0.1. Introduction 1

Numerical Center Manifold MethodsK. Bohmer

boehmer@Mathematik.Uni-Marburg.deUniversitat Marburg

Fachbereich Mathematik und InformatikHans Meerwein Strasse, Lahnberge

D-35032 MarburgGermany

dedicated to the 60th birthday of my good friendProf. Dr. Bernold Fiedler

AMS subject classification: 65M,20M05,35K,35K58,37G99,37M99,37N30,65M99Keywords: Parabolic problems, local existence, uniqueness, generators, dynamicalsystems, semigroups, elliptic,sectorial,complementing conditions, general numerical

methods, center manifolds

Abstract This paper summarizes the first available proof and results for generalfull, so space and time discretizations for center manifolds of nonlinear parabolic prob-lems. They have to admit a local time dependent solution (a germ) near the bifurcationpoint. For the linearization (A,B) of the nonlinear elliptic part and the boundary con-dition we require: A has a trivial unstable manifold, to be sectorial and for (A,B) thecomplementing condition is valid. The two last condition hold for A : Wm,p

0 (Ω,Rq)→W−m,p, 1 ≤ m, q, satisfying the Legendre-Hadamard condition with Amann [3] crite-ria for both conditions, and A in appropriate divergence form for m > 1. By the ac-tive research, the class of problems satisfying these conditions is strongly growing. Thegeneralized Agmon e.al. systems require other techniques to prove these conditions.Then essentially all the up-to-date space discretizations, except meshfree methods, withgeometric time discretization yield converging numerical center manifolds. Here I sum-marize results of my upcoming monograph and strongly generalize my earlier papers.

0.1 IntroductionThe aim of this paper is a summary of the proof for convergence for general full, so spaceand time discretizations of center manifolds. It applies to the up to date numerical meth-ods, finite elements without and with crimes and adaptivity, discontinuous Galerkin,difference, spectral and wavelet methods and a large class of nonlinear parabolic prob-lems. With more than 500 papers on center manifolds for partial differential equations,we only give a short survey on the history and analytical aspects and then turn to theirnumerical realization.

Essential for the analysis of local bifurcations and the originating dynamical sce-narios of parabolic problems is the reduction to low-dimensional ordinary differentialequations for the center manifolds. They were introduced in the sixties by Pliss [45] andKelley [33]. Due to Lanford’s [37] contributions, this theory has been applied extensively

2

to the study of bifurcation problems and dynamical systems of ordinary differentialequations, in particular, in connection with the normal form theory. The extension toordinary differential equations in infinite dimensional spaces, so to parabolic equationsstarted with Carr [17] and Henry [31]. There the elliptic part is the sum of a linear anda nonlinear operator A and R. [17] allows an A generating different semigroups, e.g.strongly continuous, R has a uniformly continuous derivative and R(0) = 0, R′(0) = 0.Under the usual conditions for the spectrum of A he proves the existence of a centermanifold. [31] discusses analytical semigroups with the sectorial generator A and itsfractional powers. His approach to center manifolds is motivated by many interestingapplications, e.g., reaction diffusion and Navier-Stokes systems.

In the meantime there are different approaches: Generalizing [17], Bates and Jones[5] prove invariant manifold theorems for a similar problem with a continuous semigroupand Lipschitz continuous R with applications to the nonlinear Klein-Gordon and to FitzHugh-Nagumo equations. Vanderbauwhede [50] and with Iooss [51] generalize the centermanifold theory in finite-dimensional systems in [50] to infinite-dimensional systems.Instead of the usual sectorial operators they consider some elementary spectral theoryof closed linear operators and avoid the use of semigroups and semiflows. They apply itto the classical Navier-Stokes equations and these equations in a cylinder. In the firstproblem the part to the right of the imaginary axis of the spectrum of the associatedlinear operator is bounded and the Cauchy problem is well posed for t > 0, however forthe second the spectrum of the linear operator is unbounded as well to the left as to theright of the imaginary axis and the Cauchy problem has no meaning.

Combinations of center manifolds with stable and unstable invariant manifolds arestudied for ordinary differential equations in Guckenheimer/Holmes [26,27], Iooss/Adel-mayer [32] and Kuznetsov [36]. Chow and Hale [20], Hale and Kocak [29] study partialdifferential equations. Finally Haragus and Iooss [30] present a very up-to-date bookon local bifurcations, the originating center manifolds, and normal forms in infinite-dimensional dynamical systems.

Many of these problems cannot be solved exactly. So appropriate discretizationmethods are mandatory. The implications of discretization for ordinary differentialequations is well understood, cf. e.g., Beyn, Lorenz and Zou, [6–11,21,54,55], Ma, [42],Sieber and Krauskopf, [48], Lynch, [41], Choe and Guckenheimer, [18]. [11] have shownthe existence of an invariant manifold of the discretization close to the center manifoldof the differential equation without, however, studying smoothness properties of thismanifold, which are needed for the analysis of bifurcations.

Beyond many other related results of Fiedler in this direction, a particularly in-teresting example is due to him and Scheurle [24]. They discretize homoclinic orbits byone-step discretizations of order p and stepsize ε. This can be viewed as time-ε maps forthe autonomous ordinary differential equations x(t) = f(λ, x(t))+εpg(ε, λ, t/ε, x(t)), x ∈N, λ ∈ Λ, with analytic f, g and ε-periodic g in t. This is a rapidly forced nonautonomoussystem. The authors study the behavior of a homoclinic orbit Γ for ε = 0, λ = 0, underdiscretization. Under generic assumptions their Γ becomes transverse for positive ε. The

0.1. Introduction 3

transversality effects are estimated from above to be exponentially small in ε. For exam-ple, the length l(ε) of the parameter interval of λ for which Γ persists can be estimatedby l(ε) ≤ Cexp(2πη/ε), where C, η are positive constants. The coefficient η is related tothe minimal distance from the real axis of the poles of Γ(t) in the complex time domain.Likewise, the region where complicated, chaotic dynamics prevail is estimated to be ex-ponentially small, provided the saddle quantity of the associated equilibrium is nonzero.The results are visualized by high precision numerical experiment, showing that, dueto exponential smallness, homoclinic transversality becomes practically invisible undernormal circumstances, already for only moderately small discretization steps.

For parabolic problems the first systematic study of time discretizations of centermanifolds is due to Lubich/Ostermann [38]. They study for a sectorial A the case of atrivial unstable manifold and an “approximate” center manifold with eigen values µ with|Re µ| < δ, instead of the usual |Re µ| = 0. They give a new and simpler proof thanHenry [8], directly generalized to numerical time, but not to space discretizations. Thistreatment is conceptually more closely related to Vanderbauwhede and Iooss [50,51].

But there are still many problems open. We want to present some of the necessaryanswers in this paper, summarizing relevant results of my [16] and strongly generalizingmy earlier papers [13,14]. This paper is organized as follows.

In Section 0.2 we discuss linear and nonlinear elliptic equations or systems of order2 or 2m. Essential for center manifolds is the linearization (A,B) of the nonlinear ellipticpart and the boundary condition there. A has to be sectorial, to have a trivial unstablemanifold and for (A,B) the complementing condition has to be satisfied. This holds forA : Wm,p(Ω,Rq) → W−m,p, 1 ≤ m, q, satisfying the Legendre-Hadamard, the Dirichletor other boundary conditions with new criteria for the complementing condition andsome generalized Agmon e.al. [1] systems. Since research in these areas is very active, theclass of problems satisfying these conditions is strongly growing. We extend the ellipticto linear and nonlinear parabolic operators in Section 0.3. Here different solutions,semigroups and generators are discussed. Now we are ready for their local dynamics viacenter manifolds introduced in Section 0.4. The parabolic systems have to admit a localtime dependent solution (a germ) near the bifurcation point and satisfy the above (A,B)conditions. The splitting of their local dynamics due to Lubich/Ostermann [38] plays acentral role. We determine the asymptotic expansion via the homologic equation and therecursive definition of the systems for solving this equation. Next we define the essentialpart, the space discretization in Section 0.5. It yields for the asymptotic expansion ofthe parameterization of the center manifold convergence for the necessary discrete to theexact terms for essentially all up-to-date space discretizations, except meshfree methods,with geometric time discretization. Then a careful monitoring of the the discrete normalforms until determinacy, so for sufficiently many terms, allows the classification of thelocal dynamical scenarios and the final time discretization and its convergence in Section0.6. For the latter the geometric integration methods in Hairer/Lubich/Wanner [28] areappropriate and applied to the small dimensional resulting system (36). Equivariancemaintaining methods are mentioned.

