Optimization and Convergence of Numerical Attractors for Discrete-Time Quasi-Linear Lattice System

1ex plus .25ex minus .1ex 1ex plus .25ex minus .1ex OPTIMIZATION AND CONVERGENCE OF NUMERICAL1 ATTRACTORS FOR DISCRETE-TIME QUASI-LINEAR LATTICE2 SYSTEM∗ 3 YANGRONG LI†, SHUANG YANG †,AND TOM´ AS CARABALLO ‡ 4 Abstract. Existence and connection of numerical attractors for discrete-time p-Laplace lattice5 systems via the implicit Euler scheme are proved. The numerical attractors are shown to have an6 optimized bound, which leads to the continuous convergence of the numerical attractors when the7 graph of the nonlinearity closes to the vertical axis or when the external force vanishes. A new type8 of Taylor expansion without Fr´echet derivatives is established and applied to show the discretization9 error of order two, which is crucial to prove that the numerical attractors converge upper semi-10 continuously to the global attractor of the original continuous-time system as the step size of the11 time goes to zero. It is also proved that the truncated numerical attractors for finitely dimensional12 systems converge upper semi-continuously to the numerical attractor and the lower semi-continuity13 holds in special cases.14 Key words. discrete-time equation, numerical attractor, p-Laplace lattice, finite-dimensional15 approximation; semi-continuity of attractors16 AMS subject classifications. 65L20, 35B40, 37L6017 1. Introduction. We study the numerical scheme of attractors as well as solu-18 tions for the p-Laplace lattice dynamical system (LDS)19 dui(t) dt =ν(Apu(t))i+f(ui(t)) + gi, i ∈Z,(1.1)20 21 where ν > 0, p≥2, u= (ui)i∈Z, and the discrete p-Laplace operator is defined by22 (Apu)i=|ui+1 −ui|p−2(ui+1 −ui)−|ui−ui−1|p−2(ui−ui−1), i ∈Z.23 24 As one knows, a LDS has many applications in fluid dynamics, chemistry and25 neural networks, see [3,5,18,32]. The p-Laplace LDS (1.1) is the space-discretization26 of the corresponding p-Laplace partial differential equation (defined on the real line),27 while the dynamics of the (deterministic or stochastic) p-Laplace PDE was studied in28 [8,11,12,13,23,26,27,28,29,30,33,37].29 As preliminaries, we show in Section 2 that the LDS (1.1) has a positively invariant30 ball Br∗(0) and a global attractor Ain `2, where the dissipative condition of f∈31 C(R,R) is different from those in [10,14,15,36] and given by32 α:= inf s6=0 −f(s) s>0.(1.2)33 34 ∗Submitted to the editors DATE. Funding: Li and Yang have been supported by National Natural Science Foundation of China grant 11571283. Yang has been partially supported by China Scholarship Council (No. 202006990054) for 1 year’s study abroad at the University of Sevilla. Caraballo has been supported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00. †School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR China ([email protected],[email protected]). ‡Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, C/Tarfia s/n, 41012-Sevilla, Spain ([email protected]). 1 This manuscript is for review purposes only. 2Y. LI, S. YANG AND T. CARABALLO In this paper, we mainly consider the numerical scheme in the discrete-time sense.35 Using the step size := tn+1 −tnof the time to discretize the LDS (1.1), we obtain36 the p-Laplace implicit Euler scheme (IES)37 u n,i =u n−1,i +ν(Apu n)i+f(u n,i) + gi,∀n∈N, i ∈Z.(1.3)38 39 As pointing out by Han, Kloeden, Sonner [17], the IES (1.3) with p= 2 is not40 globally solvable for a common step size (see Lemma 5.3 for the reason if p > 2).41 Instead the global solvability, we will prove in Theorem 3.1 that, for sufficiently small42 step sizes, the IES (1.3) is uniquely solvable when the initial datum belongs to the43 positively invariant ball Br∗(0). In the recursive proof of Theorem 3.1, we use a new44 method of enlarging the radius to overcome the difficulty that the ball Br∗(0) is no45 longer a positively invariant set under the operator defined by the right-hand side of46 (1.3). The proof is more careful and technical than in [17] even for the case of p= 2.47 For the later purpose, we have to consider the numerical approximation of so-48 lutions as →0. In this respect, Kloeden and Lorenz [25] (see also Jentzen et al.49 [4,20,21,22]) have introduced the method of the Taylor expansion by using the50 Fr´echet derivatives of the linear Laplacian Ap(p= 2) and f. However, the nonlinear51 operator Ap(p > 2) has not a Fr´echet derivative.52 To overcome the above difficulty, we establish a new