Conservative upwind ﬁnite-element method for a simpliﬁed ...epshteyn/selectedpapers.html/paperSaito.pdf · Finite-element approximation for a non-linear parabolic–elliptic system

IMA Journal of Numerical Analysis (2007) 27, 332−365doi:10.1093/imanum/drl018Advance Access publication on October 3, 2006

Conservative upwind finite-element method for a simplified Keller–Segelsystem modelling chemotaxis

NORIKAZU SAITO†

Faculty of Human Development, University of Toyama, 3190 Gofuku,Toyama 930-8555, Japan

[Received on 1 June 2005; revised on 12 April 2006]

Finite-element approximation for a non-linear parabolic–elliptic system is considered. The system de-scribes the aggregation of slime moulds resulting from their chemotactic features and is called a simpli-fied Keller–Segel system. Applying an upwind technique, first we present a finite-element scheme thatsatisfies both positivity and mass conservation properties. Consequently, if the triangulation is of acutetype, our finite-element approximation preserves the L1 norm, which is an important property of theoriginal system. Then, under some assumptions on the regularity of a solution and on the triangulation,we establish error estimates in L p × W 1,∞ with a suitable p > d, where d is the dimension of a spatialdomain. Our scheme is well suited for practical computations. Some numerical examples that validateour theoretical results are also presented.

Keywords: finite-element method; parabolic–elliptic system; non-linear evolution equation; analyticsemigroup; conservation of positivity; conservation of mass.

1. Introduction

We consider a non-linear parabolic–elliptic system for the functions u = u(x, t) and v = v(x, t) of(x, t) ∈ Ω × [0, J ): ⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

∂u∂t = ∇ · (k∇u − λu∇v) in Ω × (0, J ),

0 = v − v + u in Ω × (0, J ),

k ∂u∂ν − λu ∂v

∂ν = 0, ∂v∂ν = 0 on ∂Ω × (0, J ),

u|t=0 = u0 on Ω,

(1.1)

where Ω ⊂ Rd (d = 2, 3) is a bounded domain with the boundary ∂Ω , ν is the outer unit normal vector

to ∂Ω , ∂/∂ν represents differentiation along ν on ∂Ω , u0 = u0(x) 0 is a given non-identically zerofunction defined on Ω and J , k, λ are positive constants.

The system (1.1) describes the aggregation of slime moulds resulting from their chemotactic fea-tures, where u denotes the density of the cellular slime moulds and v is the concentration of the chemicalsubstance. Replacing the constants k, λ in the first equation of (1.1) with functions k(u, v), λ(v) andtaking

λ0∂v

∂t= v − λ1(v)v + λ2(v)u (λ0 > 0 : a constant, λ1, λ2 : functions of v)

instead of the second equation, we obtain the original system proposed by Keller & Segel (1970). Ifdiffusion of the chemical substance takes place much faster than that of the cellular slime moulds,

†Email: [email protected]

c© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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CONSERVATIVE UPWIND FEM FOR A SIMPLIFIED KELLER–SEGEL SYSTEM 333

the original Keller–Segel system can be reduced to (1.1). On the other hand, in some other contexts, weoften obtain problems that possess quite similar structure to that of (1.1) including transient drift diffusionof electrons in semiconducting media (Biler et al., 1994; Jungel, 2001) and gravitational interaction ofparticles (Biler & Nadzieja, 1994). Therefore, the results of this paper are of use not only in mathemat-ical biology but also in engineering and theoretical physics.

The mathematical study for (1.1) is well developed. In particular, Biler (1998) and Yagi (1997)showed unique solvability locally in time when ∂Ω and u0 are sufficiently smooth. Apart from thesefundamental results, many researchers elucidated the behaviour of solutions: relevant properties includeaggregation, blow-up, chemotactical collapse and so on. Two survey articles, Horstmann (2003) andHorstmann (2004), review those mathematical results. A salient property of a solution (u, v) to (1.1) isthe conservation of the L1 norm:

‖u(t)‖L1(Ω) = ‖u0‖L1(Ω), t ∈ [0, J ], (1.2)

which is an immediate consequence of the conservation of positivity

u(x, t) > 0, (x, t) ∈ Ω × (0, J ], (1.3)

and the conservation of total mass∫Ω

u(x, t)dx =∫

Ωu0(x)dx, t ∈ [0, J ]. (1.4)

Actually, the steady-state problem corresponding to (1.1) is reduced to a non-local eigenvalue problemby (1.2). Moreover, the value of ‖u0‖L1(Ω) is crucial to the global existence and blow-up of a solution.For example, in the case of d = 2, it is known that if ‖u0‖L1(Ω) < 4πλ−1, the solution exists globallyin time, whereas if ‖u0‖L1(Ω) > 8πλ−1 and

∫Ω |x − x0|2u0(x)dx 1 with some x0 ∈ Ω , then the

solution blows up in finite time. Furthermore, quantization of the blow-up mechanism can be observed.For further details, see Suzuki (2005).

The present paper has dual purpose. The first is to propose a finite-element scheme for (1.1) thatsatisfies the discrete analogues of (1.3) and (1.4). As stated above, the L1 conservation is an essentialrequirement; it is desirable that approximate solutions preserve this property. In a continuous problem,derivations of (1.3) and (1.4) are simple. Indeed, (1.3) is a result of the maximum principle and (1.4)follows from

d

dt

∫Ω

u(x, t)dx =∫

Ω∇ · (k∇u − λu∇v)dx =

∫∂Ω

(k∂u

∂ν− λu

∂v

∂ν

)dS = 0.

In a discrete problem, however, some difficulties arise. First, as is well-known, we should choose ourdiscretization parameters very carefully in order to guarantee the discrete version of (1.3). In a pre-vious paper, Saito & Suzuki (2005), we discussed that issue and proposed finite-difference schemesthat satisfy discrete versions of both (1.3) and (1.4). Those schemes made use of an upwind differencescheme and semi-implicit time discretization with step-size control. That is, at every discrete time steptn = τ1 + · · · + τn , we adjust the time increment τn to obtain a positive solution. We intend to extendthis strategy to finite-element methods. In doing so, however, we confront another issue. Upwind-typefinite-element approximations usually destroy the conservation of total mass. To surmount this obs-tacle, Baba & Tabata (1981) proposed a conservative upwind finite-element approximation for a linearconvection–diffusion equation. We combine our previous strategy with the upwind approximation ofBaba & Tabata. Consequently, we realize (cf. Theorems 2.1 and 2.2) a finite-element approximationthat satisfies discrete versions of both (1.3) and (1.4) for an arbitrary h > 0, the granularity parameter of

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the spatial discretization, if the triangulation is of acute type. (Ω is assumed to be a convex polygonaldomain.) Moreover, our scheme is well suited for practical computations.

The second purpose is to develop an error analysis. Let (unh, v

nh ) be a solution of the finite-element

scheme proposed below at a discrete time step tn . In our scheme, the point-wise value of ∇vn−1h is used

to determine the upwind nodal points at tn . Therefore, we need to pay attention to the error vnh − v(tn)

in the W 1,∞(Ω) norm (uniformly in tn). On the other hand, we have (cf. (6.13) and (6.14))

‖v(tn) − vnh‖W 1,∞(Ω) Ch1−d/p‖u(tn)‖L p(Ω) + C‖u(tn) − un

h‖L p(Ω)

for some p ∈ (d, µ). Here, µ > d is the shape constant of Ω; it is defined through the condition (R)described below, which can be thought of as a quantification of the W 2,p regularity property of a linearelliptic problem. In any case, it is crucial to study the error u(tn) − un

h in the L p norm (uniformly intn). For these reasons, we shall develop the L p theory for (1.1) and its finite-element approximation. Infact, if the triangulation is of acute type and is quasi-uniform, under some regularity hypotheses on u,we have (cf. Theorem 2.3)

sup0tnJ

‖u(tn) − unh‖L p(Ω) C(h1−d/p + τ), (1.5)

where τ = maxn τn . Because the upwind approximation employed in this paper corresponds to a one-sided difference approximation, the rate of convergence with respect to the spatial discretization isexpected to be O(h) at best. However, such a rate of convergence is not achieved in (1.5). That shortfallstems from the lack of regularity of solutions of a linear elliptic problem in a polygonal domain. There-fore, on considering (1.1) in a smooth domain, we can deduce (cf. Section 7) a refined estimate of theform

sup0tnJ

‖u(tn) − unh‖L p(Ω) C(h + τ). (1.6)

In order to derive (1.5) and (1.6), we formulate (1.1) and its finite-element approximation as singleequations for u and un

h , respectively, and apply analytical (holomorphic) semigroup theory in Banachspaces. Actually, as proved in Crouzeix & Thomee (2001), if the triangulation is of acute type, theoperator Ah , a finite-element approximation of − + 1 of the lumped mass type, becomes sectorialon a finite-element space Xh,p equipped with a modified L p norm. In particular, −Ah generates ananalytic semigroup on Xh,p. (The precise meaning of these symbols will be given in Section 4.) Wethen make use of Duhamel’s principle, fractional powers of operators and the smoothing property ofthe semigroup. Lemma 4.4, which is an application of the Heinz–Krein inequality (Lemma A.1), alsoplays an important role. Although semigroup theory is somewhat abstract, several L p estimates can bederived in quite a formal manner. Furthermore, our method of analysis is a discrete analogue of thestandard approach for solving non-linear evolution problems. It also provides a theoretical frameworkfor the approximation error analysis of some other non-linear evolution problems.