4

This approach for full discretizations is formulated for the first time for generaloperators in Bohmer [13, 14]. It applies to all the nonlinear parabolic problems andtheir discretizations presented in [15, 16] under appropriate conditions for the nonlin-earity. Consequently, the many results for the dynamics of, e.g., saddle node, trans-critical, pitchfork, cusp, Hopf bifurcations, documented, e.g., in Govaerts [25], Guck-enheimer/Holmes [27], Kuznetsov [35], Mei [43] or the many original papers on thesesubjects, apply to parabolic equations as well.

0.2 Linear and Nonlinear Elliptic Operators

0.2.1 Definitions of linear elliptic operators

We define linear and nonlinear elliptic operators essentially for its weak form and givesome results, c.f. [15]. The u(x) are ∈ Rq or ∈ Cq for q ≥ 1, and usually we omit thevector symbol ~u or the uT = (u1, .., uq). Evaluated in x ∈ Ω the following ∂αu, ∂αv,Aαβ∂

βu,Aα∂αu are ∈ Rq and the Euclidean product (r, s)q in Rq, here and below e.g.

(Aαβ∂βu, ∂αv)q are well defined under the conditions (1) - (2 ) in u ∈ Wm,p or W 2m,p

with 1 < p <∞. Wm,p=2 = Hm. Mind in Cq the scalar product (r, s)q =∑qj=1 rjsj ∈ C,

is linear in r and sesquilinear in s, used in the following definitions.

Open bounded Ω ∈ Rn, n ≥ 1, ∂Ω ∈ C0,1,m, q ≥ 1, 1 < p <∞, 1

p+

1

p′= 1, with

α = (α1, . . . , αn), α, β ∈ Nn0 , |α| =n∑j=1

αj , |β| ≤ m, ∂αu = (∂1)α1

. . . (∂n)αn

u, for (1)

u ∈ U := Up := Wm,p0 (Ω,Rq), v ∈ V := Up′ : the standard ‖u‖U , and 〈u, v〉U×V .

We define the differential operator A and its Ap by the highest order terms, with

Aαβ ∈ L∞(Ω,Rq×q), a.e. ∀x ∈ Ω : |Aαβ(x)| ≤ Φ0,∀u ∈ U , v ∈ V,V ′ := W−m,p,

A : D(A) := U → V ′, 〈Au, v〉V′×V = a(u, v) =

∫Ω

∑|α|,|β|≤m

(Aαβ∂βu, ∂αv)qdx (2)

with principal parts ap(u, v) = 〈Apu, v〉V′×V =

∫Ω

∑|α|,|β|=m

(Aαβ∂βu, ∂αv)qdx.

Then solve for u0 ∈ U : Au0 = f or a(u0, v) = 〈Au0, v〉V′×V = 〈f, v〉V′×V (3)

:=

∫Ω

∑|α|≤m

(fα, ∂αv)qdx ∀v ∈ V := Up′ for f := (fα)|α|≤m ∈W−m,p(Ω,Rq),

below with parameters λ ∈ Λ ⊂ Rd in aij , Aαβ : Ω× Λ→ Rq×q, q ≥ 1,

Aαβ continuous in λ and satisfying (7),(8) uniformly in λ ∈ Λ. (4)

This (4) is important for bifurcation and local dynamics of center manifolds.

0.2. Linear and Nonlinear Elliptic Operators 5

We always consider pairs of differential and boundary operators (A,B), mainlyDirichlet, often included in U ,V, cf. [15, 16], Amann [3] for general cases. So ∀x ∈ ∂Ωor in a subset ∂Ω1 with nonempty interior or different conditions in subsets we require

Dirichlet or general operator BDu := (Bju := ∂ju/∂νj)m−1j=0 = fD or in [3] Bu = f.(5)

Obviously all bilinear forms in (1) - (3) are continuous. More generally a continuousbilinear form induces a unique bounded linear operator A s.t.

A ∈ L(U ,V ′) : a(u, v) = 〈Au, v〉V′×V ∀ u ∈ U , v ∈ V, ‖A‖V′←U ≤ Cb. (6)

Sometimes we replace the above partials ∂ju, ∂αu, u = ∇0u, ∂u = (∂1, . . . , ∂n)u =∇u,∇ku = ∂αu, |α| = k,∇≤ku, . . . , by reals or vectors. Whenever in a formula the∇0, ∂j , ∂α, ∂ = ∇1,∇k are not applied to a function or a term, we interpret the ∇0, ∂j ,∂α, .. as ∈ Rn, the ∂ = (∂1, . . . , ∂n) = ∇,∇2, . . . , as ∈ Rn×q,Rn2×q, .. c.f. (7).

An elliptic (we omit uniformly) operator satisfies for q = 1 the strong Legendre(delete the η−terms in (7)) and for q > 1 the strong Legendre-Hadamard condition, so

∃0 < ψ0 < Ψ0 ∈ R+ : ∀x ∈ Ω a.e. ∀∂ = (∂1, . . . , ∂n) ∈ Rn, ∂α =

n∏j=1

(∂j)αj ∈ R, (7)

η ∈ Cq, q,m ≥ 1 : ψ0|∂|2m|η|2 ≤ ηTAp(x, ∂)η := ηT∑

|α|=|β|=m

Aα,β∂α∂βη ≤ Ψ0|∂|2m|η|2.

For a coercive principal part Ap we impose for m > 1 beyond (1) - (3), (7), mind (4):

assume for |α| = |β| = m > 1 for q ≥ 1 : Aα,β ∈ C(Ω,Rq×q). (8)

0.2.2 Linear elliptic operators are sectorial

We study center manifolds via analytic semigroups generating sectorial operators andvice versa. So we introduce and restrict the further discussion to them. Theorem 0.5discusses weak elliptic sectorial operators A. Most parabolic solutions satisfy the strong,and consequently the weak form as well, c.f. Section 0.3.

Definition 0.1. Analytic semigroup: For Banach spaces, U ,V, a family S(t) : U → U∀t ∈ R+, defined on a sector Σ := Σε := λ ∈ C : λ 6= 0, | arg(λ)| < ε, with 0 < ε < π/2,c.f. Figure 1, is called a (linear strongly continuous) analytic semigroup iff

S(0) = I : U → U , ∀s, t ≥ 0 : S(s+ t) = S(s)S(t), S(t) ∈ L(U ,U),

∀u ∈ U : limt→0+

S(t)u = u, ∀t ≥ 0 ∀u ∈ U : t→ S(t)u is analytic. (9)

For S(t)t≥0 its (infinitesimal) generator −A : D(A) ⊂ U → U , is defined by thefollowing limit, e.g., for a weak elliptic A : D(A) ⊂ V ⊂ U → U , as

A : D(A) ⊂ (U or V)→ U ∀ u ∈ D(A) : −Au := limt→+0

S(t)u− ut

∈ U exists. (10)

6

Σϑ

ϑRe

Im

Figure 1. Shaded sector contained in the resolvent set of Aλ0.

Definition 0.2. cf. Figure 1: Resolvent, resolvent set and sectorial operators aredefined for A : D(A) ⊂ (U or V) → U as (A − λI)−1 and in ρ(A) := λ ∈ C : ∃(A −λI)−1 these resolvents do exist. The operator A for a real ( [31] p 18, pp 21) or acomplex ( [53] p 1096) Banach space, U or V, is called sectorial, if and only if

1. A is linear, graph closed and densely defined on U , hence D(A)‖·‖U or V

= U or V.

2. There exist real c,M, ϑ ∈ R,M ≥ 1, 0 < ϑ < π/2 s.t. the open sector 1

Σc,ϑ := λ ∈ C : c 6= λ, ϑ < | arg(λ− c)| ≤ π, ⊂ ρ(A), hence (11)

∀λ ∈ Σc,ϑ : ∃ (A− λI)−1 and additionally ‖(A− λI)−1‖ ≤M/|λ− c|. (12)

For parabolic problems a combination of Theorems 0.3-0.5, [22,31,40,47,53] is important.

Theorem 0.3. Sectorial operators generate analytic semigroups and ”vice versa”:

1The following results are, with minor changes, e.g., the curve C in (13), correct for the open [53]and closed [31] sectors: Σc,ϑ := ϑ < | arg(λ− c)| · · · and Σc,ϑ := ϑ ≤ | arg(λ− c)| · · · .

