On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation

Mod´elisation Math´ematique et Analyse Num´erique Vol. 35, N^o3, 2001, pp. 389–405

ON THE CONVERGENCE OF A LINEAR TWO-STEP FINITE ELEMENT METHOD FOR THE NONLINEAR SCHR ¨ODINGER EQUATION^∗

Georgios E. Zouraris

Abstract. We discretize the nonlinear Schr¨odinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves theL² norm. We prove optimal ordera priorierror estimates in theL² andH¹norms, under mild mesh conditions for two and three space dimensions.

Mathematics Subject Classification. 65M12, 65M60.

Received: January 14, 2000.

1. Introduction

Letd= 1,2 or 3, and Ω⊂R^dbe a bounded domain with smooth boundary∂Ω. Fort^∗>0, we setI_∗:= [0, t^∗] and consider the following initial and boundary value problem for the nonlinear Schr¨odinger equation:

wt= i∆w+ if(|w|²)w in Ω×I_∗,

w= 0 on∂Ω×I_∗, (1.1)

w(x,0) =w⁰(x) forx∈Ω,

where w⁰ : Ω−→C and f ∈ C³([0,+∞);R) are given functions. The nonlinear Schr¨odinger equation, often, appears as a model in mathematical physics (see [1, 7, 8, 16, 22, 26]); for more information on the theory and applications we refer to [5, 6, 10, 14, 15, 17, 19, 21, 25] and the references therein. In the sequel, we will assume that problem (1.1) admits a unique solution which is sufficiently smooth for our purposes.

1.2. Notation and preliminaries

For integer s ∈ N⁰, we denote by H^s(Ω) the Sobolev space consisting of complex-valued functions which, along with their distributional derivatives of order up tos, are inL²(Ω), and byk · k^sthe corresponding norm.

The inner product onL²(Ω) =H⁰(Ω) is denoted by (·,·) and the associated norm byk · k. H0¹(Ω) consists of

Keywords and phrases. Nonlinear Schr¨odinger equation, two-step time discretization, linearly implicit method, finite element method,L²andH¹error estimates, optimal order of convergence.

∗Work supported by the EU-TMR project HCL # ERBFMRXCT960033 and the EU-HCM scheme Reaction Diffusion Equa- tions # ERBCHRXCT930409.

1Department of Numerical Analysis and Computing Science (NADA), Royal Institute of Technology (KTH), 10044 Stockholm, Sweden.

Present address: Centre de Recherche en Math´ematiques de la D´ecision (CEREMADE), UMR CNRS 7534, Universit´e de Paris IX-Dauphine, Place du Mar´echal de Lattre-de-Tassigny, 75775 Paris Cedex 16, France.

c EDP Sciences, SMAI 2001

the functions ofH¹(Ω) that vanish at∂Ω in the sense of trace. In addition, form∈Nand foru,v∈(L²(Ω))^m we use (u, v) :=Pm

j=1(uj, vj) and kuk:={(u, u)}^1/2. Finally, we setH:=H₀¹(Ω)∩C(Ω), and denote by| · |^p the standard norm onL^p(Ω) forp≥2 (i.e. | · |²=k · k).

We mention the Poincar´e–Friedrichs inequality

kvk ≤ CΩk∇vk, ∀v∈H₀¹(Ω), (1.2) and the Sobolev–type inequalities

|v|∞≤ C^1,∞k∇vk²¹kvk¹², ∀v∈H₀¹(Ω), d= 1, (1.3) and

|v|^s≤ C^2,sk∇vk^s−s2kvk^2s, ∀s∈[2,+∞), ∀v∈H₀¹(Ω), d= 2. (1.4) For an integerr≥2, let {Sh}h∈(0,1)be a family of finite dimensional subspaces ofHsatisfying

χinf∈S_h

kv−χk+hkv−χk¹ ≤C h^skvk^s, ∀v∈H^s(Ω)∩H₀¹(Ω), s= 2, . . . , r, ∀h∈(0,1).