Before concluding this section, we briefly discuss some other results that are related to numericalmethods for (1.1). Nakaguchi & Yagi (2002) presented finite-element/Runge–Kutta discretizations forthe original Keller–Segel system without any numerical results. They also established error estimatesin the H1+ε norm, ε ∈ (0, 1/2), for a sufficiently small J , though they devoted little attention toconservation of the L1 norm of approximate solutions. Marrocco (2003) discussed mixed finite-elementapproximations for (1.1) and offered various numerical examples, but a convergence analysis was notundertaken. Finally, as stated above, we presented finite-difference approximations to (1.1) enjoyingdiscrete positivity and mass conservation properties in Saito & Suzuki (2005), although a convergenceanalysis was not performed in that study either.

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The plan of this paper is as follows:

1. Introduction

2. Main results

3. Conservation laws (proof of Theorems 2.1 and 2.2)

4. Some preliminary results

5. Lemmas concerning b and bh

6. Error estimates (proof of Theorem 2.3)

7. Remarks on error estimates of the order h

8. Numerical examples

A. Proof of Lemma 4.4

Notation. We follow the notation of Adams & Fournier (2003). We put W m,p = W m,p(Ω), Hm =W m,2, L p = L p(Ω), ‖·‖m,p = ‖·‖W m,p , ‖·‖p = ‖·‖L p for m ∈ N and p ∈ [1, ∞]. The standard innerproduct in L2 is denoted by (·, ·). We put, for p ∈ [1, ∞),

Wp =v ∈ W 2,p

∣∣∣∣ ∂v

∂ν= 0 on ∂Ω

.

We set [a]± = max0, ±a for a ∈ R. The d-dimensional Lebesgue measure of O ⊂ Rd is denoted

by meas(O) = measd(O). For a Banach space X , its dual space is denoted as X∗. Generic positiveconstants depending on Ω are denoted as C , C ′ and so forth. In particular, C does not depend on thediscretization parameters h and τ described below. If it is necessary to specify the dependence on otherparameters, say α, β, then we write them as Cα,β or C(α, β). However, if the contribution of thoseparameters is not necessary for our argument, we omit indicating them. We shall use the same symbolI to indicate the identity operator on any space.

2. Main results

We assume that Ω is a convex polyhedral domain with the boundary ∂Ω . In order to present our finite-element scheme, we introduce a weak formulation of (1.1). Since the bilinear form (∇v, ∇χ) + (v, χ)is continuous and coercive on H1 × H1, for a given f ∈ L2, there is a unique solution v ∈ H1 of

(∇v, ∇χ) + (v, χ) = ( f, χ) ∀ χ ∈ H1.

By elliptic regularity, we have v ∈ W2 and ‖v‖2,2 C‖ f ‖2. We write v = G f to express this cor-respondence and call G : L2 → W2 ⊂ L2 the Green operator associated with − + 1 on L2. Weintroduce a trilinear form b on L2 × H1 × H1 defined as

b(w, u, χ) = −∫

Ωu∇Gw · ∇χ dx (w ∈ L2, u ∈ H1, χ ∈ H1),

which is well-defined:

|b(w, u, χ)| ‖u‖3‖∇Gw‖6‖∇χ‖2 (Holder’s inequality)

‖u‖1,2‖∇Gw‖1,2‖∇χ‖2 (Sobolev’s inequality)

‖u‖1,2‖w‖2‖∇χ‖2 (elliptic regularity)

for w ∈ L2, u ∈ H1, χ ∈ H1.

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A weak form of (1.1) is represented as follows: find u ∈ C1([0, J ]: H1) such that⎧⎪⎨⎪⎩

(du(t)

dt , χ)

+ (k∇u(t), ∇χ) + λb(u(t), u(t), χ) = 0 ∀ χ ∈ H1, ∀ t ∈ (0, J ),

u(0) = u0 ∈ H1.

(2.1)

We proceed to the presentation of our finite-element approximation. Let Th = Thh↓0 be aregular family of triangulations Th of Ω:

1. Th is a set of closed d-simplices (elements), and Ω = ∪T |T ∈ Th;2. any two elements of Th meet only in entire common faces or sides or in vertices;

3. there exists a positive constant γ1 such that

hT γ1ρT ∀ T ∈ Th ∈ Thh,

where

hT = diam(T ) (≡ the diameter of T ),

ρT = maxdiam(S)|S is a ball included in T .As the granularity parameter, we have employed h = maxhT |T ∈ Th. Let Ph = Pi N

i=1 be theset of all vertices of Th , N being a positive integer. (Although N depends on h, we shall not explicitlyexpress the dependence.) We divide Ph into two subsets Pi NI

i=1 ⊂ Ω and

PNI +iNB

i=1 ⊂ ∂Ω , where

N = NI + NB . With Pi , we associate a function φi ∈ C(Ω) such that φi is linear on each T ∈ Th andφi (Pj ) = δi j , where δi j denotes Kronecker’s delta. We define

Xh = the vector space spanned by φi Ni=1

and regard it as a closed subspace of H1. We also consider the space Xh , which is equipped withthe topology induced from L2, and denote it using the same symbol Xh . The barycentric domain Di

corresponding to Pi is defined as

Di =⋃

T∈Si

x ∈ T | φTj (x) φT

i (x)(Pj ∈ V (T ), Pj = Pi ),

where Si = S ∈ Th |Pi ∈ S, V (T ) = Pj ∈ Ph |Pj ∈ T and φTi d+1

i=1 are the barycentriccoordinates of T with respect to Pi . Let φi ∈ L∞ be the characteristic function of Di . We introduce aHilbert space Xh ⊂ L2 spanned by φi N

i=1. The operator Mh : Xh → Xh is defined as

Mhvh =N∑

i=1

vh(Pi )φi (vh ∈ Xh),

which is called the lumping operator. We put

(vh, χh)h = (Mhvh, Mhχh) (vh, χh ∈ Xh).

Thereby, (·, ·)1/2h is equivalent to ‖·‖2 on Xh (see Section 4.2). The discrete Green operator Gh : Xh →

Xh is defined as follows: For any fh ∈ Xh , we consider the unique solution vh = Gh fh ∈ Xh of

(∇vh, ∇χh) + (vh, χh) = ( fh, χh) ∀ χh ∈ Xh .

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To state our approximation of the trilinear form b, we need additional notation. Let Pi , Pj ∈ Ph . IfPi = Pj and they share an edge, we set

Si jh = T ∈ Th |Pi , Pj ∈ T ;

Γi j = ∂ Di ∩ ∂ D j and

νi j = the outer unit normal vector to Γi j with respect to Di .

Otherwise, we set Si jh = ∅, Γi j = ∅ and νi j = 0. The restrictions of Γi j and νi j on T ∈ Si j

h are denotedby Γ T

i j and νTi j , respectively. Then we introduce the functionals β±

i j on Xh as

β±i j (wh) =

∫Γi j

[∇Ghwh · νi j ]± dS =∑

T∈Si jh

meas(Γ Ti j )[(∇Ghwh)T · νT

i j ]±.

At this stage, writing

Λi = Pj ∈ Ph |Si jh = ∅,

we introduce a trilinear form bh on Xh × Xh × Xh by

bh(wh, uh, χh) =N∑

i=1

χh(Pi )∑j∈Λi

uh(Pi )β+i j (wh) − uh(Pj )β

−i j (wh) (wh, uh, χh ∈ Xh).

Although this is a direct application of the scheme of Baba & Tabata (1981), we briefly state its deriv-ation in a formal manner:

b(wh, uh, χh) =∫

Ω∇(uh∇Gwh)χh dx (integration by parts)

≈N∑

i=1

χh(Pi )

∫Di

∇(uh∇Gwh)dx (χh ≈ Mhχh)

=N∑

i=1

χh(Pi )∑j∈Λi

∫Γi j

uh∇Gwh · νi j dS (divergence theorem)

≈N∑

i=1

χh(Pi )∑j∈Λi

∫Γi j

uh∇Ghwh · νi j dS (G ≈ Gh).

Then, the integral on Γi j is approximated by considering the upwind nodal points as follows:∫Γi j

uh∇Ghwh · νi j dS ≈ uh(Pi )β+i j (wh) − uh(Pj )β

−i j (wh).

Consequently, we obtain the trilinear form bh .As a discretization of the time variable, we take positive constants τ1, τ2, . . . , and put

tn = τ1 + · · · + τn (n 1).

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We introduce the backward Euler difference quotient by setting

∂τn wnh = wn

h − wn−1h

τn(wn

hn1 ⊂ Xh).

Now we can state our finite-element scheme for (2.1): Find unhn0 ⊂ Xh such that⎧⎨

⎩(∂τn un

h, χh)h + (k∇un

h, ∇χh) + λbh(un−1h , un

h, χh) = 0 ∀ χh ∈ Xh, ∀ n 1,

u0h = u0h,

(2.2)

where u0h ∈ Xh is a suitable approximation of u0.

REMARK 2.1 The first equality of (2.2) includes two linear systems. Thus, at each time step, we initiallysolve a linear system for Ghun−1

h and then we solve one for unh . We note that Ghun

h is an approximationof v(tn) = Gu(tn).

REMARK 2.2 The semi-implicit time discretization in (2.2) is closely related to the reproduction ofLyapunov’s property, which is another important feature of the system (2.1). For further details andother methods of time discretization, we refer to Saito & Suzuki (2005).

We are in a position to state our main theorems. The first one is related to the discrete version of(1.4).