type of Taylor expansions53 without Fr´echet derivatives and give the continuous-time error of solutions for the54 LDS (1.1) (see Lemma 4.1). Using this continuous-time error, we can show the dis-55 cretization error of order two for the solutions between LDS (1.1) and IES (1.3), see56 Theorem 4.2. Our method is suitable for a wider class of discrete-time equations even57 if the operators are not Fr´echet differential.58 From Section 5, our main purpose is to study the numerical scheme of attractors,59 which is a relative new subject (introduced by Han, Kloeden, Sonner [17], see also60 [38]) in both Numerical Analysis and Dynamical Systems [31]. More precisely, we61 study the discrete approximation of the global attractor Afor LDS (1.1) in terms of62 numerical attractors for IES (1.3) and its finitely dimensional truncated system.63 We prove in Theorem 5.2 that the discrete semigroup, generated from the IES64 (1.3), possesses a unique connected numerical attractor Afor sufficiently small step65 sizes. In the proof, we need to recursively estimate the tail of solutions for all n∈N,66 where the usual cut-off function technique (see [1,2,6,7,16,19,35,39,40]) is still67 valid in the discrete-time case.68 Furthermore, we prove in Theorem 5.4 that Ahas an optimized bound given69 by kgk/α. This bound is crucial to prove that the numerical attractor converges70 continuously (upper and lower) to zero as the graph of fcloses enough to the y-axis71 and as g→0, respectively. This subject of optimization and convergence of numerical72 attractors is new in the literature.73 In Theorem 6.1, we establish the upper semi-continuity from the numerical attrac-74 tor Ato the global attractor Aas →0, where the discretization error of solutions75 in Theorem 4.2 plays a crucial role in the proof.76 In Section 7, we study the finitely dimensional approximation of the numerical77 attractor. For this end, we truncate the IES (1.3) on the (2m+1)-dimensional Euclid78 space to obtain the truncated numerical scheme with the periodic boundary condition,79 see the model (7.2). We then prove in Theorem 7.4 that the truncated IES (7.2) has80 an attractor denoted by A,m, and that A,m converges upper semi-continuously to81 the numerical attractor Aas m→ ∞. If the viscosity is zero, i.e. ν= 0, the lower82 semi-continuity from A,m to Astill holds as proved in Theorem 7.6.83 This manuscript is for review purposes only. OPTIMIZATION AND CONVERGENCE OF NUMERICAL ATTRACTORS 3 Fig. 1.Convergence paths and bounds of attractors. In a word, we have established a convergence path from A,m to the global at-84 tractor Athrough A. In fact, there is another convergence path from A,m to A85 through Am, where Amis the global attractor for the truncated system of the LDS86 (1.1) on the (2m+1)-dimensional Euclid space. All convergence paths and optimized87 bounds of attractors are displayed in FIG. 1.88 2. Positively invariant ball and global attractor for p-Laplace lattice.89 The discrete p-Laplace operator Ap(p≥2) can be formally written as90 Apu=−B∗(|Bu|p−2Bu),(Bu)i:= ui+1 −ui,(B∗u)i:= ui−1−ui,91 92 where u= (ui)i∈Z,|u|q= (|ui|q)i∈Zand uv = (uivi)i∈Z. By [14], we have (Apu, u) =93 −kBukp p, where k·kq(omitting the subscript if q= 2) denotes the norm in the Banach94 space95 `q:= {u= (ui)i∈Z:kukq q=X i∈Z|ui|q<∞}, q ≥1.96 97 We assume that g= (gi)i∈Z∈`2and f:R→Ris locally Lipschtz continuous,98 i.e. for each r > 0, there is Lr≥0 (increasingly in r) such that99 |f(s1)−f(s2)| ≤ Lr|s1−s2|,∀|s1| ≤ r, |s2| ≤ r,(2.1)100 101 and the dissipative condition (1.2) holds. Note that both (2.1) and (1.2) imply that102 f(s)s≤ −αs2,∀s∈R, f(0) = 0,(2.2)103 104 and the Nemytskii operator F:`2→`2, F(u)=(f(ui))i∈Zis bounded and locally105 Lipschtz continuous.106 Now, the p-Laplace LDS (1.1) is rewritten as an abstract form107 du(t) dt =νApu(t) + F(u(t)) + g, t > 0, u(0) = u0∈`2,(2.3)108 109 This manuscript is for review purposes only. 4Y. LI, S. YANG AND T. CARABALLO where ν > 0, p > 2. Although the dissipative condition ((1.2) or equivalently (2.2))110 is different from those in [10,14], one can similarly prove that the p-Laplace LDS111 (2.3) has a unique solution u∈C([0,∞), `2), which generates a continuous semigroup112 (semi dynamical system) defined by113 S(t) : `2→`2, S(t)u0=u(t;u0),∀t≥0, u0∈`2.114 115 Lemma 2.1. The semigroup S(·)has a positively invariant absorbing ball116 Br∗(0) := {x∈`2:kxk ≤ r∗:= p1 + kgk2/α2}.117 118 Proof. By the inner product of (2.3) with u(t), using (2.2) we obtain119 d dtkuk2=−2νkBukp p−2αX i∈Z f(ui)ui+ 2(g, u)≤ −αkuk2+kgk2 α.120 121 The Gronwall lemma yields122 ku(t)k2≤e−αtku0k2+kgk2 α2(1 −e−αt),∀t≥0.