Forh∈(0,1), we define the discrete Laplacian ∆h:Sh−→Sh by

(∆hϕ, χ) =−(∇ϕ,∇χ), ∀ϕ, χ∈Sh, an elliptic projection operatorRh:H¹(Ω)−→Shby

(∇Rhv,∇χ) = (∇v,∇χ), ∀v∈H¹(Ω), ∀χ∈Sh,

and finallyPh will be theL²–projection operator ontoSh. The elliptic projectionRh has the following approx- imation property (cf.,e.g., [23])

kRhv−vk+hkRhv−vk¹≤CRh^skvk^s, ∀v∈H^s(Ω)∩H₀¹(Ω), s= 2, . . . , r, ∀h∈(0,1), (1.5) and obviously satisfies

k∇Rhvk ≤ k∇vk, ∀v∈H¹(Ω), ∀h∈(0,1). (1.6) We will say thatf has the property (D), if there exists %≥1 such that

|f⁰(x²)x| ≤C_D 1 +x^%−1

, ∀x >0. (1.7)

1.3. The numerical method

Leth∈(0,1),N ∈N,k:= ^t_N^∗, tⁿ:=nk forn= 0, . . . , N, andt^1/2:=t⁰+^k₂. Let, also,W_h⁰∈Sh be a given approximation ofw⁰. First, we construct an approximationW_h^1/2∈Shofw(·, t^1/2) by

W_h^1/2−W_h⁰ (k/2) = i∆h

W_h^1/2+W_h⁰ 2

! + iPh

"

f(|Wh⁰|²) W_h^1/2+W_h⁰ 2

!#

. (1.8a)

Then, form= 1, . . . , N, we define an approximationW_h^m∈Sh tow(·, t^m) recursively by W_h¹−W_h⁰

k = i∆h

W_h¹+W_h⁰ 2

+ iPh

f(|W_h^1/2|²)

W_h¹+W_h⁰ 2

(1.8b)

and

W_hⁿ−W_hⁿ⁻² 2k = i∆h

W_hⁿ+W_hⁿ⁻² 2

+ iPh

f(|W_hⁿ⁻¹|²)

W_hⁿ+W_hⁿ⁻² 2

, n= 2, . . . , N. (1.8c) Remark 1.1. By dropping (1.8a) and settingW_h^1/2 =W_h⁰ in (1.8b), an alternative method is obtained. For this method, our analysis also applies, butW_h¹ is a suboptimal order approximation ofw(·, t¹) in theH¹ norm.

This is the reason that the “fractional” step (1.8a) has been introduced.

Remark 1.2. Leth∈(0,1). For givenϕ∈Sh andλ >0, we defineTh(λ, ϕ;·) :Sh−→Sh by Th(λ, ϕ;χ) :=χ−ikλ∆hχ−iλkPh(f(|ϕ|²)χ) for χ∈Sh.

To ensure that the method (1.8) is well-defined i.e. the existence and uniqueness of W_h^1/2 and {W_hⁿ}^Nn=1

, it is enough to show that Th(λ, φ;·) is invertible. Let ψ ∈ Sh be such that Th(λ, ϕ;ψ) = 0. Then, we have Re(Th(λ, ϕ;ψ), ψ) = 0, or Re

hkψk²+ iλkk∇ψk²−iλk(f(|ϕ|²),|ψ|²) i

=kψk² = 0, which yieldsψ= 0. Hence, we conclude thatTh(λ, ϕ;·) is one–to–one. SinceTh(λ, ϕ;·) is linear and the spaceSh has finite dimension, the fact that it is one-to-one yields its invertibility.

Remark 1.3. Taking theL² inner product of (1.8a) withW_h^1/2+W_h⁰, of (1.8b) withW_h¹+W_h⁰ and of (1.8c) withW_hⁿ+W_hⁿ⁻², and then real parts, we conclude thatkW_h^1/2k=kW_h⁰kandkW_h<sup>`</sup>k=kW<sub>h</sub><sup>0</sup>kfor `= 0, . . . , N. Thus the method (1.8) isL² conservative.

1.4. Main results and relations to previous work

The time discretization in (1.8c) in conjuction with a finite difference method for the space discretization is proposed in [9] for the numerical approximation of a nonlinear Schr¨odinger equation in one space dimension and with periodic boundary conditions. An optimal order error bound ofO(k²+h²) in a discreteL² norm is, also, given, only in the case of a cubic Schr¨odinger equation, wheref(x) =λxandλ∈R. This convergence result is based on the fact that the method conserves a discrete Hamiltonian which ford= 1 yields boundedness of the numerical approximations in the discrete L^∞ norm by a constant which is independent of the partition of the time and space intervals (cf. (5) in [9] and Rem. 2.22 in Sect. 2.5). In the case of a general nonlinearity this conservation property fails and thus a different technique is needed to prove convergence.