THEOREM 2.1 (Conservation of total mass) Let unhn0 ⊂ Xh be a solution of (2.2). Then

(unh, 1)h = (u0h, 1)h (2.3)

for n 0.

The second theorem is related to the well-posedness of our scheme including the discrete version of(1.3). To state it, we put

κh = minT∈Th

κT (κT = the minimal perpendicular length of T )

and introduce the following conditions.(H1) Acuteness. It is assumed that

maxcos(∇φTi , ∇φT

j )|1 i, j d + 1 0 ∀ T ∈ Th ∈ Th,

where φTi d+1

i=1 represent the barycentric coordinates of T with respect to the vertices of T .

REMARK 2.3 As stated in Section 3, (H1) guarantees the non-positivity of (∇φi , ∇φ j ) for i = j . See(3.2). For d = 2, (H1) is equivalent to a statement that each triangle of Th is a right-angle or an acutetriangle. For d = 3, (H1) is satisfied if, and only if, all angles spanned by two faces of each tetrahedron ofTh are less than or equal to π/2. Furthermore, for d = 3, (H1) is equivalent to the following condition:Given T ∈ Th , let P ∈ T and F ⊂ T be a vertex and the opposite face to P , respectively. Then the footof the perpendicular from P to a plane S, including F , is always included in F .

THEOREM 2.2 (Well-posedness and conservation of positivity) Let (H1) be satisfied. Suppose thatu0h ∈ Xh is non-negative and is not identically constant. Take τ > 0 and ε ∈ (0, 1]. Then, (2.2) with a

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time step-size control

τn = min

τ,

3(d − 1)εκh

λd2(d + 1)‖∇Ghun−1h ‖∞

admits a unique solution unhn0 ⊂ Xh such that un

h > 0 for n 1.

Combining this with Theorem 2.1, we immediately obtain

COROLLARY 2.1 Let unhn0 ⊂ Xh be a solution of (2.2) as in Theorem 2.2. Then,

‖unh‖1 = ‖u0h‖1

for n 0.

REMARK 2.4 Because all norms are equivalent on Xh , we have

‖∇Ghun−1h ‖∞ ch‖Ghun−1

h ‖1,2

ch · C‖un−1h ‖2

c′h‖un−1

h ‖1 = c′h‖u0h‖1,

where ch , c′h are positive constants depending on h. Therefore, there is a c′′

h > 0 such that τn minτ, c′′

h. Thus, τn never converges to zero as n increases, and therefore the algorithm always works.Consequently, un

h actually exists for all n 1. Further, if we assume (H2) and (R) described below, thenby making use of (4.2) and (4.31), we obtain a more explicit estimate as

τn min

τ,

Cκhhd(1−1/p)

‖u0h‖1

with some p ∈ (d, µ) and C = C(d,Ω, γ2) > 0.

To state our convergence theorem, we require additional conditions:(H2) Quasiuniformity assumption. There exists a positive constant γ2 such that

γ2h hT ∀ T ∈ Th ∈ Th.(R) Elliptic regularity. There exists µ ∈ (d, ∞) such that the following holds true: For any p ∈ (1, µ)and f ∈ L p(Ω), the linear elliptic problem

−v + v = f in Ω,∂v

∂ν= 0 on ∂Ω (2.4)

admits a unique solution v ∈ Wp that satisfies

‖v‖2,p C‖ f ‖p (2.5)

with a constant C = C(p,Ω) > 0.

REMARK 2.5 When Ω ⊂ R2 is a convex polygon, (R) is always satisfied. On the other hand, when

Ω ⊂ R3 is a convex polyhedron, it is satisfied, if all edges and all vertices of Ω are sufficiently small that

they do not produce singularities. See, for more complete descriptions, Theorems 8.2.1.2 and 8.2.2.8 ofGrisvard (1985).

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REMARK 2.6 Under condition (R), for any p ∈ (1, µ), we introduce an operator G p: L p → Wp definedby v = G p f , where v is the unique solution of (2.4) for a given f ∈ L p. Then the operator G should beunderstood precisely as G = G p with some p ∈ (1, µ). If we do not assume (R), then we understandthat G = G2.

Because ∂Ω is not smooth, we cannot treat classical solutions. Instead, we consider L p solutions,which are well discussed in Suzuki & Senba (2004) and Suzuki (2005). Thus, we make the followingregularity assumption on a solution u of (2.1):

u ∈ C([0, J ] : Wp), u′ ∈ C([0, J ] : W 1,p) ∩ Cσ ([0, J ]: L p) (2.6)

for some p 2, σ ∈ (0, 1], and put

α1,p = supt∈[0,J ]

‖u(t)‖2,p, α2,p = supt∈[0,J ]

‖u′(t)‖1,p, α3,p = supt,s∈[0,J ]

‖u′(t) − u′(s)‖p

|t − s|σ ,

where u′ = du/dt .

THEOREM 2.3 (Error estimates) Suppose that (H1), (H2) and (R) are satisfied. Assume that (2.1) admitsa unique solution u satisfying (2.6) for some p ∈ (d, µ) and σ ∈ (0, 1]. Moreover, let u0h ∈ Xh bechosen as

‖u0 − u0h‖p α0,ph1−d/p, (2.7)

with some α0,p = α0,p(u0) > 0. Then, there exist positive constants h0, τ0 depending on Ω , J , k, λ, p,σ , γi ’s and αi,p’s such that we have the error estimates

sup0nl

‖u(tn) − unh‖p C1(h

1−d/p + τσ ), (2.8)

sup0nl

‖Gu(tn) − Ghunh‖1,∞ C2(h

1−d/p + τσ ) (2.9)

for h ∈ (0, h0) and τ ∈ (0, τ0), where l = l(τ, h) = maxn ∈ N| tn < J , unhn0 ⊂ Xh is the solution

of (2.2) as in Theorem 2.2. Furthermore, the constants C1 and C2 can be taken as

Ci = C(J + 1)(α0,p + α21,p + α2

2,p + α3,p) exp[C ′(1 + α21,p)J ], i = 1, 2,

where C and C ′ are positive constants that depend only on Ω , k, λ, γi ’s, h0 and τ0.

3. Conservation laws (proof of Theorems 2.1 and 2.2)

We recall some standard results. Put

mi = meas (Di ),

κi = minκT |Pi ∈ T, T ∈ Th,ai j = (∇φ j , ∇φi ),

bi j (wh) = bh(wh, φ j , φi ) (wh ∈ Xh)

for 1 i, j N .

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We haveN∑

i=1

ai j =N∑

i=1

a ji = 0. (3.1)

Under Assumption (H1), we have (cf. Fujii, 1973)

ai j 0 (i = j), 0 <aii

mi d + 1

κ2i

. (3.2)

Moreover, under (H1), we readily verify that the N × N matrix A = [ai j ] is irreducible. As stated inLemma 2.1 of Baba & Tabata (1981), for wh ∈ Xh , we have

N∑i=1

bi j (wh) = 0, (3.3)

bi j (wh) 0 (i = j), 0 bii (wh)

mi 2d

κi‖∇Ghwh‖∞. (3.4)

Furthermore, for wh ∈ Xh and 1 i, j N , we set

βi j (wh) = β+i j (wh) − β−

i j (wh). (3.5)

Then, we easily obtain β±i j (wh) = β∓

j i (wh), βi j (wh) + β j i (wh) = 0 and

|βi j (wh)| meas (Γi j )‖∇Ghwh‖∞. (3.6)

Now we can state the following proof.Proof of Theorem 2.1. Substituting χh = φi into (2.2), we obtain

miun

h(Pi ) − un−1h (Pi )

τn= −

N∑j=1

kai j + λbi j (un−1h )un

h(Pj ).

Summing this with respect to i from 1 to N and using (3.1) and (3.3), we obtain

1

τn

(N∑

i=1

unh(Pi )mi −

N∑i=1

un−1h (Pi )mi

)= −

N∑j=1

N∑i=1

kai j + λbi j (un−1h )un

h(Pj ) = 0,

which implies (2.3). Proof of Theorem 2.2. It suffices to verify the following fact: For a given 0 wh ∈ Xh and δ > 0satisfying

δ 3(d − 1)κh

λd2(d + 1)‖∇Ghwh‖∞, (3.7)

there is a unique 0 < uh ∈ Xh such that(uh − wh

δ, χh

)h

+ (k∇uh, ∇χh) + λbh(wh, uh, χh) = 0 ∀ χh ∈ Xh . (3.8)

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As before, (3.8) implies that

uh(Pi ) − wh(Pi )

δ+

N∑j=1

kai j + λbi j (wh)

miuh(Pj ) = 0.

This is written equivalently as

Cu = w, (3.9)

where u = (uh(P1), . . . , uh(PN ))T, w = (wh(P1), . . . , wh(PN ))T and C = [ci j ] with

cii = 1 + δ

mi(kaii + λbii (wh)) (1 i N ),

ci j = δ

mi(kai j + λbi j (wh)) (1 i, j N , i = j).

