2||ψ¹j^/2(e·U)||²L²(Ω)

≥ −1

8 ψ_j^1/2|e|^{2 2}

L²(Ω)−2 ψ¹_j^/2(e·U) ²

L²(Ω)+ 2||ψj^1/2(e·U)||²L²(Ω)

=−1

8 ψ_j^1/2|e|^{2 2}

L²(Ω)=−1

8 ψ_j^1/4e ⁴

L⁴(Ω).

The lemma follows from a combination of the two estimates.

Lemma 5.3. Suppose that Assumption I holds. For all ε∈(0, ε₀),1≤m≤M, andj= 1,2, ..., dthere holds ε⁻²

f(u(tm))−f(Um);ψm,jem

≥ −λ0(1−ε²)||ψm,j^1/2em||²L²(Ω)

−(1−ε²)||∇(ψm,j^1/2em)||²L²(Ω)−3c₁ε⁻²||ψ¹m,j^/3em||³L³(Ω)− ||ψm,j^1/2em||²L²(Ω).

Proof. As in the previous proof we abbreviateu=u(tm), U =Um, e=em, andψj =ψm,j forj = 1,2, ..., d.

Given anya, b∈Cthere holds

(f(a)−f(b))·(a−b)≥f(a)(a−b)·(a−b)−3|a||a−b|³ and

f(a)(a−b)·(a−b)≥ −|a−b|².

We thus have for allθ∈[0,1], f(u)−f(U);ψje

≥

f(u)e, ψje

−3

|u||e|³;ψj

= (1−θ)

f(u)e, ψje

−3c₁||ψ¹j^/3e||³L3(Ω)+θ

f(u)e;ψje

≥(1−θ)

f(u)e, ψje

−3c₁||ψ¹_j^/3e||³L3(Ω)−θ||ψ_j^1/2e||²L2(Ω). Applying the spectral estimate of Proposition 2.3 to v:=ψ¹_j^/2ewe infer

f(u)e;ψje

=

f(u)v;v

≥ −λ0ε²||v||²L²(Ω)−ε²||∇v||²L²(Ω).

Choosingθ=ε²we verify the assertion of the lemma.

The following lemma gives bounds on the higher norms of the error that arose in the previous estimates.

Lemma 5.4. Suppose that there exists a positive function ρ : R → R such that ρ ≤ ρ² and

||∇em||L²(Ω) ≤ ρε := ρ(ε) for all 1 ≤ m ≤ M and all ε ∈ (0, ε₀). Then there exists an ε-independent constant c₄>0 such that for allε∈(0, ε₀)and all 1≤m≤M there holds

||em||³L³(Ω)+||em||⁴L⁴(Ω)≤c₄ρ²_εk²||dtem||²L²(Ω)+c₄ρε||em−1||L²(Ω)||∇em−1||²L²(Ω).

Proof. Employing the estimate ||v||²L⁴(Ω) ≤ C||v||L²(Ω)||∇v||L²(Ω) for v ∈ H₀¹(Ω;C) and the bounds C||em||L²(Ω)≤ ||∇em||L²(Ω)≤ρε we infer for 1≤m≤M

1

8||em||⁴L⁴(Ω)≤ ||em−em−1||⁴L⁴(Ω)+||em−1||⁴L⁴(Ω)

=k⁴||dtem||⁴L⁴(Ω)+||em−1||⁴L⁴(Ω)

≤Ck⁴||dtem||²L²(Ω)||dt∇em||²L²(Ω)+||em−1||⁴L⁴(Ω)

=Ck²||dtem||²L2(Ω)||∇(em−em−1)||²L2(Ω)+||em−1||⁴L4(Ω)

≤Cρ²_εk²||dtem||²L²(Ω)+||em−1||⁴L⁴(Ω)

≤Cρ²_εk²||dtem||²L2(Ω)+C||em−1||²L2(Ω)||∇em−1||²L2(Ω). Similarly, using||v||³L3(Ω)≤C||v||²L4(Ω)||v||L2(Ω) and||v||L4(Ω)≤C||∇v||L2(Ω), we derive

1

4||em||³L3(Ω)≤ ||em−em−1||³L3(Ω)+||em−1||³L3(Ω)

=k³||dtem||³L3(Ω)+||em−1||³L3(Ω)

≤Ck³||dtem||²L⁴(Ω)||dtem||L²(Ω)+||em−1||³L³(Ω)

≤Ck³||dtem||²L²(Ω)||dt∇em||L²(Ω)+||em−1||³L³(Ω)

≤Cρεk²||dtem||²L²(Ω)+||em−1||³L³(Ω)

≤Cρεk²||dtem||²L²(Ω)+C||em−1||L2(Ω)||∇em−1||²L²(Ω).

This proves the lemma.

The next lemma gives a bound on the time discretization residual.