The paper at hand is devoted to the convergence analysis of the method (1.8). We prove an optimal order error bound of O(k²+h^r) in the L² norm and of O(k²+h^r−1) in the H¹ norm, whithout conditions when d= 1, and under the following mild mesh conditions

p|ln(h)|h^r−1≤C2,a and p

|ln(h)|(k³² +h^r)≤C2,b when d= 2, (1.9) or

h^r−1≤C3,a

ph and k³² +h^r≤C3,b

ph when d= 3; (1.10)

here C2,a, C2,b, C3,a and C3,b are constants which depend only on the solution and the data, and h is the minimum of the diameter of the elements of the partition of Ω over which the finite element space is constructed (see Th. 2.14, Th. 2.15 and Rem. 2.16). Also, if d= 2 andf satisfies (D), then we prove an optimal order error bound O(k²+h^r) in theL² norm and a suboptimal one ofO(k³² +h^r−1+k⁻¹²h^r) in theH¹ norm, provided that

k²+h^r≤C√

k, (1.11)

where C is a constant which depends only on the solution and the data (see Th. 2.21).

Usually, the analysis of numerical methods for the nonlinear Schr¨odinger equation is based on inverse in- equalities between norms on the finite element spaces for quasiuniform or local quasiuniform partitions of Ω (see, e.g., [2, 3, 11–13, 18, 20, 24]). Here, when d= 1, ord= 2 and f has a polynomial growth satisfying (D), we obtain, for the method (1.8), optimal order of convergence in the L² norm, avoiding inverse inequalities or assumptions on the finite element spaces (ase.g. theH¹-boundedness assumption of theL²-projectionPh used in [12]) besides those of Section 1.2. However, for general f and d = 2 or 3, we need an inverse inequality between the L^∞and theH¹ norm, and thereforehappears in (1.9) and (1.10).

To arrive at the mesh conditions (1.9), (1.10) or (1.11), we prove convergence estimates in the L² and H¹ norm, for the approximations generated by amodified schemewhich is a nonlinear perturbation of (1.8) at the linearized term (see (Λ) and (Υ) in Sect. 2.3). Then the mesh conditions, exhibited above, are introduced to ensure that the modified approximations are bounded in theL^∞ or in theH¹ norm, by a constant independent of the discretization parameters. Having this boundness property the modified scheme coincides with (1.8) and hence the convergence estimates for it hold also for (1.8). The analysis here has been inspired from the works [12] and [24], but there the methods under consideration, the techniques used and the results obtained are different.

The analysis and the results of the paper extend, easily, to the method obtained substituting (1.8c) by W_hⁿ−W_hⁿ⁻²

2k = i∆h

W_hⁿ+W_hⁿ⁻² 2

+ iPh

h

f(|W_hⁿ⁻¹|²)W_hⁿ⁻¹i

which is a nonconservative implicit-explicit method and, as (1.8), yields only one linear system of algebraic equations at every time level, but the matrix remains unchanged.

An overview of the paper is as follows. Section 2 is divided in five parts. In Section 2.1, we prove some function inequalities often used in the convergence analysis, and in Section 2.2 present a consistency result for the time discretization. Section 2.3 contains the definition of the modified schemes and Section 2.4 the convergence theorems for a general function f. Finally, in Section 2.5 we investigate the special case where d= 2 andf satisfies the property (D).

2. Convergence analysis

We present here some function inequalities that we will often use later.

Lemma 2.1. Foru1, u2∈C(Ω)andg∈C¹([0,+∞);R), we have kg(|u1|²)−g(|u2|²)k ≤ sup

x∈I(u₁,u₂)|g⁰(x)|(|u1|∞+|u2|∞)ku1−u2k (2.1) with I(u1, u2) :=

0,max{|u1|²_∞,|u2|²_∞} .

Lemma 2.2. Foru1,u2,w1,w2∈C(Ω)andg∈C²(R;R), we have kg(u1)−g(u2)−g(w1) +g(w2)k ≤ sup

|x|∈I1(u1,u2,w1,w2)|g⁰⁰(x)| |w1−w2|∞ ku1−w1k+ku2−w2k

+ sup

|x|∈I2(u1,u2)

|g⁰(x)| k(u1−u2)−(w1−w2)k, (2.2)

with I1(u1, u2, w1, w2) :=

0,max{|u1|∞+|u2|∞,|w1|∞+|w2|∞}

andI2(u1, u2) :=

0,|u1|∞+|u2|∞ .