If we can show that

ci j 0 (i = j), cii > 0; (3.10)

C is irreducibly diagonally dominant, (3.11)

then we apply Corollary 3.20 of Varga (2000) and obtain that C is regular and that every entry of C−1

is positive. These yield the desired fact.Now we verify (3.10) and (3.11). First (3.10) is a direct consequence of (3.2) and (3.4). In order to

check (3.11), let 1 i, j N with i = j . We have by (3.1) that

|cii | −N∑

j =i

|ci j | = 1 + δ

mi

N∑j=1

(kai j + λbi j (wh))

= 1 + δλ

mi

N∑j=1

bi j (wh). (3.12)

Because

bi j (wh) = −∑k∈Λi

β−ik(wh)δ jk =

−β−i j (wh) ( j ∈ Λi )

0 ( j ∈ Λi ),

for i = j , we obtain, by (3.3), (3.5) and (3.6), that

N∑j=1

bi j (wh) = bii (wh) +N∑

j =i

bi j (wh) = −∑j∈Λi

b ji (wh) +∑j∈Λi

bi j (wh)

=∑j∈Λi

[β−j i (wh) − β−

i j (wh)] =∑j∈Λi

[β+i j (wh) − β−

i j (wh)]

−‖∇Ghwh‖∞∑j∈Λi

meas(Γi j ). (3.13)

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We recall (cf. Lemma 2.1 of Baba & Tabata, 1981) that there is ζi ∈ (0, 1] such that

1

mi

∑j∈Λi

meas(Γi j ) ζid2(d + 1)

3(d − 1)κi.

In particular, we can take ξi = 1 for 1 i NI and ζi ∈ (0, 1) for NI + 1 i N . Hence, by (3.7),(3.12) and (3.13),

|cii | −N∑

j =i

|ci j |⎧⎨⎩ 0 (1 i NI ),

> 0 (NI + 1 i N ).

This implies that C is diagonally dominant with strict inequalities holding for NI + 1 i N . Finally,we verify the irreducibility. In view of (3.2) and (3.4), ai j < 0 implies ci j < 0 for i = j . Hence, theirreducibility of C is reduced to that of A = [ai j ]. The proof is completed.

4. Some preliminary results

In this section, we collect some preliminary results, which will be useful in subsequent sections.

4.1 Sobolev and inverse inequalities

We shall make use of

‖v‖∞ C‖v‖1,p (p ∈ (d, ∞], v ∈ W 1,p). (4.1)

In what follows, we often treat functions of W 1,p, p > d, as those of C(Ω). Next, let T ∈ Th .Assumption (H2) implies the inverse inequality

‖vh‖Wl,p(T ) Chm−l+min0, d

p − dq

T ‖vh‖W m,q (T ) (vh ∈ Xh), (4.2)

where p, q ∈ [1, ∞] and 0 m l are integers. It is readily verifiable that

maxx,y∈T

|vh(x) − vh(y)| Ch1−d/pT ‖∇vh‖L p(T ) (p ∈ [1, ∞], vh ∈ Xh). (4.3)

These facts are explained more thoroughly in Adams & Fournier (2003), Brenner & Scott (2002) andCiarlet (1978).

4.2 Some auxiliary operators

The Lagrange interpolation operator πh : C(Ω) → Xh is defined by πhv(Pi ) = v(Pi ) for all Pi ∈ Ph .For T ∈ Th , we have

‖πhv − v‖L p(T ) + h‖∇(πhv − v)‖L p(T ) Ch2‖v‖W 2,p(T ) (p ∈ (d/2, ∞], v ∈ W 2,p). (4.4)

Moreover, under (H2), we have (cf. Theorem 4.4.20 of Brenner & Scott, 2002)

‖πhv − v‖L∞(T ) + h‖∇(πhv − v)‖L∞(T ) Ch2−d/p‖v‖W 2,p(T ) (p ∈ (d, ∞], v ∈ W 2,p). (4.5)

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We frequently use the L2 and H1 projection operators Ph : L2 → Xh and Rh : H1 → Xh , which aredefined as

(Phv − v, χh) = 0 ∀ χh ∈ Xh, (4.6)

(∇Rhv − ∇v, ∇χh) + (Rhv − v, χh) = 0 ∀ χh ∈ Xh . (4.7)

LEMMA 4.1 Under (H2), we have

‖Phv‖p C‖v‖p (p ∈ [1, ∞], v ∈ L p), (4.8)

‖Phv‖1,p C‖v‖1,p (p ∈ [1, ∞], v ∈ W 1,p), (4.9)

‖Phv − v‖p Ch2‖v‖2,p (p ∈ (d/2, ∞], v ∈ W 2,p). (4.10)

Proof. Inequalities (4.8) and (4.9) are attributed to Douglas et al. (1975) and Crouzeix & Thomee(1987). Inequality (4.10) follows from (4.4) and (4.8). LEMMA 4.2 Let (H2) and (R) be satisfied. Then we have

‖Rhv‖1,p C‖v‖1,p (p ∈ (1, ∞], v ∈ W 1,p), (4.11)

‖Rhv − v‖1,p Ch‖v‖2,p (p ∈ (1, ∞], v ∈ W 2,p), (4.12)

‖Rhv − v‖p Ch2‖v‖2,p (p ∈ (µ/(µ − 1), ∞), v ∈ W 2,p), (4.13)

‖Rhv − v‖1,∞ Ch1−d/p‖v‖2,p (p ∈ (d, ∞], v ∈ W 2,p). (4.14)

Proof. The derivation of (4.11)–(4.13) is the same as that shown in Chapter 8 of Brenner & Scott (2002).Therein, the case of the Dirichlet boundary condition was considered explicitly. To show (4.14), we notethat ‖Rhv − v‖1,∞ ‖Rhv − πhv‖1,∞ + ‖πhv − v‖1,∞ C‖πhv − v‖1,∞ for v ∈ W 1,∞ by (4.11).Hence, (4.14) follows from (4.5).

Let M∗h be the adjoint operator of Mh in L2, and set

Kh = M∗h Mh . (4.15)

Thereby, we have

C‖vh‖p ‖Mhvh‖p C ′‖vh‖p (p ∈ [1, ∞], vh ∈ Xh). (4.16)

Moreover, if (H2) is satisfied, then

C‖vh‖p ‖Khvh‖p C ′‖vh‖p (p ∈ [1, ∞], vh ∈ Xh), (4.17)

and

‖Mhvh − vh‖p Ch‖∇vh‖p (p ∈ [1, ∞], vh ∈ Xh). (4.18)

See Chapter 5 of Fujita et al. (2001) for (4.16), (4.17), and Baba & Tabata (1981) for (4.18). Inequality(4.17) is valid for p = 2 without (H2).

REMARK 4.1 Most results that are recalled in this paragraph are valid for a sufficiently small h, or moreprecisely, for h ∈ (0, h∗) with some suitable h∗ = h∗(Ω, γ1, γ2) > 0. However, in order to simplify thepresentation, we omit its precise definition here. Similarly, in what follows, h will be taken from (0, h∗)with some suitable h∗ = h∗(Ω, γ1, γ2) > 0.

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4.3 Discrete Laplace operator

Throughout this paragraph, we suppose that (R) is satisfied. We recall the operator G p: L p → Wp

defined by v = G p f , where v is the unique solution of (2.4) for a given f ∈ L p. We set Ap = G−1p ,

which is nothing but the L p realization of − + I with the Neumann boundary condition

D(Ap) = Wp, Apv = −v + v (v ∈ D(Ap)). (4.19)

We simply write G and A to express G p and Ap, respectively, if there is no possibility of confusion(see Remark 2.6). The closed linear operator A is sectorial in L p. Therefore, its fractional powers Aα ,α ∈ (0, 1), are defined in a natural way, and −A generates the analytic semigroup e−t At0. From thegeneral theory, we have

D(A1/2) = W 1,p, (4.20)

‖Aα e−t A‖p Ct−α (t > 0, α ∈ (0, 1)). (4.21)

For these facts, see Henry (1981) and Tanabe (1979). We introduce the operators Lh and Ah of Xh →Xh defined as

Lhuh = fh ⇔ (∇uh, ∇χh) + (uh, χh) = ( fh, χh) ∀ χh ∈ Xh,

Ahuh = fh ⇔ (∇uh, ∇χh) + (uh, χh)h = ( fh, χh)h ∀ χh ∈ Xh .

Obviously, we have Lh = G−1h and

Ah = K −1h Lh . (4.22)

Moreover,

Lh Rhv = Ph Av (v ∈ D(A)), (4.23)

or equivalently,

RhGw = Gh Phw (w ∈ L p). (4.24)

We collect some operator theoretical properties of Ah . To this end, in this paragraph, we regard anyfunction space as a complex valued one, and propose a re-definition:

(u, v) =∫

Ωu(x)v(x) dx

(u ∈ L p, v ∈ Lq ,

1

p+ 1

q= 1

).

For p ∈ [1, ∞), we introduce the discrete L p norm

‖vh‖h,p =(∫

ΩMhπh |vh(x)|p dx

)1/p

(vh ∈ Xh).

It is readily verifiable that

C‖vh‖h,p ‖vh‖p C ′‖vh‖h,p (vh ∈ Xh), (4.25)

|(vh, χh)h | ‖vh‖h,p‖χh‖h,q

(vh, χh ∈ Xh,

1

p+ 1

q= 1

). (4.26)

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We regard Xh as a Banach space equipped with the norm ‖·‖h,p and indicate it by Xh,p. In particular,Xh,2 forms a Hilbert space with respect to the inner product (·, ·)h . Furthermore, the operator norm inXh,p is denoted by the same symbol ‖·‖h,p. For instance,

‖Ah‖h,p = supvh∈Xh

‖Ahvh‖h,p

‖vh‖h,p.

LEMMA 4.3 Let p ∈ (1, ∞), and suppose that (H1) is satisfied. Then,

(i) Ah is sectorial in Xh,p, and its fractional powers Aαh , α ∈ (0, 1), are defined.