(2.4)123 124 For all u0∈ Br(0) with arbitrary radius r > 0, we have125 ku(t, u0)k2≤e−αtr2+kgk2 α2(1 −e−αt)≤1 + kgk2 α2= (r∗)2 126 127 if t≥2 αlog r. Hence Br∗(0) is an absorbing ball.128 By (2.4) again, for all u0∈ Br∗(0) and t≥0,129 ku(t, u0)k2≤e−αt1 + kgk2 α2+kgk2 α2(1 −e−αt) = e−αt +kgk2 α2≤1 + kgk2 α2= (r∗)2.130 131 Hence Br∗(0) is also positively invariant under S(·).132 We remark here that the larger radius p1+2kgk2/α2was used in [17] for p= 2.133 By the technique of a cut-off function (see e.g. [1,39]), one can give the uniform134 estimates of the tail of the solution on the ball Br∗(0), which leads to the existence of135 a global attractor. The proof is similar to the one given in [14].136 Theorem 2.2. The semigroup S(·), generated from the p-Laplace lattice, pos-137 sesses a unique global attractor A ⊂ Br∗(0).138 3. Numerical solutions and discrete semigroup on a ball. The implicit139 Euler scheme for the p-Laplace LDS (2.3) with the step size  > 0 can be read as140 u n=u n−1+νApu n+F(u n) + g, u 0=u0∈`2.(3.1)141 142 Note that there does not exist a common step size such that (3.1) is solvable for143 all initial data (see Lemma 5.3 later or see [17,31] in the case of p= 2). So, we will144 restrict (3.1) on the ball Br∗(0) to ensure the existence of a discrete-time dynamical145 system for at least one step size.146 We need to use the local Lipschitz continuity of the discrete p-Laplace operator147 kApu−Apvk ≤ Lp,rku−vkand kApuk ≤ Lp,rkuk,∀u, v ∈ Br(0),(3.2)148 149 where Lp,r := (p−1)22prp−2depends increasingly on r≥0, see [34] for the proof.150 This manuscript is for review purposes only. OPTIMIZATION AND CONVERGENCE OF NUMERICAL ATTRACTORS 5 Theorem 3.1. There is ∗>0such that, for each ∈(0, ∗]and u0∈ Br∗(0),151 the IES (3.1)has a unique solution such that152 u n(u0)∈ Br∗(0),∀n∈N,where r∗=p1 + kgk2/α2.153 154 Proof. We recursively prove the theorem in four steps.155 Step 1. In the case of n= 1, we find an ∗>0 such that the IES (3.1) has a156 solution157 u 1(u0)∈ Br∗+1(0),∀∈(0, ∗], u0∈ Br∗(0),158 159 where the radius is temporarily enlarged (from r∗to r∗+ 1).160 For each  > 0 and u0∈ Br∗(0), we define an operator M u0:`2→`2by161 M u0(x) = u0+νApx+F(x) + g, ∀x∈`2.(3.3)162 163 We prove that M u0maps Br∗+1(0) into itself if is small enough. Since Apis164 bounded, it follows from the second inequality of (3.2) that165 νkApxk ≤ νLp,r∗+1kxk ≤ (r∗+ 1)νLp,r∗+1,∀x∈ Br∗+1(0).166 167 By the local Lipschitz continuity (2.1) and f(0) = 0, we obtain168 kF(x)k ≤ Lr∗+1kxk ≤ (r∗+ 1)Lr∗+1,∀x∈ Br∗+1(0).169 170 Hence, for all u0∈ Br∗(0) and x∈ Br∗+1(0), we have171 kM u0(x)k≤ku0k+(νkApxk+kF(x)k+kgk)172 ≤r∗+(r∗+ 1)(νLp,r∗+1 +Lr∗+1) + kgk.173 174 We define an essential constant by175 ∗:= 1 (r∗+ 1)(νLp,r∗+1 +Lr∗+1) + kgk.(3.4)176 177 Then kM u0(x)k ≤ r∗+ 1 for all ∈(0, ∗], u0∈ Br∗(0) and x∈ Br∗+1(0).178 We then prove that, for each ∈(0, ∗] and u0∈ Br∗(0), the mapping M u0:179 Br∗+1(0) → Br∗+1(0) is contractive. Indeed, by the local Lipschitz continuity in (2.1)180 and (3.2), for all x, y ∈ Br∗+1(0),181 kM u0(x)−M u0(y)k ≤ (νkApx−Apyk+kF(x)−F(y)k)(3.5)182 ≤(νLp,r∗+1 +Lr∗+1)kx−yk.183 184 If ∈(0, ∗], where ∗is the constant defined by (3.4), then185 (νLp,r∗+1 +Lr∗+1)≤∗(νLp,r∗+1 +Lr∗+1)186 =νLp,r∗+1 +Lr∗+1 (r∗+ 1)(νLp,r∗+1 +Lr∗+1) + kgk≤1 r∗+ 1 <1.187 188 By the contraction mapping principle, for each ∈(0, ∗] and u0∈ Br∗(0), the189 mapping M u0:Br∗+1(0) → Br∗+1(0) has a unique fixed point190 u 1∈ Br∗+1(0) such that M u0(u 1) = u 1,191 192 This manuscript is for review purposes only. 6Y. LI, S. YANG AND T. CARABALLO which is the unique solution of the IES (3.1) for n= 1 in Br∗+1(0).193 Step 2. We further prove that the unique solution in Step 1 satisfies194 u 1(u0)∈ Br∗(0),∀∈(0, ∗], u0∈ Br∗(0).195 196 For this purpose, we take the inner product of the equation (3.1) for n= 1 by u 1 197 to obtain198 ku 1k2= (u0, u 1) + ν(Apu 1, u 1) + (F(u 1), u 1) + (g, u 1).(3.6)199 200 By (2.2),201 (F(u 1), u 1) = X i∈Z f(u 1,i)u 1,i ≤ −αku 1k2.202 203 The Young inequality implies204 (u0, u 1)≤1 2ku0k2+1 2ku 1k2and (g, u 1)≤ 2αkgk2+α 2ku 1k2.205 206 Since (Apu 1, u 1) = −kBu 1kp p≤0, it follows from (3.6) and the above estimates that207 ku 1k2≤1 1 + αku0k2+ αkgk2.