0.2. Linear and Nonlinear Elliptic Operators 7

1. A sectorial operator, A : D(A) ⊂ (U or V)→ U , on a real or complex U , generatesan analytic semigroup, denoted as and given by the Dunford integral

S(t) = e−Att>0, and e−At =1

2πi

∫C

(A+ λI)−1eλtdλ independent of the (13)

C ⊂ −Σc,ϑ 3 λ, arg λ|λ|→∞ → ±(π − ϑ), analytically extendable to −Σc=0,ϑ.

2. For each x ∈ U , t > 0, the e−Atx ∈ D(−A), so smoothing w0 ∈ U 7→ u(t) ∈ D(A).

3. Conversely, let −A be the generator of an analytic semigroup, S(t)t>0. Then Ais sectorial, and −A uniquely generates the original S(t)t>0.

A : Wm,p0 (Ω,Rq) → W−m,p is closed, sectorial, proved in [16] for p = 2,m, q ≥

1, 1 < p <∞, 1 = q ≤ m and (7). Amann [3]2 considers 1 < p <∞, 1 = m ≤ q and usesthe notations: Ap ∈ H(Lp) if −A is the infinitesimal generator of a strongly continuousanalytic semigroup S(t) = e−Att>0, on Lp. On ∂Ω he combines Dirichlet BD andNeumann type BN conditions for different equations in the system and components of∂Ω. So with the matrix diag(δ1 . . . δq), δj ∈ 0, 1, his B is = δBN + (1 − δ)BD. Hecalls (A,B) normally elliptic if B satisfies the complementing condition for A, cf. [3] forgood criteria, and its principal part Ap is elliptic (7). He claims, p.20, that these resultsremain correct as well for 1 ≤ m, q, and A in appropriate divergence form structure.With the spectrum σ he proves under these conditions Theorems 0.5 and 0.4.

σ(

(Aαβ(x)∂β , ∂α)q)|α|,|β|=m

⊂[Re z > 0

]:= z ∈ C; Re z > 0 ∀x ∈ Ω, 0 6= ∂ ∈ Rn. (14)

Theorem 0.4. Normally elliptic pairs (A,B) : [3],p.20,21. Under the previous condi-tions the pair (A,B) is normally elliptic, according to Amann [3], here and in Theorem0.5 even for 1 ≤ m, q, and A in appropriate divergence form structure.

Theorem 0.5. Sectorial A, [3] p.27: Normally elliptic (Ap, Bp) imply Aq ∈ H(Lq), q ∈(p, p′), so Aq is sectorial. For Ap ∈ H(Lp), the Ap is normally elliptic with the Bp.

This result does apply to the wide class of problems in (2) ff., but not to the moregeneral elliptic systems in the sense of Agmon/Douglis/Nirenberg [1]. For those, e.g.,the linearization of the Navier–Stokes operator and other equations in in fluid dynamicsother techniques have to be used, c.f. Kirchgassner and Kielhofer [34] and Denk, Hieberand Pruss [46].

0.2.3 Nonlinear elliptic operators

Bifurcation and the interesting (local) dynamics only exist for nonlinear PDEs. So weextend the previous linear to nonlinear operators in weak and mention strong forms for

2thanks to Robert Denk I found this paper strongly related to my problems

8

the order 2m,m ≥ 1 and for systems q ≥ 1, cf. [15]. With (2) - (7) we consider for theweak form nonlinear functions Gwm for n > 1.

Gwm :=(Awα , |α| ≤ m

): w = (x,∇≤m) ∈ D(Gwm)→ Rq, (15)

D(Gwm) = Uo = Ω× Umo , Umo ⊂ RNm×q, Nm := (nm+1 − 1)/(n− 1).

Then the weak and divergence form for wu(·) = (·,∇≤mu(·)) ∈ D(Gwm) a.e. are

(G(u), v)L2 :=∑|α|≤m

∫Ω

Aα(u)∂αvdx, Aα(u) := Awα (wu) ∈ L2 the weak and (16)

Gd(u)(·) :=∑|α|≤m

(−1)|α|∂α(Aα(u)), Aα ∈ C |α| the divergence form.

Replace Gwm, wu,Um0 by Gw2m, wu(·) := (·,∇≤2mu(·)),U2m0 ⊂ RN2m×q to define the strong

Gs(u)(·) := Gw2m(wu(·)), D(Gs) := u : Ω→ Rq∀x ∈ Ω a.e.wu(x) ∈ D(Gw2m). (17)

Definition 0.6. Nonlinear differential operators, ellipticity, coercivity: cf. [15], p 80.

1. A nonlinear PDE of order 2 or 2m – indicated as 2 / 2m – for u has the formform (15),(16) or (17). For a fully nonlinear Gs(u) in (17) its Gw2m is non linearin ∇2 / ∇2m. The form (15),(16) for Gwm, G,Gd is called a quasilinear. If all theAα are independent of u, but A0(u)(·) = A0(·, u(·)) then it is called semilinear.

2. These G, similarly Gd, Gs are called elliptic in u ∈ D(G) if G is differentiable nearu and G′(u) is elliptic, So only the principal part of G′(u) matters. This impliesthe coercivity and ellipticity under appropriate Garding conditions.

So we admit these nonlinear elliptic operators for the parabolic equations in (18), despiteexistence and uniqueness results only exist under very specific conditions.

0.3 Linear and Nonlinear Parabolic OperatorsWe start them with Ω, A,G in (1)- (3), (15) - Definition 0.6, u0, u in the state space U ,with often included boundary conditions for ∀t ∈ (t0, T ) with initial condition in t0 as

du0

dt+Au0 = f(t) or

du0

dt+G(u0) =

du0

dt+G′(u0)(u− u0) +R(u− u0) = f(t, u),(18)

special f, u0(t0) = w0 ∈ V ⊂ U on Ω with elliptic A,G′(u0) and R(v) = O(‖v‖2), so

u0 = u(w0, t). These A,G′(u0) are assumed independent of the time, so we consider(slightly generalized) autonomous systems (18). We say that (18) generates an evolutionoperator Φt if for appropriate w0 ∈ V ∃T = T (w0) > 0 s.t. u0(t + t0) := Φtw0 solves(18) for 0 ≤ t < T = T (w0). The family Φtt is called flow for (18).

0.3. Linear and Nonlinear Parabolic Operators 9

0.3.1 Solutions of parabolic problems and analytic semigroups

We introduce different types of solutions for parabolic problems. Much less is known forparabolic than for elliptic problems. We interpret (18) and (19) as linear or nonlinearordinary differential equations for elements and operators in Banach function spaces,with u,A,G, f, . . . , f(u) ∈ U ,V, C(Ω), . . . , defined on Ω. We usually omit or indicateby · the reference to x ∈ Ω, but include the time t in u(t), f(t), (t, u(t)), t ∈ (0, T ), inour generalized autonomous systems (18), studied in Henry [31], Carr [17], Pazy [44],Amann [3], Zeidler [53], Lunardi [40], Evans [23], Engel,Nagel [22], Raasch [47].

For linear, semi-, quasi- and fully non- linear problems many results for differenttypes of solutions, their relation to semigroups and generators are known. We cite somerelated to the generator −A, of an analytic semigroup S(t)t>0, [16], and start with:

du

dt(t) +Au(t) = f(t) or = f(t, u(t)) for t ∈ (t0, T ), u(t0) = w0 on Ω for (19)

As : D(As) =: W 2m,p(Ω,Rq) ∩Wm,p0 → U := Lp → U , 1/p+ 1/p′ = 1, q ≥ 1,

A = Aw : D(A) := Wm,p0 ⊂ V := Lp ⊂ U := W−m,p → U , often p = 2. (20)

For analytic semigroups S(t) = e−Att>0 by (13) we introduce a generalizedformula of variation of constants, cf. [52], defining different types of solution by

u0(t) := S(t− t0)w0 +

∫ t

t0

S(t− s)f(s)ds = e−A(t−t0)w0 +

∫ t

t0

e−A(t−s)f(s)ds (21)

and replace the f(s) here and in (22), (23) e.g. by f(s, u(s)), cf. (19) or other choices.More precisely the f : (t0, T )→ U is replaced by f : (t0, T )×OU → U , with OU either= U or an open subset of U . We require additionally u0(t) ∈ OU ∀ t ∈ [t0, T ). Or choosethe R(u− u0) in (18) or R(u, λ) in (25). The different conditions for f are listed below.