(ii) Ah and Aαh , α ∈ (0, 1), are positive and self-adjoint inXh,2.

(iii) if (H2) is also satisfied, for any θ ∈ (0, 1) and τ j nj=1, τ j > 0, we have

‖r(τn Ah) · · · r(τ1 Ah)Aθh‖h,p Cθ (τn + · · · + τ1)

−θ , (4.27)

where r(τ j Ah) = (I + τ j Ah)−1.

Proof. (i) See Crouzeix & Thomee (2001). (ii) is easy to verify (see also Saito, 2004a). (iii) followsfrom Theorem 4 of Saito (2004b). REMARK 4.2 Since Ah is a linear operator on a finite-dimensional space, (i) and (ii) hold simultane-ously. Such a fact generally fails in infinite-dimensional spaces. In fact, A2 is self-adjoint in L2, whereasAp is not self-adjoint for p = 2.

REMARK 4.3 Inequality (4.27) is equivalently written as

‖Aθhr(τn Ah) · · · r(τ1 Ah)‖h,p Cθ (τn + · · · + τ1)

−θ , (4.28)

which is a discrete analogue of (4.21).

The following lemma, which is a discrete analogue of (4.20), will be proved in Appendix A.

LEMMA 4.4 Suppose that (H1), (H2) and (R) are satisfied. Then,

‖vh‖1,p C‖Aθhvh‖h,p (vh ∈ Xh) (4.29)

for p ∈ (µ/(µ − 1), µ) and θ ∈ (1/2, 1]. (When p = 2, we can take θ = 1/2 without (R).)

REMARK 4.4 We do not know whether the restriction on p and the condition (R) are necessary or notfor (4.29) to hold. See also Remark A.1. We note that these assumptions produce no obstruction in ourarguments. Actually, in Section 6, we suppose that p ∈ (d, µ). Hence, the conjugate exponent of p is inthe interval (µ/(µ − 1), d/(d − 1)).

REMARK 4.5 As a readily obtainable consequence of (4.29), we deduce that if (H1), (H2) and (R) aresatisfied, then

‖∇ A−θh vh‖p ‖A−θ

h vh‖1,p C‖vh‖h,p (vh ∈ Xh) (4.30)

for p ∈ (µ/(µ − 1), µ) and θ ∈ (1/2, 1].

LEMMA 4.5 Suppose that (H2) and (R) are satisfied. Then, for p ∈ (d, µ),

‖vh‖1,∞ C‖Lhvh‖p (vh ∈ Xh). (4.31)

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Proof. By virtue of (4.11), (4.1) and (2.5), we have

‖Rhv‖1,∞ C‖v‖1,∞ C‖v‖2,p C‖Av‖p

for v ∈ D(A). Let vh ∈ Xh . Choosing v = A−1vh , we obtain by (4.23)

‖L−1h vh‖1,∞ C‖vh‖p,

which implies (4.31).

4.4 Dual norms

Let p, q ∈ (1, ∞) such that 1p + 1

q = 1. It is evident thatXh,p = X ∗h,q . Therefore,

‖vh‖h,p C supχh∈Xh

(vh, χh)h

‖χh‖h,q(vh ∈ Xh). (4.32)

On the other hand, we consider the space Xh equipped with ‖ · ‖p and regard it as a Banach space Xh,p.Then, we have Xh,p = X∗

h,q and

‖vh‖p C supχh∈Xh

(vh, χh)

‖χh‖q(vh ∈ Xh). (4.33)

4.5 A lemma concerning Kh

LEMMA 4.6 Let (H1), (H2) and (R) be satisfied. Then,

‖A−θh (K −1

h − I )vh‖h,p Ch2‖∇vh‖p (4.34)

for p ∈ (µ/(µ − 1), µ), θ ∈ (1/2, 1] and vh ∈ Xh .

Proof. In the same way as in the proof of Lemma 15.1 of Thomee (1997), we have

|((Kh − I )vh, χh)| = |(Mhvh, Mhχh) − (vh, χh)| Ch2‖∇vh‖p‖∇χh‖q (4.35)

for vh, χh ∈ Xh and 1p + 1

q = 1. Hence, by (4.25) and (4.30),

|(A−θh (K −1

h − I )vh, χh)h | = |((K −1h − I )vh, A−θ

h χh)h |

= |((I − Kh)vh, A−θh χh)|

Ch2‖∇vh‖p‖∇ A−θh χh‖q

Ch2‖∇vh‖p‖χh‖h,q

for vh, χh ∈ Xh . This, together with (4.32), yields (4.34).

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4.6 Discrete inequality of Volterra type

LEMMA 4.7 Taking positive constants τ1, . . ., τl (l ∈ N), putting tn = τ1 + · · · + τn for 1 n l andt0 = 0, suppose that a sequence znl

n=0 ⊂ R satisfies

0 < zn c1 + c2

n∑j=1

τ j

(tn − t j−1)r(z j−1 + z j ) (1 n l), (4.36)

where c1, c2 and r ∈ (0, 1) are positive constants. Then we have

zn c1c3 exp

(c4c

11−r2 tn

)(0 n l),

where c3 and c4 are positive constants depending only on r .

Proof. Setting Z = max0nl zn , we have by (4.36) that

zn c1 + 2Zc2

∫ tn

0(tn − s)−r ds = c1 + 2Zc2t1−r

n B(1 − r, 1),

where B(·, ·) denotes Euler’s B-function. The rest of the proof is exactly the same as that of Lemma 6.5of Okamoto (1982).

5. Lemmas concerning b and bh

LEMMA 5.1 Let (R) be satisfied. Let p ∈ (d, µ) and 1p + 1

q = 1. Then,

|b(w − Phw, u, χh)| Ch2‖w‖2,p‖u‖∞‖∇χh‖q (5.1)

for w ∈ W 2,p, u ∈ L∞, χh ∈ Xh , and

|b(w, u, χh − Mhχh)| Ch‖w‖p‖u‖1,p‖∇χh‖q (5.2)

for w ∈ L p, u ∈ W 1,p, χh ∈ Xh .

Proof. Let w ∈ W 2,p, u ∈ L∞ and χh ∈ Xh . Then, by (2.5) and (4.10),

|b(w − Phw, u, χh)| ‖u‖∞‖∇G(w − Phw)‖p‖∇χh‖q

‖u‖∞‖G(w − Phw)‖2,p‖∇χh‖q

‖u‖∞‖w − Phw‖p‖∇χh‖q

‖u‖∞Ch2‖w‖2,p‖∇χh‖q

which implies (5.1). On the other hand, by integration by parts,

b(w, u, χh − Mhχh) =∫

Ω∇ (u∇Gw) · (χh − Mhχh)dx

for w ∈ L p, u ∈ W 1,p and χh ∈ Xh . Since p > d, we can perform an estimation using (2.5):

‖∇ · (u∇Gw)‖p C‖u‖1,p‖∇Gw‖1,p C‖u‖1,p‖w‖p.

Combining these with (4.18), we obtain (5.2).

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Next we derive some estimates for bh . Putting

Γh = Γi j = ∂ Di ∩ ∂ D j |1 i, j N

, (5.3)

we observe that, for wh, uh, χh ∈ Xh ,

bh(wh, uh, χh) =∑

Γi j ∈Γh

[χh(Pi ) − χh(Pj )][β+i j (wh)uh(Pi ) − β−

i j (wh)uh(Pj )]

=∑

Γi j ∈Γh

[χh(Pi ) − χh(Pj )][σ+i j (wh)uh(Pi ) − σ−

i j (wh)uh(Pj )]βi j (wh), (5.4)

where

σ+i j (wh) = sgn β+

i j (wh), σ−i j (wh) = 1 − σ−

i j (wh). (5.5)

LEMMA 5.2 Let p, q ∈ (1, ∞) with 1p + 1

q = 1. Then, under (H2),

|bh(wh, uh, χh)|

C‖∇Ghwh‖p‖uh‖∞‖∇χh‖q ,

C‖∇Ghwh‖∞‖uh‖p‖∇χh‖q

(5.6)

for wh, uh, χh ∈ Xh .

Proof. For 1 i, j N , we put

hTi j = maxhT |T ∈ Si jh . (5.7)

(If Si j

h = ∅, then hTi j = 0.)

Let wh, uh, χh ∈ Xh . By (5.4) and (4.3), we can deduce that

|bh(wh, uh, χh)|∑

Γi j ∈Γh

|χh(Pi ) − χh(Pj )| · |σ+i j (wh)uh(Pi ) − σ−

i j (wh)uh(Pj )| · |βi j (wh)|

∑

Γi j ∈Γh

Ch1− d

qTi j

‖∇χh‖Lq (Ti j )‖uh‖L∞(Γi j )‖∇Ghwh‖L∞(Γi j )meas (Γi j )

C‖uh‖∞∑

Γi j ∈Γh

h1− d

qTi j

‖∇χh‖Lq (Ti j )h− d

pTi j

‖∇Ghwh‖L p(Ti j )hd−1Ti j

C‖uh‖∞∑

T∈Th

‖∇χh‖Lq (T )‖∇Ghwh‖L p(T )

C‖uh‖∞‖∇χh‖q‖∇Ghwh‖p.

Similarly, we obtain

|bh(wh, uh, χh)| C‖∇Ghwh‖∞∑

T∈Th

‖∇χh‖Lq (T )‖uh‖L p(T )

C‖∇Ghwh‖∞‖uh‖p‖∇χh‖q ,

which completes the proof.