(3.7)208 209 Since u0∈ Br∗(0), it follows from (3.7) that210 ku 1k2≤1 1 + α1 + kgk2 α2+ αkgk2≤1 + kgk2 α2= (r∗)2,211 212 which means that u 1∈ Br∗(0) as desired.213 Step 3. We show that the solution is unique globally. Let ∈(0, ∗] and u0∈214 Br∗(0). By Step 1, the solution u 1(u0) is unique in Br∗+1(0). By Step 2, there is215 not a solution outside Br∗(0) and thus the solution u 1(u0) is unique in `2. So far, the216 theorem for n= 1 has been proved.217 Step 4. Suppose the theorem holds for a certain n, that is, for each ∈(0, ∗]218 (where ∗is still the constant given by (3.4)) and u0∈ Br∗(0), the n-th IES (3.1) has219 a unique solution u n(u0)∈ Br∗(0). We then define a mapping by220 M u n(x) = u n+νApx+F(x) + g, ∀x∈ Br∗+1(0),221 222 where u n∈ Br∗(0) instead of u0∈ Br∗(0) in (3.3). Repeating the process in Step223 1, we know that, for each ∈(0, ∗], the mapping M u n:Br∗+1(0) → Br∗+1(0) is224 well-defined and contractive, which implies the existence of a unique fixed point u n+1 225 in Br∗+1(0).226 Repeating the estimates in Step 2, we obtain an analogue inequality of (3.7)227 ku n+1k2≤1 1 + αku nk2+ αkgk2.(3.8)228 229 By the recursive hypothesis u n∈ Br∗(0), we infer from (3.8) that u n+1 ∈ Br∗(0), which230 is the unique solution of the (n+ 1)-th IES (3.1). The recursive proof is complete.231 Remark 3.2. The above proof is more careful than the proof of [17, Lemma 2]232 even in the case of p= 2 . In fact, Br∗(0) may not be positively invariant under the233 operator M u0(although it is invariant under the solution mapping, see [17, Lemma234 1]). To overcome this difficulty, we enlarge the radius r∗to r∗+ 1 such that Br∗+1(0)235 is positive invariant under M u0with a possible maximal size ∗.236 This manuscript is for review purposes only. OPTIMIZATION AND CONVERGENCE OF NUMERICAL ATTRACTORS 7 The following result shows the generation of a discrete-time dynamical system (see237 [24]), which has better properties than the continuous system. The proof is standard.238 Corollary 3.3. For each ∈(0, ∗], where ∗is given by (3.4), the unique239 solution of the IES (3.1)in Br∗(0) generates a discrete semigroup given by240 S(n) : Br∗(0) → Br∗(0), S(n)u0=u n(u0),∀n∈N0, u0∈ Br∗(0).241 242 Lemma 3.4. For ∈(0, ∗]and n∈N, the operator S(n)is Lipschitz continuous243 in Br∗(0).244 Proof. Let n= 1 and u0, v0∈ Br∗(0). By (3.5), the solutions u 1=S(1)u0and245 v 1=S(1)v0satisfy246 ku 1−v 1k≤ku0−v0k+(νkApu 1−Apv 1k+kF(u 1)−F(v 1)k)247 ≤ ku0−v0k+(νLp,r∗+1 +Lr∗+1)ku 1−v 1k,248 249 which further implies that for all ∈(0, ∗],250 ku 1−v 1k ≤ ku0−v0k 1−(νLp,r∗+1 +Lr∗+1)≤ku0−v0k 1−∗(νLp,r∗+1 +Lr∗+1),251 252 where ∗(νLp,r∗+1 +Lr∗+1)<1 in view of (3.4). By the semigroup property,253 kS(n)u0−S(n)v0k ≤ ku0−v0k (1 −∗(νLp,r∗+1 +Lr∗+1))n 254 255 for all n∈N. The proof is complete.256 4. Generalized Taylor expansion and discretization error. To study the257 convergence of attractors, we need to estimate the discretization error of solutions,258 for which we need to develop a generalized Taylor expansion.259 4.1. Generalized Taylor expansion for continuous-time error. According260 to the method in [17,25], one must consider the Taylor expansion of LDS (2.3) starting261 from u(tn+1;u0) and going back to u(tn;u0) as follows:262 u(tn) = u(tn+1)+(−)Hp(u(tn+1)) + 1 2(−)2DHp(u(θ))263 264 where tn+1 −tn=,θ∈(tn, tn+1), the operator Hp:`2→`2is given by265 Hp(x) := νApx+F(x) + g, ∀x∈`2 (4.1)266 267 and DHpdenotes the Fr´echet derivative (perhaps formal) of Hp. If p= 2, then268 A:= Apis a bounded linear operator, which has a Fr´echet derivative given by itself,269 and thus, by the method as in [21], one can clearly write the Fr´echet derivative as270 DH(x) = (νA + diag(f0(xi))H(x) for x∈`2. However, if p > 2, then the nonlinear271 operator Aphas not a Fr´echet derivative (even the original function y=|s|p−2sis not272 differential in R).273 To overcome the difficulty, we give an alternative for the second order Taylor274 expansion of LDS (2.3) without Fr´echet derivatives, which will be useful for estimating275 the discretization error in the next subsection.276 Lemma 4.1. Let u(·;u0)be the solution of LDS (2.3),tn+1 −tn= > 0,tn≥0.277 Then, for each u0∈ Br(0) with any radius r > 0, there is M(u0)∈`2such that278 u(tn;u0) = u(tn+1;u0)−Hp(u(tn+1;u0)) + M(u0),(4.2)279 kM(u0)k ≤ Cr,∀u0∈ Br(0),(4.3)280 281 This manuscript is for review purposes only. 8Y. LI, S. YANG AND T. CARABALLO where Cris increasing in r(but independent of ) and the operator Hp:`2→`2is282 well-defined by (4.1).283 Proof. The first order Taylor expansion of LDS (2.3) can be read as284 u(tn) = u(tn+1)−du dt (θ) = u(tn+1)−Hp(u(θ))285 =u(tn+1)−Hp(u(tn+1)) + (Hp(u(tn+1)) −Hp(u(θ))),286 287 where θ∈(tn, tn+1). Hence (4.2) follows if we put288 M(u0) := Hp(u(tn+1;u0)) −Hp(u(θ;u0)).289 290 To prove (4.3), we assume without loss of generality that r > kgk/α (otherwise,291 one can use r+kgk/α instead of r), and claim that Br(0) is a positively invariant set292 for the semigroup S(·). Indeed, by (2.4), for all t≥0 and u0∈ Br(0),293 ku(t;u0)k2≤e−αtku0k2+kgk2 α2(1 −e−αt)≤e−αtr2−kgk2 α2+kgk2 α2≤r2.