The technique for elliptic is employed for parabolic problems, their bifurcation andnumerical methods as well, cf. Thomee [49] and Section 0.5. So for the weak equationin (20) determine a weak solution u0 ∈ D(A) and u0(t0) = w0 ∈ V in (22), (23) s.t.

∀v ∈Wm,p′

0 (Ω,Rq) a.e.∀t ∈ (t0, T ) :d〈u0(t), v〉D(A)×Wm,p′

0

dt+ a(u0(t), v) = 〈f(t), v〉.... (22)

The Carr [17] solutions use the adjoint operator A∗ : D(A∗) ⊂ U∗ → U∗ ofA : D(A) ⊂ U → U and the pairing 〈·, ·〉 between U and its dual. [17] defines u0 by

∀v ∈ D(A∗) a.e. t ∈ (t0, T ) :d〈u0(t), v〉

dt+ 〈u0(t), A∗v〉 = 〈f(t), v〉, u0(t0) = w0.(23)

Definition 0.7. Mild, classical, weak solutions: Assume S : (t0, T )→ U with generator−A, t0 < T <∞, u0 : [t0, T )→ U , w0 ∈ V and f(s) or generalizations in (19), (21) ff.

1. Then the function u0 in (21) is called a mild solution for (19).

10

2. A u0 ∈ C1((t0, T ];U) ∩ C((t0, T ],D(A)) satisfying (19) for all t ∈ (t0, T ) andu0(t0) = w0 it is called a classical (or strong) solution.

3. A u0 ∈ C[t0, T ), with absolutely continuous, so differentiable, 〈u0(t), v〉U×Wm,p′0

∀v ∈

Wm,p′

0 on [t0, T ] with d〈u0(t), v〉D(A)×Wm,p′0

/dt ∈ L1(t0, T ) satisfying (22) is called

a weak solution. [17] calls u0 satisfying (23) weak solution as well.

4. A u0 ∈ C[t0, T ), with 〈u0(t), v〉∀v ∈ D(A∗) absolutely continuous, so differentiables.t. d < u0(t), v > /dt ∈ L1(t0, T ) satisfies (23) is called a distributional solution.

Theorem 0.8. Existence, uniqueness of different solutions for linear equations andanalytic semigroups S(t): Assume u0 : [t0, T )→ U , T <∞, w0 ∈ U . Then

1. for the strong As = A in (20) a mild solution u0 satisfies u0(t) ∈ D(As) for allt ∈ (t0, T ), hence u0 is a weak solution for Aw as well and we only use A here.

2. d(e−At)/dt = −Ae−At and there exists at most one classical solution u0 of (19).Each classical is a mild and a weak solution.

3. for continuous f : [t0, T )→ U the u0 in (21) with limt→t0+0 u0(t) = w0 ∈ V is thewell defined unique mild solution for (19).

4. for f ∈ C1[t0, T ), w0 ∈ D(A) the uniquely existing mild solution u0 in (21) satisfiesu0(t) ∈ D(A) ∀t > t0, and is a classical solution of (19).

5. for f ∈ L1(t0, T ) the weak and the distributional solution u0 for (22) and (23),resp., uniquely exist and are obtained by (21).

6. for f ∈ L2([t0, T ]), a unique weak solution u0 : [t0, T ]→ H1(Ω), hence m = 1, for(22) exists.

7. for a locally Holder continuous f in [t0, T ] with∫ ρt0‖f(s)‖ds <∞ for some ρ > 0

and w0 ∈ U , the mild solution u0 in (21) uniquely exists, is a classical and weaksolution with u0 ∈ C[t0, T ) ∩ C1(t0, T ). This u0(t) ∈ D(A) ∀t ∈ (t0, T ) representsthe well known smoothing from w0 ∈ U into u(t) ∈ D(A) for t > t0.

8. for the special A ∈ L(U ,U), w0 ∈ D(A) = U , and continuous f , the mild solutionin (21) is a classical solution of (21) as well.

The following two examples of semilinear parabolic equations show an essentialdifference between elliptic and parabolic nonlinear problems. For elliptic and somenonlinear parabolic problems still Hilbert, Sobolev and Holder spaces are appropriate as

0.4. Center Manifold for the Local Dynamics 11

in Theorem 0.9. For most parabolic cases these do not fit any more, cf. Theorem 0.10, 2.Then for fractional powers (A+aI)α, 0 ≤ α with invertible sectorial A : D(A) ⊂ U → U ,bounded a ≥ 0, so Re σ(A+ aI) > 0, often spaces with equivalent ||u||α are introduced

Uα := D((A+ aI)α), the new ||u|| := ||u||α := ‖(A+ aI)αu‖U , (24)

cf. e.g. [3, 22,39,40,53]. We only mention that, but do not define them explicitly

Theorem 0.9. [17], p 116: Distributional solutions for semilinear equations: Let−A generate an analytic semigroup, replace f(t) in (23) by f(t, u) : U → U , Lipschitzcontinuous in u. Then a unique distributional solution u0 exists with u0 ∈ C([t0, T );U).

Theorem 0.10. [31], p. 53: Solutions for semilinear equations: Let −A generate ananalytic semigroup S(t), f = f(t, v) in lines below (21), be locally Holder and Lipschitzcontinuous for small t0 − tj , u0 − uj , j = 1, 2, in a neighborhood O′ ⊂ OU of u0, s.t.

∃L > 0, θ > 0 : ∀(t, v), (s, w) ∈ O′ : ||f(t, v)− f(s, w)|| ≤ L(|t− s|θ + ||w − v||α).

1. Then for any (t0, w0) ∈ O, there exists T = T (t0, w0) > 0 such that (19) has aunique strong, simultaneously mild solution u0 on (t0, t0 + T ).

2. Conversely, if u0 ∈ C(t0, t1) → Uα, for some ρ > 0 :∫ t0+ρ

t0||f(t, u0(t))||dt < ∞,

and if (21) holds for t0 < t < t1, then u0 is a strong solution of (19) in t0 < t < t1.

3. Under these conditions each strong is a weak solution as well.

0.4 Center Manifold for the Local Dynamics

0.4.1 Splitting according to local dynamics

The goal of this paper, center manifolds and their numerical approximation, originatein a stationary bifurcation point (u0 ≡ 0, λ0) of the elliptic part, so G(0, λ0) = 0 ofnonlinear parabolic problems. An equilibrium, G(u0) = 0, is called stable, if for everyneighborhood V0 of u0 there exists a neighborhood V1 ⊂ V0, s.t. for every u1 ∈ V1 thesolution u(u1, t) of (18) with w0 replaced by u1 stays in V0 as long as it exists. If V1

can be chosen s.t. all these solutions exist for all t > 0 and u(u1, t) → u0 for t → ∞we call u0 asymptotically stable. A local invariant manifold of (18) is a subset M of Uin which the solution u(t) of (18) remains for some 0 < t ≤ T = Tu(0) and ∀u(0) ∈ M.If T =∞ ∀u(0) ∈ M, then M is called a (global) invariant manifold. A local or globalcenter manifold is tangential to N c(λ0), c.f. (27), at (u0 ≡ 0, λ0) and locally or globallyinvariant. Navier-Stokes problems admit only local center manifolds.

Lubich and Ostermann develop in [39] Section 2, a new analytical approach forcenter manifolds for their time discretizations. We summarize the results without and

12

with parameters in this and the end of the last Section. This yields a solid basis for theproof of the full, so space and time discretization of center manifolds in Section 0.5.

For the abstract evolution equation (25) in a Banach space U with A(λ), uniformlysectorial in λ, and D(A) independent of λ, we consider, below again omitting the λ,

du

dt+G(u, λ) =

du

dt+A(λ)u+R(u, λ) = 0, λ ∈ Λ ⊂ Rd, u ∈ D(A) ∩ Du(R) ⊂ U .