Proof. (2.2) follows directly from the following expansion g(u1)−g(u2)−g(w1) +g(w2) = (w1−w2)

Z 1 0

g⁰((1−τ)u2+τ u1)−g⁰((1−τ)w2+τ w1) dτ +

(u1−u2)−(w1−w2) Z ¹

0

g⁰((1−τ)u2+τ u1)dτ.

Lemma 2.3. Foru1,u2,w1,w2∈C(Ω), we have

|u1|²− |u2|²− |w1|²+|w2|² ≤ 2|w1−w2|∞ku2−w2k

+B(u1, u2, w1, w2)k(u1−w1)−(u2−w2)k, (2.3) with B(u1, u2, w1, w2) := max{|u1|∞+|w1|∞,|u2|∞+|w2|∞}.

Proof. We obtain (2.3) observing that

(|u1|²− |w1|²)−(|u2|²− |w2|²) = Ren

(u1−w1)−(u2−w2)

(u1+w1) +(u2−w2)

(u1−w1)−(u2−w2) +2(u2−w2)(w1−w2)

o .

Lemma 2.4. Let F :C²−→Cbe defined byF(z, ω) :=fe(|z|)ω where fe(x) :=f(x²) for x∈R. If d= 2andf satisfies(1.7), then, for u1,u2,w1,w2∈C(Ω)withu1−w1,u2−w2∈H₀¹(Ω), we have

kF(u1, u2)−F(w1, w2)k ≤C ku1−w1k+ku2−w2kh 1 +

X2 j=1

|wj|^%_∞+k∇(uj−wj)k^%i

. (2.4)

Proof. Letu1,u2, w1,w2∈C(Ω) withu1−w1,u2−w2∈H₀¹(Ω). Observing that

|F(z1, z2)−F(ω1, ω2)| ≤C |z1−ω1|+|z2−ω2|h

1 + |ω1|+|ω2|+|z1−ω1|+|z2−ω2|%i forz1,z2, ω1,ω2∈C, and using (1.4), we obtain

kF(u1, u2)−F(w1, w2)k ≤ChZ

Ω

1 + (|w1|+|w2|)^2%

|u1−w1|+|u2−w2|2

dx +

Z

Ω

|u1−w1|+|u2−w2|2%+2

dxi1/2

≤C h

1 +|w1|^2%_∞+|w2|^2%_∞

ku1−w1k²+ku2−w2k²

+ku1−w1k²k∇(u1−w1)k^2%+ku2−w2k²k∇(u2−w2)k^2%i1/2

which yields (2.4).

2.2. Consistency

We continue by presenting a consistency result concerning the time discretization.

Forn= 0, . . . , N, we defineσⁿ∈L²(Ω) by w^`∗n−w^jn∗

λnk = i

2∆(w^`∗n+w^jn∗) + i

2f(|w^i∗n|²)(w^`∗n+w^j∗n) +σⁿ, (2.5) where w^1/2 :=w(·, t^1/2), w^m := w(·, t^m) for m = 0, . . . , N, λ0 := 1/2, λ1 := 1, λm := 2 form = 2, . . . , N,

`<sup>∗</sup><sub>0</sub> := 1/2,`^∗_m:=mfor m= 1, . . . , N, i^∗₀ := 0,i^∗₁:= 1/2,i^∗_m:=m−1 form= 2, . . . , N, andj₀^∗:= 0, j₁^∗ := 0, j_m^∗ :=m−2 for m= 2, . . . , N. Then, using Taylor expansions, we arrive at

1≤maxn≤Nkσⁿk+kkσ⁰k ≤Ck² and max

4≤n≤Nkσⁿ−σⁿ⁻²k ≤Ck³. (2.6) 2.3. Modified Schemes

Modifying properly the linearized term in the numerical method (1.8), we construct two modified schemes, (Λ) and (Υ), which we will use later in the convergence analysis. A modified scheme is connected to a real parameterδ >0 and a given norm ofH, and it is not a numerical method. When the approximations that the scheme furnishes are bounded in that norm byδ, then they coincide with those that (1.8) produces provided, of course, that the initial approximation is the same. Even that the original method (1.8) is linear, the modified scheme will be nonlinear. Hence, we cannot ensure the existence of the modified approximations following the argument of Remark 1.2. For this reason, we shall employ the following Brouwer-type fixed-point lemma, for a proof of which we refer to [3].

Lemma 2.5. Let (X,(·,·)_X) be a finite-dimensional inner product space and k · kX the associated norm. Let µ: X −→X be continuous and assume that there exists α >0such that for every z∈ X with kzkX =αthere holdsRe(µ(z), z)_X ≥0. Then, there exists az^∗∈ X such thatµ(z^∗) = 0 andkz^∗kX ≤α.