(4.4)294 295 By the local Lipschitz continuity of Apand F, we obtain296 kM(u0)k=kν(Apu(tn+1)−Apu(θ)) + (F(u(tn+1)) −F(u(θ)))k297 ≤(νLp,r +Lr)ku(tn+1)−u(θ)k.298 299 By the first order Taylor expansion again, we have300 u(tn+1)−u(θ)=(tn+1 −θ)du dt (ˆ θ)=(tn+1 −θ)Hp(u(ˆ θ))301 = (tn+1 −θ)(νApu(ˆ θ) + F(u(ˆ θ)) + g)302 303 for some ˆ θ∈(θ, tn+1). By the local Lipschitz continuity of Apand Fagain, it follows304 from (4.4) that305 ku(tn+1)−u(θ)k≤|tn+1 −θ|(νLp,r +Lr)ku(ˆ θ)k+kgk≤(r(νLp,r +Lr) + kgk),306 307 which further implies that for all u0∈ Br(0),308 kM(u0)k ≤ r(νLp,r +Lr)2+kgk(νLp,r +Lr)=: Cr,309 310 where Cris obviously increasing in r. The proof is complete.311 4.2. Discretization error of order two. We now use the generalized Taylor312 expansion in Lemma 4.1 to estimate the discretisation error of solutions when the313 initial data are restricted on the ball Br∗(0).314 Theorem 4.2. Let u(t;u0)and u n(u0)be the solutions of LDS (2.3)and IES315 (3.1)respectively, where u0∈ Br∗(0). We have the discretisation error of order 2:316 ku(;u n(u0)) −u n+1(u0)k ≤ 2Cr∗,∀∈(0, ∗], n ∈N0.(4.5)317 318 Furthermore, for each T > 0, there is a CT,r∗>0such that319 ku(tn;u0)−u n(u0)k ≤ CT,r∗,∀tn:= n ∈[0, T],  ∈(0, ∗].(4.6)320 321 This manuscript is for review purposes only. OPTIMIZATION AND CONVERGENCE OF NUMERICAL ATTRACTORS 9 Proof. Both (4.2) and (3.1) can be rewritten as322 u(tn+1) = u(tn) + Hp(u(tn+1)) −M(u0) and u n+1 =u n+H(u n+1),323 324 where Hp=νAp+F+gI as given in (4.1). From the difference between the above325 two equalities, we know that the discretisation error ∆ n(u0) := u(tn;u0)−u n(u0)326 satisfies the following equation:327 ∆ n+1 = ∆ n+(Hp(u(tn+1)) −Hp(u n+1)) −M(u0).328 329 Taking the inner product with ∆ n+1 yields330 k∆ n+1k2= (∆ n+(Hp(u(tn+1)) −Hp(u n+1)) −M(u0),∆ n+1)331 ≤(k∆ nk+kHp(u(tn+1)) −Hp(u n+1)k+kM(u0)k)k∆ n+1k,332 333 which further implies334 k∆ n+1k≤k∆ nk+kHp(u(tn+1)) −Hp(u n+1)k+kM(u0)k.(4.7)335 336 Since Br∗(0) is positively invariant under both S(t) and S(n) (see Lemma 2.1 and337 Theorem 3.1), it follows that u(tn+1), u n∈ Br∗(0), and thus we see from the local338 Lipschitz continuity of Apand Fthat339 kHp(u(tn+1)) −Hp(u n+1)k ≤ νkApu(tn+1)−Apu n+1k+kF(u(tn+1)) −F(u n+1)k340 ≤(νLp,r∗+Lr∗)ku(tn+1)−u n+1k= (νLp,r∗+Lr∗)k∆ n+1k.341 342 By Lemma 4.1,kM(u0)k ≤ Cr∗for all u0∈Br∗(0), and thus (4.7) yields343 k∆ n+1k≤k∆ nk+(νLp,r∗+Lr∗)k∆ n+1k+2Cr∗.344 345 Denote by ˆ Lr∗:= νLp,r∗+Lr∗. By (3.4), for all ∈(0, ∗], ˆ Lr∗≤∗ˆ Lr∗<1 and346 thus347 k∆ n+1k ≤ 1 1−ˆ Lr∗k∆ nk+2Cr∗,∀n∈N0,  ∈(0, ∗],(4.8)348 349 where Cr∗is 1/(1 −∗ˆ Lr∗)-times bigger than the original constant. Since ∆ 0= 0,350 we infer from (4.8) that ku(;u0)−u 1(u0)k=k∆ 1k ≤ 2Cr∗. Using u nas an initial351 datum in the above formula, we obtain the discretization error (4.5) of order 2.352 On the other hand, for all tn=n ∈[0, T], by the recursive inequality (4.8) and353 ∆ 0= 0, we have354 k∆ nk ≤ 2Cr∗ n−1 X j=0 1 (1 −ˆ Lr∗)j.(4.9)355 356 Since ˆ Lr∗<1 and n≤T/, it follows that357 ˆ Lr∗ n−1 X j=0 1 (1 −ˆ Lr∗)j=1−(1 −ˆ Lr∗)n (1 −ˆ Lr∗)n−1≤(1 −ˆ Lr∗)−(n−1) ≤(1 −ˆ Lr∗)−T ↑eTˆ Lr∗ 358 359 as ↓0, where the last limit is deduced from the basic limit (1 + 1/k)k↑eas k→ ∞.360 By (4.9),361 k∆ nk ≤ Cr∗ ˆ Lr∗ ˆ Lr∗ n−1 X j=0 1 (1 −ˆ Lr∗)j≤Cr∗eTˆ Lr∗ ˆ Lr∗ =: CT,r∗,362 363 for all ∈(0, ∗], tn=n ∈[0, T] and u0∈Br∗(0). Hence (4.6) holds true.364 This manuscript is for review purposes only. 16 Y. LI, S. YANG AND T. CARABALLO Since {xk}is bounded in `2, its truncation on `2(|i| ≤ I) = R2I+1 is also bounded.591 Hence the truncated sequence of {xk}has a finite δ-net in R2I+1, which together592 with (6.3) implies that the sequence {xk}has a finite 2δ-net and thus it is relatively593 compact in `2.594 Case 2: inf k= 0. Passing to a subsequence, we assume k→0. By the upper595 semi-continuity as in Theorem 6.1, we have596 d`2(xk,A)≤d`2(Ak,A)→0 as k→ ∞.597 598 Since Ais compact, we see from [9, lemma 2.3] that the sequence {xk}has a convergent599 subsequence.600 Remark 