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LEMMA 5.3 Suppose that (H2) and (R) are satisfied. Let p, q ∈ (1, ∞) with 1p + 1

q = 1, and letr ∈ (d, µ). Then we have

|bh(Phw, uh, χh)| C‖w‖r‖uh‖p‖∇χh‖q (5.8)

for w ∈ Lr , uh, χh ∈ Xh .

Proof. Let w ∈ Lr . Then in view of (4.24) and (4.11), we have that

‖Gh Phw‖1,∞ = ‖RhGw‖1,∞ C‖Gw‖1,∞ C‖Gw‖2,r C‖w‖r ,

which, together with (5.6), yields (5.8). LEMMA 5.4 Suppose that (H2) and (R) are satisfied. Let p ∈ (d, µ) and 1

p + 1q = 1. Then,

|b(Phw, u, χh) − bh(Phw,πhu, χh)| Ch1−d/p‖w‖p‖u‖1,p‖∇χh‖q (5.9)

for w ∈ L p, u ∈ W 1,p and χh ∈ Xh .

Proof. Let w ∈ L p, u ∈ W 1,p, χh ∈ Xh . We divide

b(Phw, u, χh) − bh(Phw,πhu, χh)

= b(Phw, u, χh − Mhχh) + b(Phw, u, Mhχh) − bh(Phw,πhu, χh)≡ J1 + J2.

Because, by (5.2), we have |J1| Ch‖w‖p‖u‖1,p‖∇χh‖q , it is sufficient to prove that

|J2| Ch1−d/p‖w‖p‖u‖1,p‖∇χh‖q . (5.10)

To this end, we write as wh = Phw, σ±i j = σ±

i j (wh), ui = u(Pi ) and χi = χh(Pi ) for the sake ofsimplicity, and put

βi j =∫

Γi j

u(∇Gwh · νi j )dS, β ′i j =

∫Γi j

∇Gwh · νi j dS.

We have

b(wh, u, Mhχh) =N∑

i=1

χi

∑j∈Λi

∫Γi j

u(∇Gwh · νi j )dS =∑

Γi j ∈Γh

(χi − χ j )βi j .

This, together with (5.4), implies that

J2 =∑

Γi j ∈Γh

(χi − χ j )[βi j − (σ+i j ui + σ−

i j u j )β′i j ]

+∑

Γi j ∈Γh

(χi − χ j )(σ+i j ui + σ−

i j u j )(β′i j − βi j (wh))

≡ J21 + J22.

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To estimate J21, we apply (cf. Lemma 3.1 of Baba & Tabata, 1981)∫U

|v(P) − v(x)|dx Chd−d/pT ‖v‖W 1,p(T ) (T ∈ Th, p ∈ (d, ∞), v ∈ W 1,p),

where P is any point in T and U is the intersection of T and any hyper-plane through P . Noting thatσ+

i j + σ−i j = 1, we have, by (4.3),

|J21|∑

Γi j ∈Γh

|χi − χ j | ·∣∣∣∣∣∫

Γi j

[σ+i j (u(x) − ui ) + σ−

i j (u(x) − u j )](∇Gwh · νi j )dS

∣∣∣∣∣

∑Γi j ∈Γh

Ch1−d/qTi j

‖∇χh‖Lq (Ti j )‖∇Gwh‖∞hd−d/pTi j

‖u‖W 1,p(Ti j )

Ch‖Gwh‖2,p

∑T∈Th

‖∇χh‖Lq (T )‖u‖W 1,p(T )

Ch‖Phw‖p‖u‖1,p‖∇χh‖q

Ch‖w‖p‖u‖1,p‖∇χh‖q ,

where Ti j is defined as (5.7). Now set v = G Phw; then Rhv = Gh Ph Phw = Gh Phw by (4.24). Hence,in view of (4.14), we have

‖Gh Phw − G Phw‖1,∞ Ch1−d/p‖v‖2,p Ch1−d/p‖w‖p,

which implies

|β ′i j − βi j (wh)| Ch1−d/p‖w‖pmeas (Γi j ) Chd−d/p‖w‖p.

Therefore,

|J22|∑

Γi j ∈Γh

Ch1−d/qTi j

‖∇χh‖Lq (Ti j )‖u‖∞Chd−d/p‖w‖p

Ch1−d/p‖u‖∞‖w‖p

∑T∈Th

hd/p‖∇χh‖Lq (T )

Ch1−d/p‖u‖∞‖w‖p

⎛⎝ ∑

T∈Th

hd

⎞⎠

1/p

‖∇χh‖q

Ch1−d/p‖u‖1,p‖w‖p‖∇χh‖q .

Thus, we obtain (5.10). Before concluding this section, we introduce two operators associated with bh . In view of the second

inequality of (5.6) with p = q = 2, for a fixed wh ∈ Xh , we can define the operator B1h : Xh → Xh as

(vh, B1hχh) = bh(wh, vh, χh) ∀ vh, χh ∈ Xh .

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Then, the operator Bh : Xh → Xh is defined as the adjoint operator of B1h in L2. Thus,

(Bhvh, χh) = bh(wh, vh, χh) ∀ vh, χh ∈ Xh . (5.11)

Below we will write Bh(wh) to express the dependence of Bh on wh .

6. Error estimates (proof of Theorem 2.3)

This section is devoted to the proof of Theorem 2.3. Throughout this section, we suppose that (H1),(H2) and (R) are satisfied, and fix p ∈ (d, µ), θ ∈ (1/2, 1). Then q ∈ (µ/(µ − 1), d/(d − 1)), whereq = p/(p −1). We recall that G is understood as G = G p. We simply write αi = αi,p for i = 0, . . . , 3.Further, the contributions of k and λ are not essential for the discussion below; we take k = λ = 1without loss of generality.

Let u(t) be a solution of (2.1), and let unhn1 be a solution of (2.2), as in Theorem 2.2. First, we

shall derive (2.8). We decompose the error into the form of

u(tn) − unh = ηn

h + wnh (6.1)

for tn < J , where ηnh = ηh(tn), ηh(t) = u(t)−Uh(t), Uh(t) = Rhu(t), wn

h = Unh −un

h and Unh = Uh(tn).

Since, from (4.12) and (4.13), we obtain

‖ηnh‖p + h‖ηn

h‖1,p Ch2α1, (6.2)

it suffices to prove that there exist C0, h0, τ0 > 0 such that

‖wnh‖p C0(h

1−d/p + τσ ) (h ∈ (0, h0), τ ∈ (0, τ0)). (6.3)

To this end, using (2.2), we observe that(∂τn w

nh , χh

)h + (∇wn

h , ∇χh) + (wnh , χh)h

= (∂τn Un

h , χh)h + (∇Un

h , ∇χh) + (Unh , χh)h + bh(un−1

h , unh, χh) − (un

h, χh)h

for any χh ∈ Xh ; equivalently,

∂τn wnh + Ahwn

h = ∂τnUnh + AhUn

h + K −1h Bh(un−1

h )unh − un

h

≡ Fn, (6.4)

where Ah , Kh and Bh(·) are operators defined by (4.22), (4.15) and (5.11), respectively. Thus, putting

r(τ j Ah) = (I + τ j Ah)−1 (r(s) = (1 + s)−1),

En, j = r(τn Ah)r(τn−1 Ah) · · · r(τ j Ah)

for 1 j n, we obtain the following identity:

wnh = En,1w

0h +

n∑j=1

τ j En, j F j . (6.5)

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At this stage, we note that (2.1) is reduced to an evolution equation on L p under (2.5):

du(t)

dt+ Au(t) + B(u(t))u(t) = u(t) (t ∈ (0, J )), u(0) = u0. (6.6)

Here, A = Ap is the operator defined by (4.19). For every w ∈ L p, B(w) : W 1,p → L p is defined byB(w)u = ∇(u∇Gw). This, together with (4.23), gives

K −1h Phu′(t j ) + AhU j

h = K −1h Phu′(t j ) + K −1

h Lh · L−1h Ph Au(t j )

= K −1h Ph[u(t j ) − B(u(t j ))u(t j )],

where u′ = du/dt . Hence, F j is written as

F j = Ph∂τ j (Uj

h − u(t j )) + Ph(∂τ j u(t j ) − u′(t j )

) + (I − K −1h )Phu′(t j )

+ K −1h Ph[u(t j ) − B(u(t j ))u(t j )] + K −1

h Bh(uj−1h )u j

h − u jh .

Therefore, we have

wnh =

4∑i=0

Ii ,

where

I0 = En,1w0h −

n∑j=1

τ j En, j Ph∂τ j ηjh,

I1 =n∑

j=1

τ j En, j (K −1h Phu(t j ) − u j

h),

I2 =n∑

j=1

τ j En, j Ph(∂τ j u(t j ) − u′(t j )

),

I3 =n∑

j=1

τ j En, j (I − K −1h )Phu′(t j ),

I4 =n∑

j=1

τ j En, j K−1h [Bh(u j−1

h )u jh − Ph B(u(t j ))u(t j )].

First, since

I0 = En,1 Ph(u0 − u0h) −n−1∑j=1

En, j+1τ j Ahr(τ j Ah)Phηjh − r(τn Ah)Phηn

h ,

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we have, by (2.7), (4.8), (4.25), (4.28) and (6.2),

‖I0‖h,p 1 · C · ‖u0 − u0h‖p +n∑

j=1

1 · τ j‖Ahr(τ j Ah)‖h,pC‖η jh‖h,p + 1 · C‖ηn

h‖h,p

α0h1−d/p + C(J + 1)h2α1.