•Modified Scheme (Λ): Letδ >sup_t∈I∗|w(·, t)|∞ andgδ be an increasingC²(R;R) function, with bounded derivatives up to second order, satisfying

gδ(x) :=

x, if |x| ≤δ

2_|^x_x|δ, if |x|>2δ and gδ(x)∈

[δ,2δ] if x∈[δ,2δ]

[−2δ,−δ] if x∈[−2δ,−δ] for x∈R. Then, we define a functionγδ :C−→Cbyγδ(z) =gδ(Rez) + igδ(Imz).

Forh∈(0,1) andm= 0,1/2,1, . . . , N, let Λ^m_δ,h∈Shbe specified inductively by

Λ⁰_δ,h=Rhw⁰ (2.7a)

and

Λ^`_δ,h^∗m −Λ^j_δ,h^∗m λmk = i∆h

Λ^`_δ,h^∗m + Λ^j_δ,h^∗m 2

! + iPh

"

f(|γδ(Λⁱ_δ,h^∗m)|²)γδ

Λ^`_δ,h^∗m + Λ^j_δ,h^m∗ 2

!#

(2.7b) form= 0, . . . , N.

Remark 2.6. It is easily seen thatγδ(w(·, τ)) =w(·, τ) andγδ(^w(·,τ1)+w(₂ ^·,τ2)) =^w(·,τ1)+w(₂ ^·,τ2) forτ, τ1, τ2∈I_∗, wherewis the solution of (1.1). Thus, the consistency argument for the method (1.8) (cf. Sect. 2.2) holds also for (2.7).

Remark 2.7. Assuming that |Λ⁰_δ,h|∞ ≤ δ and |Λ^`_δ,h^∗m|∞ ≤ δ for m = 0, . . . , N, we obtain f(|γδ(Λⁱ_δ,h^∗m)|²) = f(|Λⁱ_δ,h^∗m|²) and γδ

Λ^`_δ,h^∗m+Λ^j_δ,h^∗m 2

= ^Λ

`∗ m δ,h+Λ^j_δ,h^m∗

2 for m = 0, . . . , N. Hence, if W_h⁰ = Rhw⁰, then Λ^{`</sup><sub>δ,h</sub><sup>∗m</sup> = W<sub>h</sub><sup>`∗m} for m= 0, . . . , N.

We ensure the existence of a Λ^`_δ,h^∗m ∈Sh which solves the nonlinear system in (2.7b), by an argument based on Lemma 2.5. In particular, letm∈ {0, . . . , N}, h∈(0,1), (X,(·,·)_X) = (Sh,(·,·)) and Π^m_δ,h:X −→X be an operator given by

Π^m_δ,h(χ) :=χ−Λ^j_δ,h^m∗ − i

2λmk∆hχ− i 2λmkPh

f(|γδ(Λⁱ_δ,h^∗m)|²)γδ(χ)

; Π^m_δ,h is continuous, sinceγδ is continuous andShhas finite dimension. Then, we obtain

Re(Π^m_δ,h(χ), χ)≥ kχk kχk − kΛ^j_δ,h^∗mk −λmkBδ

, ∀χ∈Sh,

where Bδ := √

2δ|Ω|^1/2sup_x∈[0,8δ2]|f(x)| and|Ω| is the area of Ω. Therefore, we have Re(Π^m_δ,h(χ), χ)>0 for everyχ ∈ Sh with kχk= 1 +kΛ^j_δ,h^m∗k+λmkBδ. Applying Lemma 2.5 with µ = Π^m_δ,h, we conclude that there exists aχ^m_δ,h∈Sh such that Π^m_δ,h(χ^m_δ,h) = 0. Thus Λ^`_δ,h^∗m = 2χ^m_δ,h−Λ^j_δ,h^∗m is a solution of (2.7b).

In Remark 2.7, we explained that when the approximations produced by (Λ) are bounded in theL^∞norm by δandW_h⁰=Rhw⁰, then they are the numerical approximations of the method (1.8). Next, we present another modified scheme that has this property for any normν onH, instead of theL^∞ one.