6.3. The uniform compactness of attractors is usually applied to prove601 the upper semi-continuity, see [28,37]. For the numerical attractors, the situation is602 reversed.603 7. Finitely dimensional approximation of numerical attractors . How to604 truncate the IES (3.1) on a finite-dimensional space? For each m∈N, the operator605 F:`2→`2has a natural truncation given by606 Fm:R2m+1 →R2m+1, Fm(x) = (f(xi))|i|≤m,∀x= (xi)|i|≤m∈R2m+1.607 608 However, it is not easy to truncate the discrete p-Laplace operators Apbecause609 Apx(x= (xi)|i|≤m) involves two unknown components xm+1 and x−m−1outside610 R2m+1. To overcome it, we use the periodic boundary conditions (see [2,17])611 xm+1 =x−mand x−m−1=xm.612 613 So, the truncation Ap,m :R2m+1 →R2m+1 of Apcan be defined by614 (Ap,mx)−m=|x−m+1 −x−m|p−2(x−m+1 −x−m)−|x−m−xm|p−2(x−m−xm),615 (Ap,mx)i= (Apx)i,∀|i|< m,616 (Ap,mx)m=|x−m−xm|p−2(x−m−xm)−|xm−xm−1|p−2(xm−xm−1)617 618 for all x= (xi)|i|≤m∈R2m+1. For p > 2, the truncated operator Ap,m is nonlinear619 and thus it is not a matrix. But Ap,m is a function of matrixes620 Ap,mx=−BT m(|Bmx|p−2Bmx),∀x∈R2m+1,621 622 where BT mis the transport matrix of Bmand623 Bm=          −1 1 0 ··· 0 0−1....... . . . . ..........0 0...−1 1 1 0 ··· 0−1          ∈(R2m+1)2.624 As in (3.2), Ap,m :R2m+1 →R2m+1 is locally Lipschitz continuous:625 kAp,mx−Ap,myk ≤ Lp,rkx−ykand kAp,mxk ≤ Lp,rkxk,∀x, y ∈ Bm r(0),(7.1)626 627 where Bm r(0) is the ball in R2m+1 and Lp,r := (p−1)22prp−2.628 Then the IES (3.1) can be truncated as follows:629 u,m n=u,m n−1+νAp,mu,m n+Fm(u,m n) + g|m, u,m 0=um 0∈R2m+1,(7.2)630 631 where g|m:= (gi)|i|≤m∈R2m+1 is the truncation of g∈`2, and the unknown is632 denoted by u,m n= (u,m n,i )|i|≤m∈R2m+1.633 This manuscript is for review purposes only. OPTIMIZATION AND CONVERGENCE OF NUMERICAL ATTRACTORS 17 7.1. Existence and bound of truncated numerical attractors. As in the634 infinite dimension case, we show that the truncated IES (7.2) with small step size is635 solvable when the initial datum belongs to some suitable balls.636 Lemma 7.1. For each r0>kg|mk/α and m∈N, there is r0>0, such that,637 for all ∈(0, r0]and um 0∈ Bm r0(0), the truncated IES (7.2)has a unique solution638 {u,m n}n∈Nwhich satisfies639 ku,m nk2≤1 1 + αku,m n−1k2+ αkg|mk2.(7.3)640 641 In particular, for all n∈N,u,m n∈ Bm r0(0).642 Proof. We recursively prove it as done in Theorem 3.1. Consider the case n= 1.643 For  > 0 and um 0∈ Bm r0(0), we denote by644 M um 0(x) = um 0+νAp,mx+Fm(x) + g|m,∀x∈ Bm r0+1(0).645 646 By the local Lipschitz continuity of Ap,m and Fm, we have647 kM um 0(x)k≤kum 0k+(νkAp,mxk+kFm(x)k+kg|mk)648 ≤r0+(r0+ 1)(νLp,r0+1 +Lr0+1) + kg|mk.649 650 Using kgkinstead of kg|mkwe put651 r0:= 1 (r0+ 1)(νLp,r0+1 +Lr0+1) + kgk,(7.4)652 653 which is independent of m. Since kg|mk ≤ kgk, it follows that for all ∈(0, r0],654 kM um 0(x)k ≤ r0+(r0+ 1)(νLp,r0+1 +Lr0+1) + kg|mk (r0+ 1)(νLp,r0+1 +Lr0+1) + kgk≤r0+ 1,655 656 which means that M um 0:Bm r0+1(0) → Bm r0+1(0) is well-defined. By the local Lipschitz657 continuity of Ap,m and Fmagain, for all ∈(0, r0] and x, y ∈ Bm r0+1(0),658 kM um 0(x)−M um 0(y)k ≤ (νkAp,mx−Ap,myk+kFm(x)−Fm(y)k)659 ≤(νLp,r0+1 +Lr0+1)kx−yk ≤ 1 r0+ 1kx−yk.660 661 Then the contraction mapping principle implies that the first equation of (7.2) has a662 unique solution u,m 1∈ Bm r0+1(0) for all ∈(0, r0].663 Now, we take the R2m+1-inner product of the truncted IES (7.2) with u,m 1, the664 result is665 ku,m 1k2=hum 0, u,m 1i+νhAp,mu,m 1, u,m 1i+hFm(u,m 1), u,m 1i+hg|m, u,m 1i.(7.5)666 667 Since Ap,m is the function of the matrix Bm, it follows that for all x∈R2m+1,668 hAp,mx, xi=−h|Bmx|p−2Bmx, Bmxi=−X |i|≤m|(Bmx)i|p≤0.669 670 Hence, by estimating other three terms in (7.5) and using the method in Theorem671 3.1, we obtain672 ku,m 1k2≤1 1 + αkum 0k2+ αkg|mk2.673 674 This manuscript is for review purposes only. 18 Y. LI, S. YANG AND T. CARABALLO Since r0>kg|mk/α and um 0∈ Bm r0(0), it follows that675 ku,m 1k2≤1 1 + αr2 0+ αkg|mk2 676 =1 1 + αr2 0−kg|mk2 α2+1 1 + αkg|mk2 α2+ αkg|mk2 677 =1 1 + αr2 0−kg|mk2 α2+kg|mk2 α2≤r2 0,678 679 which means u,m 1∈ Bm r0(0) for all ∈(0, r0]. Repeating the above process with680 u,m n−1∈ Bm r0(0) instead of um 0∈ Bm r0(0), the recursive proof is available.681 Note that the radius r∗and step size ∗in the previous sections satisfy682 r∗=r1 + kgk2 α2>kg|mk α,and ∗=r∗,683 684 where r∗is defined as in (7.4). By Lemma 7.1, for each ∈(0, ∗] and m∈N, we685 can define a discrete semigroup by686 S,m(n) : Bm r∗(0) → Bm r∗(0), S,m(n)um 0=u,m n(um 0),∀n∈N0.687 688 Theorem 7.2. For each ∈(0, ∗]and m∈N, the discrete semigroup S,m(·)689 has a (numerical) attractor A,m such that690 A,m ⊂ Bm kg|mk/α(0) and A,m is connected.