∀ λ A(λ) = ∂uG(u0 ≡ 0, λ) and R(0, λ) = 0, Ru(0, λ) = ∂uR(0, λ) = 0. (25)

Λ is a neighborhood of λ0 for a stationary bifurcation point (u0 ≡ 0, λ0) of the ellip-tic part. (25) is modified in case studies in [16] by replacing du/dt by Sdu/dt witha boundedly invertible elliptic operator S(λ). For parameter dependent problems thefollowing λ−independent A,P,Q,G, . . . in (26) and below have to be replaced againby A(λ), P (λ), Q(λ), G(u, λ), · · · So for A : D(A) → V ′, eg D(A) = Wm,p(Ω,Rq), u ∈D(A) ∩ Du(R) ⊂ U ,

du

dt+G(u) =

du

dt+Au+R(u) = 0, A = Gu(0), R(0) = 0, Ru(0) = 0. (26)

[39] have proved for (25), (26) existence, properties and the dominant role of centermanifoldsWc, if σ(A) = (σu = ∅)∪σc∪σs with AN s ⊂ N s, its stable manifoldWc and

σu = ∅, N u = 0 and N c = N c(λ) = N , κ = dimN c small, (27)

so a non-hyperbolic bifurcation point.

Figure 2. Spectrum of −A contained in the shaded area.

Numerical approximation requires generalizing the spectrum σc to eigenvalues with|Re µ| ≤ β small instead of β = 0. This N is spanned by the basis of generalized

0.4. Center Manifold for the Local Dynamics 13

eigenfunctions ϕi for A and µ ∈ σc. With the conjugate eigen-values and -vectors ϕ′i ofits dual Ad s.t. with usually omitted indices in, e.g., 〈ϕ′i, ϕj〉U ′×U = 〈ϕ′i, ϕj〉 we obtain

N = span[ϕ1, . . . , ϕκ] ⊂ U , ϕ′1, . . . , ϕ′κ ∈ U ′, 〈ϕ′i, ϕj〉 = δi,j , i, j = 1, . . . , κ, (28)

So σc ⊂ R and σs ⊂ S are subsets of a rectangle R and a sector S and a gap between,cf. Figure 2. We associate R,S, with the spectral projectors P : U → N , Q : U →M

P =1

2πi

∫∂R

(z +A)−1 dz, Q = I − P, U =(N = P (U)

)⊕(M := Q(U)

),

∀u ∈ U : v := Pu :=κ∑i=1

〈ϕ′i, u〉ϕi =:

κ∑i=1

viϕi =: (v,Φ) ∈ N ∼= Rκ and (29)

Rκ 3 v = (vi = vi)ni=1,Pu := v, w = Qu ∈M ∼= U, u = v + w ∈ N ⊕M.

The standard notations for multi-index, factorial, power, derivative, c.f. (1),(2), yields

v = (v1, . . . ,vκ), vk = vk11 . . .vkκκ ,∂vk

∂v1= k1v

k1−11 vk22 . . .vkκκ =: k1

vk

v1, · · · , |v| (30)

norm on Rκ. Splitting (25) according to P,P, Q, the sectorial A is block diagonalized as

A ∼=(B 00 L

), (31)

with L = QA|M sectorial on M ∼= U and B ∈ Rκ×κ on Rκ. B is a diagonal matrix ofJordan blocks one for each eigen value µ ∈ R. By Definition 0.2, the resolvents satisfy

|(z +B)−1| ≤ K

dist(z,R)for z 6∈ R (32)

|(z + L)−1| ≤ K

dist(z,S)for z 6∈ S and a constant K > 0.

For the above α, Uα = D(Lα) and norm we decompose the nonlinearity into

R(u) = PR(u) +QR(u) ∼=(

PR(u)

QR(u)

)= −

(f(v, w)

g(v, w)

)∈(Rκ

M

), (33)

where f : Rκ ×Uα → Rκ, g : Rκ ×Uα → U are as differentiable as R near (0, 0) s.t.

f(0, 0) = 0 , g(0, 0) = 0

∂vf(0, 0) = 0 , ∂wf(0, 0) = 0 , ∂vg(0, 0) = 0 , ∂wg(0, 0) = 0 .(34)

For studying the local center manifold for (25), [39] modify these f, g with a prob-lem depending sufficiently small ρ > 0 by a smooth cutting function

χ : Rκ → [0, 1], χ(v) = 1 for |v| ≤ ρ and χ(v) = 0 for |v| ≥ 2ρ, replace (35)

f(v, w), g(v, w) by f(χ(v)v, w), g(χ(v)v, w), maintaining (34).

14

Then [39] prove the existence of germs for a local center manifold for

dv

dt= −(Bv + PR(v + w)) = −Bv + f(v, w) (36)

dw

dt= −(Lw +QR(v + w)) = −Lw + g(v, w), with (37)

w = W (v) = W ρ(v) ≈W ι(v) =

ι∑2≤|k|

wkvk, ι ≤ ρ, wk = 0 for |k| < 2. (38)

Theorem 0.11. Exponentially attracting center manifold: Assume a non hyperbolicbifurcation point u0 ≡ 0 of the elliptic part with a sectorial linearized operator A and aspectrum split by the projectors P,Q into R and S with a gap β < `, and nonlinear termsR, f, g, Lipschitz continuous w.r.t. (v, w) ∈ Rκ ×Uα, cf. Theorem 0.5, (24),(26),(27).Finally define for a sufficiently small ρ, the cutting function χ in (35). Then (36),(37)admits a local exponentially attracting invariant center manifold, given as the graph ofa Lipschitz continuous map W : Rκ → Vα with W (0) = 0, cf. (38).

0.4.2 Asymptotic expansion in the homological equation

It is folklore that for center manifolds of finitely determined problems the determiningterms are uniquely determined, cf. Ashwin/Bohmer/Mei [4]. This is proved in Haragus/Iooss in Chapter 2, Remark 2.14: Local center manifolds are in general not unique eventhough the Taylor expansion at the origin is unique. This is due to the occurrencein the proof of a smooth cut-off function χ in (35), which is not unique. So our fulldiscretization methods studied here only converge for these restricted problems, [16].

Theorem 0.12. Uniquely determined terms of a center manifold: Assume the con-ditions of Theorem 0.11 for a ρ−determined problem. Then the coefficients wk of theasymptotic expansion of the center manifold in (38) are uniquely determined for |k| ≤ ρ.

The local dynamics of (26) would be known with the parameterization in (38)

Wc := (v, w) ∈ Rκ ×M,W,Wv : Rκ →M, w = W (v),W (0) = 0,Wv(0) = 0. (39)

Then we could split (26) into two subproblems: Insert w = W (v) into (36) and

solve v = −(Bv + PR(v +W (v)), with this v solve (37). (40)

This unrealistic procedure is avoided by recursively computing the approximationW ι(v) in (38) for W (v) by the terms up to the order ι ≤ ρ of its asymptotic expansion.Here the Lipschitz continuity of the map W : Rκ → Vα is essential. Due to the cut-offfunction in (35) we only get local results for v ≈ 0. So we recursively determine, startingwith |k| = 2, the wk, then = 3, . . . : We formulate for W (v) a characterizing, so called

0.4. Center Manifold for the Local Dynamics 15

homological equation C(W (v)) = 0 in (41): Insert ddt

w = Wv(v) ddt

v with ddt

v from

(36) into (37) and obtain with (29) - (33) this C(W (v)) and solve approximately the

C(W (v)) := −Wv(v)(Bv + PR(v +W (v))

)+(LW (v) +QR(v +W (v))

)= 0. (41)

For W (v) = W ι(v))+O(|v|ι+1) we combine appropriate |k|−linear forms wk : (Rκ)|k| →M applied to |k| ∈ N identical arguments v. We want to determine the wk ∈M

W (v) ≈W ι(v) =

ι∑2≤|k|

wkvk aymptotically satisfying (41) as C(W ι(v)) = O(|v|ι+1). (42)

Collecting the different terms for the same powers of vk in C(W (v)) we obtain

C(W ι(v)) =

ι∑|k|≥2

βkvk +O(|v|ι+1) = 0, with βk = 0 for |k| < 2 by (39). (43)

Starting with ι = 2 we determine the wk, |k| = 2, in (42) from (43) by equating theseβk = 0. Then we continue with ι = 3, . . . , inserting W ι(v) with the known wk, |k| <ι and the still unknown wk, |k| = ι into C(W (v)), Then (41), (43) are satisfied upto O(‖v‖|k|+1). This implies as an immediate consequence Theorem 0.13 1., 3.. 2.essentially is proved below at (52), cf. [16]. Here we formulate the λ−dependent case:

Theorem 0.13. A unique asymptotic expansion exists for the center manifold W (v, λ) :Under the conditions (25) - (27), cf. [39], we assume an approximate operator W ι :Rκ×Λ→M, with W ι(0, λ) = 0, (W ι)v(0, λ) = 0, let C(W ι(v, λ)) = O(‖(v, λ−λ0)‖ι+1)for small |v|, all λ near λ0, ‖(v, λ− λ0)‖ → 0 and 2 ≤ ι ≤ ρ.