•Modified Scheme (Υ): Letν be a norm onH,δ >sup_t∈I∗ν(w(·, t)) be a given constant, andξδ :R−→Rbe a continuous function defined by

ξδ(x) :=





1, if x≤2δ

−^xδ + 3, if x∈[2δ,3δ]

0, if x >3δ

for x∈R. Then, fort∈I_∗, we define a mapgν,δ(t;·) :H−→Hby

gν,δ(t;ω) :=ω ξδ(ν(ω−w(·, t))) + w(·, t) (1−ξδ(ν(ω−w(·, t)))) for ω∈ H, wherewis, always, the solution of problem (1.1).

Forh∈(0,1) andm= 0,1/2,1, . . . , N, we specify functions Υ^m_δ,h∈Sh, inductively by

Υ⁰_δ,h=Rhw⁰ (2.8a)

and

Υ^`_δ,h^∗m −Υ^j_δ,h^m∗ λmk = i∆h

Υ^`_δ,h^∗m+ Υ^j_δ,h^∗m 2

! + iPh

"

f(|gν,δ(t^i∗m; Υⁱ_δ,h^∗m)|²)gν,δ(t^{`∗m</sup>; Υ<sup>`}_δ,h^∗m) +gν,δ(t^jm∗; Υ^j_δ,h^m∗) 2

#

(2.8b) form= 0, . . . , N.

Remark 2.8. Sincegν,δ(t;w(·, t)) =w(·, t) fort∈I_∗, the consistency argument for (2.8) is the same with that for (1.8) (cf.Sect. 2.2).

Remark 2.9. Assuming thatν(Υ⁰_δ,h) ≤δand ν(Υ^{`</sup><sub>δ,h</sub><sup>∗m</sup>) ≤δ form = 0, . . . , N, we have ν(Υ<sup>0</sup><sub>δ,h</sub>−w(·, t))<2δ and ν(Υ<sup>`}_δ,h^∗m −w(·, t))≤2δ form= 0, . . . , N and t∈I_∗. Thus, we obtaingν,δ(t; Υ^{`</sup><sub>δ,h</sub><sup>∗m</sup>) = Υ<sup>`}_δ,h^∗m form= 0, . . . , N andt∈I_∗. In that case, ifW_h⁰=Rhw⁰, then a simple induction argument yields Υ^{`</sup><sub>δ,h</sub><sup>∗m</sup> =W<sub>h</sub><sup>`∗m}form= 0, . . . , N.

Lemma 2.10. Let ν be a norm on Handδ >sup_t∈I∗ν(w(·, t)). Then, fort∈I_∗ andω∈ H, we have

ν(gν,δ(t;ω))≤4δ. (2.9)

Proof. Lett∈I_∗ andω ∈ H. Ifν(ω−w(·, t))≥3δ thenν(gν,δ(t;ω)) =ν(w(·, t))≤δ. We assume, now, that ν(ω−w(·, t))<3δ. Then we haveν(ω)≤4δ, and

ν(gν,δ(t;ω)) ≤ ν(ω)ξδ(ν(ω−w(·, t))) +ν(w(·, t)) (1−ξδ(ν(ω−w(·, t))))

≤ 3δ ξδ(ν(ω−w(·, t))) +δ

≤ 4δ.

Lemma 2.11. Let ν be a norm onHand δ >supt∈I_∗ν(w(·, t)). Fort∈I_∗, the operatorgν,δ(t;·) :H−→His continuous on (H, ν).

Proof. Fort∈I_∗ andω1,ω2∈ H, we have ν gν,δ(t;ω1)−gν,δ(t;ω2)

≤ν(ω1−ω2)+

ν(w(·, t)) +ν(ω2) ξδ(ν(ω1−w(·, t)))−ξδ(ν(ω2−w(·, t))) which, together with the continuity ofξδ, yields the continuity ofgν,δ(t;·) on (H, ν).

As for (Λ), we discuss, now, the existence of a solution for the nonlinear system in (2.8b) following a similar argument. Letm∈ {0, . . . , N},h∈(0,1), (X,(·,·)_X) = (Sh,(·,·)) and Φ^m_δ,h:X −→X given by