(7.6)691 692 Proof. The existence of a unique attractor A,m follows from the compactness of693 the state space Bm r∗(0) immediately. The connection of A,m follows from the same694 method as in Theorem 5.2.695 To prove the bound of the attractor, we put r0∈(kg|mk/α, r∗) and prove that696 Bm r0(0) is an absorbing set for the semigroup S,m(·). It suffices to prove that Bm r0(0)697 absorbs the whole state space Bm r∗(0). Iterating (7.3) in Lemma 7.1, we have for all698 um 0∈ Bm r∗(0),699 ku,m nk2≤1 1 + αku,m n−1k2+ αkg|mk2 700 ≤kum 0k2 (1 + α)n+ αkg|mk2 n X j=1 1 (1 + α)j≤(r∗)2 (1 + α)n+kg|mk2 α2.701 702 Since r0>kg|mk/α, it follows that there is N∈Nsuch that for all n≥N,703 ku,m nk2≤r2 0−kg|mk2 α2+kg|mk2 α2=r2 0.704 705 Since r0< r∗, we see from (7.4) that r0> ∗. By Lemma 7.1,Bm r0(0) is positively706 invariant under S,m(·) for all 0 <  ≤∗(and thus ≤r0). Therefore,707 A,m =∩k∈N∪n≥kS,m(n)Bm r0(0) ⊂ Bm r0(0)708 709 for all ∈(0, ∗] and r0∈(kg|mk/α, r∗). Taking the limit as r0→ kg|mk/α, we obtain710 the inclusion in (7.6).711 This manuscript is for review purposes only. OPTIMIZATION AND CONVERGENCE OF NUMERICAL ATTRACTORS 19 7.2. Convergence from truncated attractor to numerical attractor. The712 following result states that the tail of any element in A,m is uniformly small in713 ∈(0, ∗] as m→ ∞.714 Lemma 7.3. For each δ > 0, there exists I(δ)∈Nsuch that for all m∈Nand715 ∈(0, ∗],716 X I(δ)≤|i|≤m|xi|2< δ, ∀(xi)|i|≤m∈ A,m,(7.7)717 718 where the sum is zero if m<I(δ).719 Proof. Taking the inner product of the truncated IES (7.2) with ξm ku,m nin R2m+1,720 where ξm k= (ξk,i)|i|≤m, we obtain721 X |i|≤m ξk,i|u n,i|2= (u,m n−1, ξm ku,m n)722 +ν(Ap,mu,m n, ξm ku,m n) + (Fm(u,m n) + g|m, ξm ku,m n).723 724 Since Ap,m and Aphave the same local Lipschitz constants, we can similarly obtain725 ν(Ap,mu,m n, ξm ku,m n)≤Cp k(r∗)p.(7.8)726 727 where the constant Cpis independent of m. Hence, by other same arguments as in the728 proof of Lemma 5.1, it follows that, for each δ > 0 and ∈(0, ∗], there are N(δ)∈N729 and I(δ)∈Nsuch that730 X I(δ)≤|i|≤m|u,m n,i |2< δ, ∀n≥N(δ).731 732 Given now x∈ A,m with arbitrary ∈(0, ∗] and m∈N. The invariance implies733 that734 x=S,m(N(δ))um 0=u,m N(δ)(um 0),for some um 0∈ Bm r∗(0),735 736 and thus737 X I(δ)≤|i|≤m|xi|2=X I(δ)≤|i|≤m|u,m N(δ),i|2< δ,738 739 which implies (7.7) as desired.740 Any x∈R2m+1 has a null-extension ˜x∈`2defined by741 ˜xi= 0,∀|i|> m, ˜xi=xi,∀|i| ≤ m.742 743 Then a set Din R2m+1 still has a null-extension set in `2denoted by e D. In this744 viewpoint, both attractors A,m and Acan be contained into the same ball Br∗(0)745 of `2.746 Theorem 7.4. For each ∈(0, ∗], the numerical attractor A,m of the truncated747 IES (7.2)upper semi-converges to the attractor Aof the IES (3.1), i.e.748 d`2(A,m,A) := d`2(e A,m,A)→0,as m→ ∞.(7.9)749 750 This manuscript is for review purposes only. 20 Y. LI, S. YANG AND T. CARABALLO Proof. Suppose (7.9) is false for a fixed ∈(0, ∗]. Then there are η0>0, a751 subsequence {mj}of {m}and xmj∈ A,mjsuch that the null-extension ˜xmjof xmj 752 satisfies753 d(˜xmj,A)≥η0,∀j∈N.(7.10)754 755 The solution of (7.2) with the initial data xmjis given by756 u,mj n=u,mj n(xmj) = S,mj(n)xmj,∀n∈N0, j ∈N.757 758 Since xmj∈ A,mj, it follows that the solution u,mj ncan be expanded as an entire759 solution, i.e. defined for all n∈Z.760 Since u,mj n∈ A,mjfor all n∈Z, it follows from Lemma 7.3 that, for each δ > 0,761 there is I(δ)∈Nsuch that762 X |i|≥I(δ)|˜u,mj n,i |2=X I(δ)≤|i|≤mj |u,mj n,i |2< δ2,∀n∈Z, j ∈N,(7.11)763 764 where ˜uis the null-extension of u. By the previous discussion, all attractors are765 contained in those balls of radius r∗and thus the double sequence766 {(˜u,mj n,i )|i|<I(δ):n∈Z, j ∈N}767 768 is bounded in R2I(δ)−1and thus it has a finite δ-net, which together with (7.11) implies769 that the double sequence770 {˜u,mj n:n∈Z, j ∈N}771 772 has a finite 2δ-net and thus it is relatively compact in `2. By a diagonal argument,773 there are {u∗ n:n∈Z} ⊂ `2and an index subsequence (denoted by itself) of {j}such774 that775 k˜u,mj n−u∗ nk → 0 as j→ ∞,∀n∈Z.