1. This implies, with unique k- linear operators wk(λ)vk, for ‖(v, λ− λ0)‖ → 0

W (v, λ) +O(‖(v, λ− λ0)‖ι+1 = W ι(v, λ) :=

ι∑2≤|k|

wk(λ)vk hence (44)

W ι represents the unique asymptotic expansion for W (v, λ) up to the order ι.

2. In particular, each system starting ι = 2, . . . , is the compact perturbation of acoercive principal part and is uniquely solvable, hence is nonsingular.

3. With B(λ), f(v, w, λ) in (31),(33) we solve the asymptotically reduced equations(36),(37) for the center manifold for ι = 2, . . . , ρ, to determinacy,

dv

dt+B(λ)v = f(v,W ι(v, λ), λ) +O(‖(v, λ− λ0)‖ι+1). (45)

The second parabolic differential equation (37) can then be (often numerically)solved with this w = W ι(v, λ) up to O(‖(v, λ− λ0)‖ι+1).

16

This result allows the transformation of (40) into normal form: We can stop com-puting the wk, |k| = 2, 3, . . . , ι when we have reached determinacy ι = ρ.

If the normal form is fully determined by second order terms, then W (v) in (40)can be neglected. In some sense, this is not only numerically the easiest possible case.Indeed the equation for the center manifold has the form, cf. (36), (37),

v = −(Bv + PR(v)) +O(|v|3), w = −Lw +O(|v|3) and we are done! (46)

0.4.3 Recursive systems for solving the homological equation

To solve (43), we insert the components vj in (29) into (45)

vi = −κ∑j=1

〈ϕ′i, Aϕj〉vj − 〈ϕ′i, R(

κ∑j=1

vjϕj +W (v))〉, i = 1, . . . , κ. (47)

This generates the matrix J and the ODE for the center manifold, cf. (29), (33), (42),

J = (Jij)κi,j=1 := 〈ϕ′i, Aϕj〉κi,j=1, v = −Jv −PR((v,Φ) +W (v)),W (v) =

ρ∑2≤|k|

wkvk. (48)

A combination with (29), (33), (37), (39), (41), Q = I − P, yields for w and C(W (v))

w =d

dtW (v) =

κ∑i=1

∂W

∂vi(v)vi = −

κ∑i=1

∂W

∂vi(v)(Jv + PR((v,Φ) +W (v))

)i, and

C(W (v)) = −κ∑i=1

ρ∑2≤|k|

kiviwkv

k( κ∑j=1

Jijvj + 〈ϕ′i, R(

κ∑j=1

vjϕj +

ρ∑2≤|ν|

wνvν〉)

(49)

+(I • −

κ∑i=1

〈ϕ′i, •〉ϕi)(A

ρ∑2≤|k|

wkvk +R(

κ∑j=1

vjϕj +

ρ∑2≤|ν|

wνvν))

+O(‖v‖ρ+1) = 0.

Collecting for C(W (v)) the powers vk yields the explicit form of their coefficientsβk in (43) and (50). It is important that we have a specific structure for βk := 0 ∀|k| < 2.The terms, linear in the wν only depend upon the unknown w` with |`| = |k|, thosenonlinear in the wν , the Gk in (50), only upon the known wν with |ν| < |k|. This willbe important for the numerical methods below. Collecting all the linear and nonlinearterms we obtain, c.f. [16],

C(W (v)) =

ρ∑2≤|k|

vk(−∑|`|=|k|

w`β`,k + (I − P )A|Mwk +Gk(wν , |ν| < |k|))

+O(hρ+1)

with k−ij := (k1, ., ki + 1, ., kj − 1, ., kκ),∑|`|=|k|

w`β`,k =

κ∑ki>0

i=1

ki(wkJii +

κ∑kj<|k|1=j 6=i

Jijwk−ij

).(50)

0.4. Center Manifold for the Local Dynamics 17

According to the strategy indicated above, we annihilate the βk∀ |k| ≤ ρ, so up todeterminacy, hence solve by (49),(50) the following equations for the wk ∈M :

βk = (I − P )A|Mwk −∑|`|=|k|

w`β`,k +Gk(wν , |ν| < |k|) = 0, ∀ 2 ≤ |k| ≤ ρ. (51)

In contrast to Liapunov-Schmidt methods these equations (51) do not allow tocompute each wk separately, but only from a sequence of bordered coupled systemsstarting with ∀k : |k| = 2, then for ∀k : |k| = 3, a.s.o. The systems have to enforcethe two conditions C(W (v)) ∈ M or βk ∈ M and wk ∈ M. The wk is coupled to thew` still to be determined with |`| = |k|, and to already known wν with |ν| < |k|. Thecoupling for different k is realized via the terms

∑|`|=|k| w`β`,k and Gk(wν , |ν| < |k|),

the latter a compact perturbations of Aw` and known functions, resp. The followingterms 〈ϕ′j , wk〉 = 0 enforce a wk ∈ U into wk ∈ M, the

∑κj=1 αk,jϕj compensates

replacing (I−P )A|Mwk in (51) by Awk. The systems for |k| = 2, 3, . . . , ρ ≤ q, are linearin the unknowns w` ∈ U , ~αk := (αk,j)

κj=1 ∈ Rκ, |k| = |`|. So we obtain

Fk(·) := Awk −∑|`|=|k|

w`β`,k +

κ∑j=1

αk,jϕj +Gk(wν , |ν| < |k|) = 0, and (52)

〈ϕ′j , wk〉 = 0, j = 1, . . . , κ, with known β`,k, recursively solve for |k| = 2, 3, . . . , ρ.

This system is a compact perturbation of a coercive elliptic system with imposedside conditions, for details cf.(60). By Theorem 0.13, we do know that the system (49),or equivalently one of the (50) - (52) is uniquely solvable for the wk, ~αk, |k| = 2, then= 3, · · · ≤ ρ. The βk with |k| = 2 are obtained by neglecting the W (v) term in R(·) anduniquely define the wk, ~αk for |k| = 2. If we procede to the βk with |k| = 3 the wk with|k| = 2 have already been computed, so the Gk(wν , |ν| < |k|)vk are known. So the wkwith |k| = 3 are uniquely determined by the βk with |k| = 3 a.s.o. So the correspondingoperators are all boundedly invertible. We summarize our discussion in

Theorem 0.14. Bounded invertible linear systems imply a unique asymptotic expan-sion until determinacy for the center manifold: Under the conditions for center man-ifolds in (26), (27), (28) and Theorem 0.13, the system (49), guarantees the uniqueapproximate asymptotic expansion W ι : Rκ → M, with W ι(0) = 0, (W ι)v(0) = 0 andC(W ι(v)) = O(‖(v)‖ι+1) for all sufficiently small |v| → 0 and up to determinacy, soι ≤ ρ for the center manifold of (26) by Theorem 0.13. Therefore the systems (52),linear in the wk, ~αk ∀k are uniquely solvable, recursively for |k| = 2, 3, . . . , ρ for thesewk, ~αk. Hence they define, recursively for |k| = 2, 3, . . . , boundedly invertible systems oflinear elliptic operators.

18

0.5 Space Discretization for Center ManifoldsThe summarized results in Section 0.4 and at the end of this Section yield a solidfoundation for the convergence proof of the full, so space and time discretization oflocal center manifolds. The information about the structure of their local dynamics isconcentrated in the asymptotic solution of (36), so of elliptic problems. Again we firstomit the fixed λ, and start solving the stationary elliptic problem, c.f. (3), (52) withnumerical methods. Only very seldom the solution of the complete problem (36), (37)for the parabolic (18) is interesting. So time discretization is only indicated at the end.

The equation in (52) for each wk, |k| = 2, . . . and the other wν with ν = |k| islinear in these unknowns, but nonlinear only in the already known wν , |ν| < |k| via thenonlinear term Gk. Therefore we reduce the discretization here to linear operators.

(52) does not live in U = Wm,p0 (Ω,Rq), but in product spaces. For |k| = ι we have

to compute sι :=∑|k|=ι 1 different wk(x) ∈ Rq with ι = |k| = 2, 3, . . . , ρ. Furthermore,

there are sικ different αk,j , and 〈ϕ′j , wk〉 = 0. We denote the linear operator for the full

system as A = Aι, based upon A := G0u = G′(u0).