Φ^m_δ,h(χ) :=χ−Υ^j_δ,h^m∗ − i

2λmk∆hχ− i 4λmkPh

h

f |gν,δ(t^i∗m; Υⁱ_δ,h^∗m)|²

gν,δ(t^`∗m; 2χ−Υ^j_δ,h^m∗) +gν,δ(t^jm∗; Υ^j_δ,h^m∗)i ,

which is continuous, since gν,δ(t^`∗m;·) is continuous (cf. Lem. 2.11) and Sh has finite dimension. Also, in Sh

the norms ν and | · |∞ are equivalent, and hence there exists an h–dependent constant Ch,∞,ν, such that

|χ|∞≤Ch,∞,νν(χ) forχ∈Sh. Using (2.9), we arrive at the following general estimates

|gν,δ(τ;ω)|∞≤Be1,h,δ:= 4δCh,∞,ν, ∀τ∈I_∗, ∀ω∈ H, and

f |gν,δ(τ1;ω1)|² gν,δ(τ2;ω2) +gν,δ(τ3;ω3)≤Be2,h,δ:= 2Be1,h,δ

p|Ω| sup

[0,(B^e_1,h,δ)²]

|f|,

forτ1, τ2, τ3∈I_∗ andω1, ω2, ω3∈ H. Thus, we obtain

Re(Φ^m_δ,h(χ), χ)≥ kχk kχk − kΥ^j_δ,h^∗mk − ^λm4^kBe2,h,δ

, ∀χ∈Sh.

Therefore, we have Re(Φ^m_δ,h(χ), χ)>0 for everyχ∈Shwithkχk= 1 +kΥ^j_δ,h^∗mk+^λm₄^kBe2,h,δ, and by Lemma 2.5, withµ= Φ^m_δ,h, we conclude the existence of aχ^m_δ,h∈Sh such that Φ^m_δ,h(χ^m_δ,h) = 0. Finally, Υ^`_δ,h^∗m = 2χ^m_δ,h−Υ^j_δ,h^∗m is a solution of (2.8b).

2.4. L²- and H¹-convergence

Next we will prove the following optimal order error estimates for the method (1.8):

0≤maxm≤NkW_h^m−w^mk ≤ CA(k²+h^r) and max

0≤m≤Nk∇(W_h^m−w^m)k ≤ CA(k²+h^r−1) (2.10) forh∈(0, h0) andN ≥N0. Hereh0∈(0,1),N0∈NandCA is a constant independent ofhandN.

The basic step in the convergence analysis is the estimation of the differencesϑ^1/2_Λ,δ,h := Λ^1/2_δ,h −Rhw^1/2 and ϑⁿ_Λ,δ,h:= Λⁿ_δ,h−Rhwⁿ,n= 0, . . . , N, where due to (2.7a), we haveϑ⁰_Λ,δ,h= 0.

Proposition 2.12. Let d= 1,2or 3, andδ >sup_t∈I∗|w(·, t)|∞. Then, there existsNδ ∈Nsuch that kϑ^1/2_Λ,δ,hk ≤CA,δ(k²+kh^r) and max

1≤m≤Nkϑ^m_Λ,δ,hk ≤CA,δ(k²+h^r), ∀h∈(0,1), ∀N ≥Nδ. (2.11) The constantCA,δ is independent of handN, but depends onδ, the solution or its derivatives, and the data.

Proof. Leth∈(0,1). Form= 0, . . . , N, we defineσ^m_Λ,δ,h∈Sh by ϑ^`_Λ,δ,h^∗m −ϑ^j_Λ,δ,h^∗m

λmk = i∆h

ϑ^`_Λ,δ,h^∗m +ϑ^j_Λ,δ,h^m∗ 2

+σ^mΛ,δ,h. (2.12)

Combining (2.12), (2.5), Remark 2.6, and (1.8), we obtain

σ^m_Λ,δ,h=−Ph(σ^m+ω^m_w) + iPhω^m_Λ,δ, m= 0, . . . , N, (2.13) with

ω^m_w :=Rh

w^`∗m−w^jm∗ λmk

−w^`∗m−w^jm∗ λmk

(2.14) and

ω^m_Λ,δ:=f(|γδ(Λⁱ_δ,h^∗m)|²)γδ

_Λ^`∗m δ,h+Λ^j_δ,h^m∗

2

−f(|w^i∗m|²)γδ

w^`∗m+w^j∗m 2

. (2.15)

Taking real parts of theL²inner product of (2.12) withϑ^{`</sup><sub>h</sub><sup>∗m</sup>+ϑ<sup>j</sup><sub>h</sub><sup>∗m</sup> and using (2.13), form= 0, . . . , N, we have kϑ<sup>`}_Λ,δ,h^∗m k²− kϑ^j_Λ,δ,h^∗m k²=−λmkh

Re(σ^m+ω^m_w, ϑ^{`</sup><sub>Λ,δ,h</sub><sup>∗m</sup> +ϑ<sup>j</sup><sub>Λ,δ,h</sub><sup>m∗</sup> ) + Im(ω<sub>Λ,δ</sub><sup>m</sup> , ϑ<sup>`}_Λ,δ,h^∗m +ϑ^j_Λ,δ,h^m∗ )i