(7.12)776 777 We then prove that {u∗ n:n∈Z}is an entire solution of the IES (3.1). As an778 entire solution, {u,mj n:n∈Z}satisfies the truncated IES (7.2) for all n∈Z:779 u,mj n=u,mj n−1+νAp,mju,mj n+Fm(u,mj n) + g|mj.780 781 We now fix i∈Z, then there is ji∈Nsuch that for all j≥jiwe have mj≥ |i|+1,782 and thus783 u,mj n,i = ˜u,mj n,i ,(Ap,mju,mj n)i= (Ap˜u,mj n)i,(g|mj)i=gi 784 785 for all j≥jiand n∈Z. Hence, the ith-component of the entire solution u,mj n 786 satisfies787 ˜u,mj n,i = ˜u,mj n−1,i +ν(Ap˜u,mj n)i+f(˜u,mj n,i ) + gi (7.13)788 789 for all n∈Zand j≥ji. By the local Lipschitz continuity of Apand F, we have790 |(Ap˜u,mj n)i−(Apu∗ n)i| ≤ kAp˜u,mj n−Apu∗ nk ≤ Lp,r∗k˜u,mj n−u∗ nk,791 |f(˜u,mj n,i )−f(u∗ n,i)|≤kF(˜u,m n)−F(u∗ n)k ≤ Lr∗k˜u,mj n−u∗ nk.792 793 This manuscript is for review purposes only. OPTIMIZATION AND CONVERGENCE OF NUMERICAL ATTRACTORS 21 Letting j→ ∞ (thus mj→ ∞) in (7.13) and using (7.12), we obtain794 u∗ n,i =u∗ n−1,i +ν(Apu∗ n)i+f(u∗ n,i) + gi,∀n∈Z.795 796 Since i∈Zis arbitrary, it follows that {u∗ n:n∈Z}is a (bounded) entire solution of797 the truncated IES (7.2). Hence, u∗ 0∈ A, which together with798 ˜xmj= ˜u,mj 0→u∗ 0as j→ ∞799 800 gives a contradiction to (7.10).801 7.3. Lower semi-continuity of numerical attractors for viscosity zero.802 We denote the restrictions of an element x∈`2and a subset D⊂`2on R2m+1 by803 x|m= (xi)|i|≤mand D|m={y∈R2m+1 :∃x∈D, s.t. y=x|m}.804 805 806 Proposition 7.5. For each ∈(0, ∗], the numerical attractor Aof the IES807 (3.1)satisfies the following lower semi-continuity:808 lim m→∞ d`2(A,A|m) = 0.(7.14)809 810 Proof. By Lemma 5.1, for each δ > 0, there are I(δ)∈Nand N(δ)∈Nsuch811 that the solution of the IES (3.1) satisfies812 kS(n)u0k`2(|i|≥I(δ)) < δ, ∀n≥N(δ), u0∈ Br∗(0).813 814 Given any x∈ A. By the invariance, we have x=S(N(δ))yfor some y∈ A.815 Hence, for all m≥I(δ),816 kx−g x|mk2=kxk2 `2(|i|≥m)=kS(N(δ))yk2 `2(|i|≥m)< δ2.817 818 Since g x|m∈] A|m, it follows that for all m≥I(δ) and x∈ A,819 d`2(x, ] A|m)≤ kx−g x|mk< δ,820 821 which further implies822 d`2(A,] A|m) = sup x∈A d`2(x, ] A|m)< δ,823 824 for all m≥I(δ). Hence the lower semi-continuity (7.14) holds as desired.825 However, A,m 6=A|mgenerally, where A,m is the truncated numerical attractor826 for the truncated IES (7.2). We only prove the lower semi-continuity in a special case827 of viscosity zero.828 Theorem 7.6. Suppose ν= 0 in both IES (3.1)and (7.2). Then, for each ∈829 (0, ∗], we have the following lower semi-convergence:830 lim m→∞ d`2(A,A,m)=0.(7.15)831 832 Proof. Given x∈ A. We know that the solution un:= u n(x) = S(n)xcan be833 expanded into an entire solution defined for all n∈Z. Hence, the entire solution834 {un:n∈Z}satisfies835 un=un−1+F(un) + g, ∀n∈Z, u0=x.(7.16)836 837 This manuscript is for review purposes only. 22 Y. LI, S. YANG AND T. CARABALLO The component form of (7.16) can be read as838 un,i =un−1,i +f(un,i) + gi,∀n∈Z, i ∈Z.(7.17)839 840 Considering the truncation of (7.17) for those components |i| ≤ m, it follows that841 un|msatisfies842 un|m=un−1|m+Fm(un|m) + gm,∀n∈Z,843 u0|m=x|m∈R2m+1,844 845 which means {un|m:n∈Z}is an entire solution of the truncated IES (7.2) with846 ν= 0. Due to the positively invariance, we know un|m∈ Bm r∗(0) and thus the entire847 solution is bounded in R2m+1, which implies848 x|m=u0|m∈ A,m.849 850 Denote by g x|mthe null-expansion of x|m, by x∈`2, we have851 lim m→∞ kg x|m−xk= lim m→∞ X |i|>m |xi|2= 0.(7.18)852 853 Suppose now the lower semi-convergence (7.15) is false. Then there is a subse-854 quence {mj}of {m}and δ0>0 such that855 d`2(A,^ A,mj)> δ, ∀j∈N,856 857 where the tilde denotes the null-expansion of the set. Furthermore, for each j∈N,858 there is yj∈ Asuch that859 d`2(yj,^ A,mj)> δ, ∀j∈N.(7.19)860 861 Since Ais compact in `2, there is an index subsequence {jk}of {j}such that yjk→x862 for some x∈ A.863 By the previous proof, we know x|m∈ A,m such that (7.18) holds. In particular,864 lim k→∞ k^ x|mjk−xk= 0,and ^ x|mjk∈^ A,mjk.865 866 Hence,867 d`2(yjk,^ A,mjk)≤ kyjk−xk+kx−^ x|mjkk868 +d`2(^ x|mjk,^ A,mjk)→0,as k→ ∞,869 870 which contradicts (7.19).871 7.4. Final Conclusions. As displaying in FIG. 1, we have established a path872 of upper semi-convergence from the truncated numerical attractor A,m to the global873 attractor Athrough the numerical attractor A, see Theorems 7.4 and 6.1.874 On the other hand, we can establish another path of upper semi-convergence from875 A,m to Athrough Am, where Amis the attractor of the following truncated LDS of876 LDS (2.3):877 du(t) dt =νAp,mu(t) + Fm(u(t)) + g|m, u(0) ∈R2m+1.878 879 This manuscript is for review purposes only. 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