We only give a short summary for simplified discretization methods and linearoperators, for a general theory with nonlinear G : D(G) ⊂ U → V ′ see [15, 16]. Oftenwe have U = V including boundary conditions. On a sequence of subspaces Uh,V ′h forh → 0 we consider Petrov-Galerkin approximations for a linear A : U → V ′. ChooseUh ⊂ U ,Vh ⊂ V, e.g. for variational crimes only approximating Uh 6⊂ U ,Vh 6⊂ V,

Uh ⊂ U ,Vh ⊂ V with dim Uh = dim Vh, limh→0

dist(u,Uh) = 0 ∀u ∈ U , (v ∈ V). (53)

Then interpolation and approximation operators Ih and Ph exist and projection oper-ators Q′h or quadrature approximations Qh ≈ Qh and discretizations are defined by

∀u ∈ U : ‖Phu− u‖Uh or ‖Qhu− u‖Uh → 0 for h→ 0, and (54)

Q′h ∈ L(V ′,V ′h) by Q′hf := f |Vh ∈ Vh′⇔ 〈Q′hf − f, vh〉hV′×Vh = 0 ∀ vh ∈ Vh,

U A−→ V ′ tested by←→ VIh, Ph

y Φhy Q

′h(Qh)y

Uh Ah−→ Vh′ tested (approximately) by←→ Vh(55)

The solutions uh0 ∈ Uh are determined, possibly with appropriate extensions Ah, Ah forvariational crimes, quadrature formulas and collocation methods, from Ah = QhAh or

Ahuh0 := Q′hAhuh0 = Q′hAh|Uhuh0 = Q′hf ⇔ 〈Ahuh0 − f, vh〉

hV′×Vh = 0 ∀vh ∈ Vh. (56)

Basic concepts are consistency in u and of order p > 0, satisfied by (55) and stability:

‖Q′hAhP

hu−Q′hAu‖Vh′ → 0 and = O(h`) e.g. = C‖u‖W `,p(Ω;Rq)h

` for h→ 0 (57)

and ∃h0, S ∈ R+ s.t. ∀uhi ∈ Uh, i = 1, 2⇒ ‖uh1 − uh2‖Uh ≤ S‖Ahuh1 −Ahuh2‖Vh′ . (58)

0.5. Space Discretization for Center Manifolds 19

All methods mentioned above are shown to be consistent, stable and satisfy Theorems0.15 - 0.17, cf. [15, 16].

Theorem 0.15. Unique converging solution: Let u0 be the exact solution of Au0 =f, A ∈ L(U ,V ′) for a boundedly invertible A. Let its discretization Ah : Uh → Vh′ ,satisfying (54)-(58), be consistent in Phu0 and stable. Then ∀0 < h ≤ h0 there exists aunique discrete solution uh0 ∈ Uh for Ahuh = f or = Q

′hf s.t. uh0 converge according to

‖uh0 − u0‖Uh ≤ S‖AhPhu0 −Q′hf‖V′h and ≤ O(h`) e.g. = C‖u‖W `,p(Ω;Rq)h

`, (59)

so of order `. This holds, e.g., for a coercive A, hence with a stable discretization Ah.

Theorem 0.16. Compact linear perturbations of invertible B remain stable: ForA,B,C ∈ L(U ,V) with boundedly invertible, A,B, A = B + C, stable Bh, and compact(low order) perturbation C, the Ah is stable as well.

Very important for nonlinear problems is their possible equivariance. It stronglyinfluences and determines the structure of the solutions. So the discretization methodshave to inherit this equivariance. In [16] this is studied in two chapters on spectralmethods for infinite groups and in ’Numerical Exploitation of Equivariance for FiniteGroups’. We summarize for all the following results:

Theorem 0.17. Inherit equivariance in numerical methods: If the chosen discretizationmethods inherits the equivariance of the elliptic or parabolic problem, for all the listedmethods and following numerical results the corresponding equivariance results remainvalid.

We apply Theorems 0.15-0.17 to the systems (51) and the bordered forms (52).By Theorem 0.14 the matrix operator for (52) is boundedly invertible. In [16] we haveproved that the equation (52) for the wk, αk,j , with the known functions Gk, representsa modified compact perturbation of a coercive, hence stable system defined by its maindiagonal in (61). So by Theorems 0.16, modified, and 0.15, all these discretization of(52) are stable and consistent with the results on convergence of eigenvalue problems ofdiscretized elliptic problems in [15]. So we get convergent approximations whν , α

hν,j for

|ν| = |k| = ι, j = 1, . . . , κ. We go into the details:To formulate (52) as a matrix equation we need an ordering for the wk, ~αk =

(αk,j)κj=1, e.g., with wk1 = w(ι,...,0), wk2 = w(ι−1,1,0...,0), wksι with sι =

∑|k|=ι 1. With

20

N = span[Φ = (ϕj)κj=1] and N d = span[Φ′ = (ϕ′j)

κj=1] the operator matrices are

Bι + Cι = Aι := (60)

Ae,k1 −βk2,k1 −βk3,k1 . . . −βksι ,k1 Φ 0 0 0−βk1,k2 Ae,k2 −βk3,k2 . . . −βksι ,k2 0 Φ 0 0

. . .. . . . . .

. . .. . .

−βk1,ksι −βk2,ksι −βk3,ksι . . . Ae,ksι 0 0 0 ΦΦ′ 0 0 . . . 0 0 0 0 00 Φ′ 0 . . . 0 0 0 0 0

. . .. . .

. . .. . .

0 0 0 . . . Φ′ 0 0 0 0

with Ae,kiwki := (A− βki,ki)wki , the identity Iκ in Rκ×κ, the principle part B of A, so

Bι :=

B 0 0 0 0 0 0 00 B 0 0 0 0 0 0

0. . .

. . . 0 0. . .

. . . 00 0 0 B 0 0 0 00 0 0 0 Iκ 0 0 00 0 0 0 0 Iκ 0 0

0. . .

. . . 0 0. . .

. . . 00 0 0 0 0 0 0 Iκ

is coercive and Aι −Bι = Cι. (61)

So (52) for the unknowns wk, ~αk := (αk,j)κj=1 for each ι = |k| = 2, 3, . . . , ρ has the form

Aι × uι,0 :=(Bι + Cι

)×

wk1wk2. . .

wksι~αk1~αk2. . .

~αksι

= −fι := −

Gk1Gk2. . .

Gksι00

. . .

0

(62)

Now we are able to formulate the discrete counterparts of our above systems. Tothis end we simply have to go through Section 0.4 and replace all the exact data bytheir discrete approximate counterparts. So e.g. in (60) – (62) the Aι = Bι + Cι

wk, ~αk = (αk,j)κj=1, N = span[Φ = (ϕj)

κj=1],N d = span[Φ′ = (ϕ′j)

κj=1] are replaced by

Ahι = Bh

ι + Chι w

hk , ~α

hk = (αhk,j)

κj=1, N h = span[Φh = (ϕhj )κj=1], (N h)d = span[Φ

′h =

0.5. Space Discretization for Center Manifolds 21

(ϕ′hj )κj=1]. So we obtain e.g, instead of the (47), (48) with Wh(vh) =

∑ρ2≤|k| w

hk (vh)k,

vhi = −κ∑j=1

〈ϕ′hi , A

hϕhj 〉vhj − 〈ϕ′hi , R(

κ∑j=1

vhj ϕhj +Wh(vh))〉, i = 1, . . . , κ., (63)

Jh = (Jhij)κi,j=1 := 〈ϕhi , Ahϕhj 〉κi,j=1, v

h = −Jhvh −PhR((vh,Φh) +Wh(vh)). (64)

Modifying (60)-(62), we obtain with uhι = (whk1 , . . . , whksι , ~α

hk1 , . . . , ~α

hksι )

t, the equa-tion

Ahι u

hι,0 = −fhι = −Q

′hι fι ∈ U

′hι . (65)

Our modified Theorems 0.16 in [16] for a generalized compact perturbation Cι ofAι applied to a boundedly invertible Bι with stable Bh

ι yields stability for Ahι and with

an extension of the above consistency the convergence ofuhι = (whk1 , . . . , w

hksι , ~α

hk1 , . . . , ~α

hksι )→ uι = (wk1 , . . . , wksι , ~αk1 , . . . , ~αksι ).