. (2.16)

By (1.5) and the Taylor formula, we conclude that

0≤maxm≤Nkω_w^mk ≤Ch^r. (2.17)

Letm∈ {0, . . . , N}. To estimateω^m_Λ,δ in theL² norm, we split it as follows:

ω^m_Λ,δ=ω_Λ,δ,1^m +ω^m_Λ,δ,2

with ω_Λ,δ,1^m :=

f(|γδ(Λⁱ_δ,h^∗m)|²)−f(|w^i∗m|²) γδ

_Λ^`∗m δ,h+Λ^j_δ,h^∗m

2

and ω_Λ,δ,2^m :=f(|w^i∗m|²) h

γδ

_Λ^`∗m δ,h+Λ^j_δ,h^∗m

2

−γδ

w^`∗m+w^jm^∗

2

i .

Using (2.1), the mean value theorem and (1.5), we obtain kω_Λ,δ,1^m k ≤ C δ(1 +δ) sup

x∈[0,8δ²]|f⁰(x)| kγδ(Λⁱ_δ,h^∗m)−γδ(w^i∗m)k

≤ Cδ sup

|x|∈[0,2δ]

|g_δ⁰(x)| h^r+kϑⁱ_Λ,δ,h^∗m k ,

and

kω_Λ,δ,2^m k ≤C sup

|x|∈[0,2δ]|g⁰_δ(x)| h^r+kϑ^`_Λ,δ,h^∗m k+kϑ^j_Λ,δ,h^∗m k .

Thus, we get the following estimate

kω^m_Λ,δk ≤Cδ(h^r+kϑ^`_Λ,δ,h^∗m k+kϑ^j_Λ,δ,h^m∗ k+kϑⁱ_Λ,δ,h^∗m k). (2.18) From (2.16), (2.17), (2.6) and (2.18), it follows that

kϑ^1/2_Λ,δ,hk+kϑ¹_Λ,δ,hk ≤Ceδk(kϑ^1/2_Λ,δ,hk+kϑ¹_Λ,δ,hk) +Cδ(k²+kh^r) (2.19a) and

(1−Ceδk)(kϑ^m_Λ,δ,hk+kϑ^m_Λ,δ,h⁻¹k)≤(1 +Ceδk)(kϑ^m_Λ,δ,h⁻¹k+kϑ^m_Λ,δ,h⁻²k) +Cδk(k²+h^r), m= 2, . . . , N. (2.19b) Finally, assuming thatCeδk≤¹3 and applying a discrete Gr¨onwall argument on (2.19a-b), we arrive at (2.11).

Proposition 2.13. Let d= 1,2or 3, and δ >sup_t∈I∗|w(·, t)|∞. Then there exists an integerNeδ ≥Nδ such that

k∇ϑ^1/2_Λ,δ,hk ≤ C^A,δ(k³² +k¹²h^r) and max

1≤m≤Nk∇ϑ^m_Λ,δ,hk ≤ C^A,δ(k²+h^r), ∀h∈(0,1), ∀N≥Neδ. (2.20) The constantC^A,δ is independent ofhandN, but depends onδ, the solution or its derivatives, and the data.

Proof. Let h∈ (0,1) and N ≥ Nδ (cf. Prop. 2.12). For m = 0, . . . , N, we define σ_Λ,δ,h^m , ω_w^m and ω^m_Λ,δ as in (2.12), (2.14) and (2.15), respectively.

Taking the L²-inner product of (2.12) byϑ^`_Λ,δ,h^∗m −ϑ^j_Λ,δ,h^∗m and then imaginary parts, we obtain

k∇ϑ^1/2_Λ,δ,hk²= 2Im(σ_Λ,δ,h⁰ , ϑ^1/2_Λ,δ,h), k∇ϑ¹_Λ,δ,hk²= 2Im(σ_Λ,δ,h¹ , ϑ¹_Λ,δ,h), (2.21) and

k∇ϑ^m_Λ,δ,hk²+k∇ϑ^m_Λ,δ,h⁻¹k²=k∇ϑ^m_Λ,δ,h⁻¹k²+k∇ϑ^m_Λ,δ,h⁻²k²+ 2Im(σ^m_Λ,δ,h, ϑ^m_Λ,δ,h−ϑ^m_Λ,δ,h⁻²), m= 2, . . . , N. (2.22)