To estimate I1, we put u j = u(t j ) and ξ j = τ j (tn − t j−1)−θ . We note that

‖Phu j − u jh‖h,p ‖Phu j − u j‖h,p + ‖u j − Rhu j‖h,p + ‖w j

h‖h,p

Ch2α1 + ‖w jh‖h,p; (6.7)

we also note thatn∑

j=1

ξ j ∫ tn

0(tn − s)−θ ds J 1−θ

1 − θ. (6.8)

These, together with (4.27) and (4.34), give

‖I1‖h,p n∑

j=1

τ j‖En, j Aθh‖h,p‖A−θ

h (K −1h Phu j − u j

h)‖h,p

Cn∑

j=1

ξ j (‖A−θh (K −1

h − I )Phu j‖h,p + ‖A−θh (Phu j − u j

h)‖h,p)

Cn∑

j=1

ξ j (h2α1 + ‖w j

h‖h,p)

Cα1 J 1−θh2 + Cn∑

j=1

ξ j‖w jh‖h,p.

We estimate I2 as follows:

‖I2‖h,p =∥∥∥∥∥∥

n∑j=1

En, j

∫ tn

t j−1

Ph[u′(s) − u′(t j−1)]ds

∥∥∥∥∥∥h,p

n∑

j=1

‖En, j‖h,p

∫ tn

t j−1

C‖u′(s) − u′(t j−1)‖pds

n∑

j=1

1 · C · τσj α3 C Jτσ α3.

Similarly, as in the case of I1, we obtain

‖I3‖h,p C J 1−θh2α2.

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We proceed to the estimation of I4. If it can be shown that

‖A−θh K −1

h [Bh(u j−1h )u j

h − Ph B(u j )u j ]‖h,p Cα21h1−d/p + Cτα1α2

+ C(1 + h2)α1‖w j−1h ‖h,p + Cα1‖w j

h‖h,p + C‖w j−1h ‖h,p‖w j

h‖h,p, (6.9)

we have

‖I4‖h,p C(α21 + α2

2)(h1−d/p + τ)J 1−θ + Cα1

n∑j=1

ξ j‖w j−1h ‖h,p

+ C(1 + α1)

n∑j=1

ξ j (1 + ‖w j−1h ‖h,p)‖w j

h‖h,p (6.10)

in the same manner as for the estimation of I1. In order to verify (6.9), it suffices to show that

|(Bh(u j−1h )u j

h − B(u j )u j , χh)| [Cα21h1−d/p + Cτα1α2

+ C(1 + h2)α1‖w j−1h ‖h,p + Cα1‖w j

h‖h,p

+ C‖w j−1h ‖h,p‖w j

h‖h,p]‖∇χh‖q (6.11)

for χh ∈ Xh . Indeed, by virtue of (4.32),

‖A−θh K −1

h [Bh(u j−1h )u j

h − Ph B(u j )u j ]‖h,p

C supχh∈Xh

(A−θh K −1

h [Bh(uj−1h )u j

h − Ph B(u j )u j ], χh)h

‖χh‖h,q

C supχh∈Xh

(Bh(uj−1h )u j

h − B(u j )u j , A−θh χh)

‖χh‖q,

which, together with (6.11) and (4.30), implies (6.9). Consequently, (6.9) is reduced to (6.11). To prove(6.11), letting χh ∈ Xh , we write it as

(B(u j )u j − Bh(u j−1h )u j

h, χh) = b(u j , u j , χh) − bh(u j−1h , u j

h, χh)

= b(u j , u j , χh) − b(Phu j , u j , χh)

+ b(Phu j , u j , χh) − bh(Phu j , πhu j , χh)

+ bh(Phu j , πhu j , χh) − bh(Phu j−1, πhu j , χh)

+ bh(Phu j−1, πhu j , χh) − bh(uj−1h , πhu j , χh)

+ bh(u j−1h , πhu j , χh) − bh(u

j−1h , u j

h, χh)

≡ J1 + J2 + J3 + J4 + J5.

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We, respectively, obtain, using Lemmas 5.1 and 5.4,

|J1| Ch2α21‖∇χh‖q , |J2| Ch1−d/pα2

1‖∇χh‖q .

Furthermore, by virtue of Lemma 5.3 and (6.7), we can deduce the following bounds:

|J3|C‖Ph(u j − u j−1)‖p‖πhu j‖p‖∇χh‖q

Cτ j supt∈[0,J ]

‖u′(t)‖p‖u j‖∞‖∇χh‖q Cτα1α2‖∇χh‖q ,

|J4|C‖Phu j−1 − u j−1h ‖p‖πhu j‖p‖∇χh‖q

C(h2‖u j−1‖2,p + ‖w j−1h ‖h,p)‖u j‖∞‖∇χh‖q

C(h2α1 + ‖w j−1h ‖h,p)α1‖∇χh‖q

and

|J5|C‖u j−1h ‖p‖πhu j − u j

h‖p‖∇χh‖q

C(‖w j−1h ‖h,p + ‖u j−1‖1,p)(h

2‖u j‖2,p + ‖w jh‖h,p)‖∇χh‖q

C(h2α21 + h2α1‖w j−1

h ‖h,p + α1‖w jh‖h,p + ‖w j−1

h ‖h,p‖w jh‖h,p)‖∇χh‖q .

Consequently, we obtain (6.11); hence (6.10) is verified.Now we can complete the proof of Theorem 2.3. Summing the estimates for I0, . . . , I4, we obtain

‖wnh‖h,p C3(h

1−d/p + τσ ) + Cα1

n∑j=1

ξ j‖w j−1h ‖h,p

+ C(1 + α1)

n∑j=1

ξ j (1 + ‖w j−1h ‖h,p)‖w j

h‖h,p, (6.12)

where C3 = C(J +1)(α21 +α2

2 +α3). We define z j = ‖wjh‖h,p and Zn = sup0 jn z j . First we assume

that

Zn−1 1.

Thereby, (6.12) implies that

zn C3(h1−d/p + τσ ) + C4

n∑j=1

ξ j (z j−1 + z j ),

with C4 = C(1 + α1). Hence, by Lemma 4.7,

zn C5(h1−d/p + τσ ) exp(C6 J ) ≡ z,

where C5 = CC3, C6 = CC24 and C is the absolute positive constant. Since C5 and C6 are independent

of h, τ and n, we can choose sufficiently small h1 > 0 and τ0 > 0 such that z 1 for h ∈ (0, h1) and

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τ ∈ (0, τ0). On the other hand, we choose a sufficiently small h2 > 0 such that z0 C‖u0 −u0h‖p 1for h ∈ (0, h2). At this stage, we set h0 = min(h1, h2). Then, since h ∈ (0, h0) and τ ∈ (0, τ0), wehave Zn 1 for all n 0 such that tn < J by induction. In conclusion, we have

‖wnh‖p C‖wn

h‖h,p C[C5 exp(C6 J )](h1−d/p + τσ ),

which implies (6.3), thereby establishing (2.8).We proceed to the proof of (2.9). We put v(t) = Gu(t), vn

h = Ghunh and wn

h = Rhv(tn) − vnh . Then,

by virtue of (2.5) and (4.14),

‖v(tn) − vnh‖1,∞ ‖v(tn) − Rhv(tn)‖1,∞ + ‖wn

h‖1,∞

Ch1−d/p‖u(tn)‖p + ‖wnh‖1,∞. (6.13)

In order to derive an estimate of wnh , we observe that

(∇wnh , ∇χh) + (wn

h , χh) = (Ph(u(tn) − unh), χh) ∀ χh ∈ Xh,

or equivalently,

Lhwnh = Ph(u(tn) − un

h).

Hence, in view of (4.31) and (4.8),

‖wnh‖1,∞ C‖Lhwn

h‖p C‖u(tn) − unh‖p. (6.14)

Therefore, (2.9) follows from (2.8) and (6.13). This completes the proof of Theorem 2.3.

7. Remarks on error estimates of the order h

In this section, we suppose that Ω ⊂ Rd is a bounded domain with a sufficiently smooth boundary ∂Ω .

Then, (1.1) admits a unique classical solution satisfying

u,∂u

∂xi,

∂2u

∂xi x j, ut , v,

∂v

∂xi,

∂2v

∂xi x j∈ C(Ω × [0, J ])

and (2.6) holds true for all p > 0 and σ = 1. Moreover, (R) is satisfied for any µ > 0 (see Suzuki &Senba, 2004; Suzuki, 2005).

We take a regular family of (curved) triangulations Thh , which ‘exactly fit the boundary’:

Ω =⋃

T∈Th

T .

The definitions of Xh , φi , Di , etc. are similar to those before with obvious modifications (see, e.g.Baba & Tabata, 1981). Under these assumptions, Schatz (1998) proved that

‖Rhv − v‖1,∞ Ch‖v‖2,∞ (v ∈ W 2,∞). (7.1)

LEMMA 7.1 Suppose that (H2) is satisfied. Let p ∈ (d, ∞) with 1p + 1

q = 1. Then, under the assump-tions described above,

|b(Phw, u, χh) − bh(Phw,πhu, χh)| Ch(‖w‖2,p + ‖Gw‖2,∞)‖u‖1,p‖∇χh‖q (7.2)

for w ∈ W 2,p, u ∈ W 1,p and χh ∈ Xh .