The equations (49)-(52) require Taylor formulas for the W (~v) and C(W (~v)). If wediscretize center manifold equations we need

Definition 0.18. A general discretization method is called linear and k times diffe-rentiably consistent, if it is classically consistent and for a k times Frechet differentiablenonlinear operator G it satisfies (66) and (67), so with an interpolation operator Ih,

(G1 +G2)h = Gh1 + Gh2 , (α G)h = α Gh ∀α ∈ R, (66)

‖(Gh)(j)(Ihu)Ihv −Q′h(G(j)(u)v)‖V′h → 0, usually ‖v‖Us = e.g.‖u‖W `,p(Ω;Rq)

‖(Gh)(j)(Ihu)Ihu1 · . . . · Ihuj −Q′h(G(j)(u)u1 · . . . · uj)‖V′h (67)

= O(‖u1 − Ihu1‖Uh · . . . ‖uj − Ihuj‖Uh

)(1 + ‖u− Ihu‖Uh) for h→ 0.

Theorem 0.19. Linear differentiably consistent discretization methods: All space dis-cretization methods for elliptic problems in [15, 16] have this property.

We summarize the corresponding results in [16], yielding stable and convergentdiscretization methods for whk ≈ wk, with appropriate perturbations in [13,14].

Theorem 0.20. Stability and convergence for the center manifold discretizations:We assume the conditions of Theorems 0.13 - 0.17, 0.19 guaranteeing existence andsmoothness of a center manifold, and the unique existence of the asymptotic expansionexpansion Φι, ι = 2, 3, . . . , ρ, until determinacy. Finally, use approximations Φh,Φ

′h forΦ,Φ′ with the same order ` as in the above consistency.

1. Then the unique discrete solutions of Ahι u

hι,0 = Q

′hι fι, (and Ah

ι uhι,0 = Q

′hι fι) erxist

and converge (and of order `) to the exact solution uι,0 of Aιuhι,0 = fι :

‖uhι,0 − Ihι uι,0‖Uhι ≤ S‖Ahι (Ihι uι,0)−Q

′hι Aιuι,0‖U ′hι (and ≤ O(h`))for h→ 0. (68)

22

2. By (62) and (68) these unique solutions uhι,0 immediately yield the asymptotic

expansion with the coefficients whk ≈ wk. This implies the convergence for theasymptotic expansion for the center manifold, hence, for ι = 2, . . . , ρ, determinacy,

(W ι)h(v) :=

ι∑2≤|k|

whkvk ∈ Uh converge to W ι(v) :=

r∑2≤|k|

wkvk and (69)

‖Q′hW ι(v)− (W ι)h(v)‖Uh ≤ O(h`) +O(‖v‖r+1) for h, |v| → 0.

3. There are interesting cases for which the Φ,Φ′ are exactly available. Then theyare used instead of the numerical approximations Φh,Φ

′h.

0.6 Converging Normal Forms and Time Discretization,Parameter Dependence and Numerical Equivariance

We aim for a normal form for the small dimensional system of ODEs in (45). Thetransformations here have to account for the time derivative as well. So the recognitionproblem for center manifolds is still not fully solved. The standard reduction to normalform is a constructive process. It inductively eliminates or transforms to a special formas many of the lower order terms starting with the original f2 := f(v,W ι(v)) againomitting λ in (45). To this end we define the `k in a sequence of transformations as

v = `k(z) = I + P ′k(z), Pk(z) ∈ Hk, `k : D(`k) ⊂ Rκ → Rκ, `′k(0) regular, yielding

`′k(z)z = v = fk(`k(z))⇐⇒ z = (`′k(z))−1fk(`(z)) for small ‖z‖ and

Hk := homog. polyn. in z of degree k with z,Hk(z) ∈ Cκ, k = 2, 3, ... (70)

Since B resembles the spectral and sectorial information for the center manifold, we leaveB and the lower order transformed terms unchanged. With the Lie bracket operatorad B and unique orthogonal complement Ok ⊂ Hk, [19, 25], we obtain Theorem 0.21

ad B : Hk → Hk, P (z)→ ad B(P (z)) := BP (z)− P ′(z)Bz (71)

Ok ⊆ Hk = Ok ⊥ ad B(Hk)⇔ (Pg(z), Pb(z)) = 0∀Pg ∈ Ok, Pb ∈ ad B(Hk).

Theorem 0.21. Normal forms and convergence: For a (center manifold) dynamicalsystem v = −Bv + fn(v) of the form (36),(70) let f, fn ∈ Cr, ad B with complementsOk as in (71). Then there exists a sequence of transformations of the form (36),(70)such that the original v = −Bv + f(v) is transformed for 2 ≤ k ≤ ρ, into a system

z = −Bz + f (2)(z) + · · ·+ f (ρ)(z) +Rι, f(k) ∈ Ok, 2 ≤ k ≤ ρ, Rι = o(‖z‖ρ). (72)

For the above discretizations the discrete counterparts vh, Bh, fh, fhk , Ohk , 2 ≤ k ≤ ρ.

converge (of order p) to the preceeding exact v, B, f, fk, Ok. So do the normal forms.

0.6. Converging Normal Forms and Time Discretization, Parameter Dependence and Numerical Equivariance23

Remark 0.22. Different strategies for the time discretization of center manifolds: 1.) We have mainly discussed the normal form of the centermanifold equation. This can be numerically solved for short times with the standard, e.g.Runge-Kutta methods with good stability and yield good enough approximations. Only ifthe full solution v +W (v) has to be determined, elliptic problems have to be solved.2.) For longer time intervals the geometric numerical integration methods and the cor-responding structure-preserving algorithms for ODEs in Hairer/Lubich/Wanner, [28],can be modified for this case.3.) Huge time intervals hardly interest for the strictly local structure of center manifolds.4.) Compared to Lubich/Ostermann’s [39] time discretization, our start with space dis-cretization yields applicable numerical methods.

We return from the previously studied problem (18) to the parameter version

du

dt+G(u, λ) =

du

dt+Au+R(u, λ), λ ∈ Rk, R(0, 0) = 0, R′(0, 0) = 0. (73)

In the standard approach, cf. [27], the λ has to be added everywhere, e.g., in u = v+w,with (v, λ), w = W∗(v, λ) and W∗(0, 0) = 0, W ′∗(0, 0) = 0 and A is replaced by G′(0, 0) =(A 00 0

), a.s.o. The invariant manifold Mc

λ for (36) is now the graph of a mapping

W∗ : (v, λ) ∈ N (λ)× Rd → Vα or we consider W as a function of the two variables

W : (v, λ) ∈ Rκ × Λ→ Vα with W (v, λ) := W∗(v, λ). (74)

Theorem 0.23. Center manifolds, smooth in parameters and initial values, cf. [31],Theorem 6.1.7: Suppose the assumptions of Theorem 0.11 hold smoothly and uniformlyin λ ⊂ Λ ⊂ Rd with fixed R ∪ S in Figure 2 and A,R ∈ Ck,1 with 0 < (k + 1)β < `.Then, the mapping W in (74), defining locally the center manifoldMc

λ for (73) near thestationary bifurcation point (u0 ≡ 0, λ0), and (v, λ) for v ∈ Rκ ∼= N = N (λ), satisfies

W ∈ Ck,1(Rκ × Λ,Vα), and Mcλ = (v, w, λ) ∈ Rκ × Vα × Λ|w = W (v, λ).

This center manifold is tangential to N (λ) near (0, λ0), but not unique and satisfies

W (0, λ) = 0, ∂vW (0, λ) = 0 ∀λ ∈ Λ.

Instead of (73) we could include the equation ∂λ/∂t = 0. This yields a nonlinearterm λ(v + w), while for fixed λ such a term is linear. This allows the center manifoldto capture dynamics of the original problem in neighborhoods of the parameter λ = λ0.The techniques for computing center manifolds for both ways and the original problemare identical. The transformations to normal form have to be updated. The normal

24

forms may change their character, when λ passes through the origin. The convergenceresults remain correct.

If the equivariance adapted numerical methods for finite and infinite groups inChapters 10 and 11 in [16] are applied the corresponding equivariance, stability, conver-gence results hold, cf. [2, 12–14].

Theorem 0.24. Equivariance for center manifolds and their discretizations: Let underthe conditions of Theorem 0.14 the original problem (26) be equivariant under somegroup. Then the reduced center manifold ODE (40) exhibits the same equivariance, thediscrete center manifold is even invariance under the group. This equivariance can besaved to the recursive systems (52) and their discretization in Section 0.5.

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