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Proof. Let w ∈ W 2,p, u ∈ W 1,p, χh ∈ Xh . We follow the notation of the proof of Lemma 5.4. Wenote first that Gw ∈ C2(Ω) follows from w ∈ C(Ω). We introduce J1, J21 and J22, as in the proof ofLemma 5.4. It suffices to derive the estimate for J22. To this end, we decompose it into

J22 =∑

Γi j ∈Γh


i j u j )(β′i j − βi j )

+∑

Γi j ∈Γh


i j u j )(βi j − βi j (wh))

≡ J ′22 + J ′′

22,

where

βi j =∫

Γi j

∇Gw · νi j dS.

Put v = Gw; thereby, Rhv = Gh Phw. By virtue of (7.1), we obtain

|βi j − βi j (wh)| Chd‖w‖2,∞;therefore,

|J ′′22| Ch‖Gw‖2,∞‖u‖1,p‖∇χh‖q .

On the other hand,

|β ′i j − βi j |Chd−1

Ti j‖∇G(w − Phw)‖∞

Chd−1Ti j

‖G(w − Phw)‖2,p

Chd−1Ti j

‖w − Phw‖p

Chd−1Ti j

h2‖w‖2,p.

Hence,

|J ′22| Ch2‖w‖2,p‖u‖1,p‖∇χh‖q .

Summing up these estimates, we obtain (7.2). With the aid of this lemma, we can derive the following error estimates:

sup0nl

‖u(tn) − unh‖p C1(h + τ), (7.3)

sup0nl

‖Gu(tn) − Ghunh‖1,∞ C2(h + τ) (7.4)

for h ∈ (0, h0) and τ ∈ (0, τ0); this can be achieved similarly to the proof of Theorem 2.3. Here, u(t)is a solution of (2.1), un

hln=0 ⊂ Xh is a solution of (2.2) as in Theorem 2.2 and u0h ∈ Xh has been

chosen as

‖u0 − u0h‖p α0,ph

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with some α0,p = α0,p(u0) > 0. Furthermore, Ci can be taken as

Ci = C(J + 1)

(α0,p + α2

1,p + α22,p + α3,p + sup

t∈[0,J ]‖v(t)‖2

2,∞

)exp[C ′(1 + α2

1,p)J ],

where C , C ′ are positive constants that depend only on Ω , k, λ, γi ’s, h0 and τ0.However, the exactly fitted triangulation is unrealistic in practical computations. Now we consider

approximations of the domain Ω . Let us introduce an approximate polyhedron Ωh for Ω such that thevertices of ∂Ωh are situated on ∂Ω and assume that there are h1 > 0 and c0 > 0 such that

dist(Ω, ∂Ωh) c0h2 (h ∈ (0, h1)). (7.5)

In general, Ωh ⊂ Ω . Let Th = Thh↓0 be a regular family of triangulations Th of Ωh introduced inSection 2. In this case, Rh : H1(Ωh) → Xh is defined by∫

Ωh

∇(Rhv − v) · ∇χh dx +∫

Ωh

(Rhv − v)χh dx = 0 ∀ χh ∈ Xh .

Let E be a total extension operator for Ω (cf. Adams & Fournier, 2003). That is, for any m ∈ N ∪ 0and p ∈ [1, ∞), E is an operator of W m,p(Ω) → W m,p(Rd) satisfying

(E u)(x) = u(x) (a.e. x ∈ Ω), ‖E u‖W m,p(Rd ) Cl,p‖u‖W m,p(Ω).

Such a total extension operator extends functions in Cm(Ω) to lie in Cm(Rd). Then, if

‖RhE v‖W 1,∞(Ωh) C‖E v‖W 1,∞(Rd ) (v ∈ C1(Ω)), (7.6)

‖RhE v − E v‖W 1,∞(Ωh) Ch‖E v‖W 2,∞(Rd ) (v ∈ C2(Ω)), (7.7)

we can obtain (7.3) and (7.4) in a similar fashion to that described above. However, unfortunately, it isnot obvious that these estimates hold true. They are left here as relevant open problems.

REMARK 7.1 Consider the Dirichlet boundary value problem for the Poisson equation −v = f in Ωand v = 0 on ∂Ω . If Ω is convex, then we actually have (7.6) and (7.7) (see Bakaev et al., 2003).

8. Numerical examples

We assume that Ω ⊂ R2 is a unit square: Ω = (0, 1)2. We take Th as a uniform mesh composed of 22

congruent right-angle triangles for ∈ N; each side of Ω is divided into intervals of the same length.Then each small square is decomposed into two equal triangles by a diagonal. We define h = √

2−1,and we take k = λ = 1 and ε = 0.9.

Figure 1 shows the results of a numerical experiment at several time levels tn with the value of ‖unh‖1.

Therein, the L1 norm of the initial value u0h is taken as ‖u0h‖1 = 50 > 8π so that the correspondingsolution u may blow up.

We can observe that a highly concentrated solution is captured successfully. Moreover, the L1 normis actually preserved. Next, we plot (tn, τn) in Fig. 2. Comparison of Figs 1 and 2 illustrates that themore the approximate solution is concentrated, the smaller τn becomes.

Finally, we consider the rate of convergence. Initial values are taken as ‖u0h‖1 = 10, 20, 50, andthe shape of u0h is illustrated in Fig. 1(i). (Note that 10, 20 < 8π .) We take τ = 10−6 and = 8, 16,

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FIG. 1. Shape of unh (λ = k = 1; ε = 0.9; τ = h/2; = 64): (i) the initial function has three peaks; (ii, iii, iv) they gather and

produce a single peak; (v, vi) the peak moves towards a corner.

32, 64, 128. Because τ/h 1, we can infer that the effect of time discretization is negligible. The exactsolution of (2.1) is not available; therefore, we employ the following technique. We define

ep(h) = sup0tnJ

‖unh − un

h/2‖p

with J = 0.003. We assume that

sup0tnJ

‖u(tn) − unh‖p = Chθ ,

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FIG. 2. Time tn versus time increment τn . (J = 0.09; k = λ = 1; ε = 0.9; τ = h/2; = 64; ‖u0h‖1 = 50.0 and u0h is illustratedin Fig. 1.)

FIG. 3. log h versus log ep(h). (J = 0.003; k = λ = 1; ε = 0.9; τ = 10−6; = 16, 32, 64, 128.) A3: γ = 10, p = 3;A4: γ = 10, p = 4; B3: γ = 20, p = 3; B4: γ = 20, p = 4; C3: γ = 50, p = 3; C4: γ = 50, p = 4, where γ = ‖u0h‖1 andthe shape of u0h is illustrated in Fig. 1(i).

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where C and θ denote positive constants. Then ep(h) C(1 + 2−θ )hθ . In Fig. 3, we plot (log h,log ep(h)) for p = 3, 4; we see from the figure that θ ≈ 1. We note that, in the present situation,(R) with µ = ∞ and (7.1) are satisfied by the Fourier series. Hence, the numerical result supports theargument of Section 6, in particular, the error estimate (7.3). We deduce from this observation that theoptimal rate of convergence is actually achieved if u and v are sufficiently regular.

Acknowledgements

I thank Prof. T. Suzuki who brought the subject to my attention and encouraged me through valuablediscussions. This work is supported by Grants-in-Aid for Scientific Research (No. 16740050), the JapanSociety of the Promotion of Science.

REFERENCES

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Appendix

A. Proof of Lemma 4.4

We require some auxiliary results. The first one is a version of the Heinz–Krein inequality on Banachspaces. The second one concerns the dual norm on W 1,p.

LEMMA A.1 Let A and B, respectively, represent operators of type (ω, M) and (ω′, M ′) on Banachspaces X and Y with bounded inverses (See, e.g. Krein, 1972; Tanabe, 1979). Let P be a boundedlinear operator from X into Y . Suppose that PD(A) ⊂ D(B) and that ‖B Pu‖Y C‖Au‖X for allu ∈ D(A). Then, for 0 α < β 1, we have PD(Aβ) ⊂ D(Bα) and ‖Bα Pu‖Y C K‖Aβu‖X forall u ∈ D(Aβ), where C = C(α, β, M, M ′) is positive constant and K = max1, ‖P‖1−α

X,Y .Proof. The proof resembles that of the proof of Lemma 1-7.3 of Krein (1972), where the case of Y = Xand P = I was stated explicitly. LEMMA A.2 Suppose that (R) is satisfied. Let p, q ∈ (1, µ) such that 1

p + 1q = 1. Then,

‖v‖1,p C supχ∈W 1,q

|(∇v, ∇χ) + (v, χ)|‖χ‖1,q

(A.1)

for any v ∈ W 1,p.

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364 N. SAITO

Proof. Note that (R) guarantees

‖u‖2,p C‖Apu‖p (u ∈ D(Ap)),

‖u‖2,q C‖Aqu‖q (u ∈ D(Aq)),

where Ap and Aq are operators defined as (4.19). Since A∗p = Aq , we can apply Theorem 3.1 of

Schechter (1963) (s = 1, m = 2) and obtain

‖v‖1,p C supχ∈W 1,q

|(Apv, χ)|‖χ‖1,q

= C supχ∈W 1,q

|(∇v, ∇χ) + (v, χ)|‖χ‖1,q

for any v ∈ D(Ap). Hence, by a density argument, we establish (A.1). REMARK A.1 When considering a smooth domain, we can obtain (A.1) for p ∈ (1, ∞) in a similarmanner as in Simader (1972).

Now we can state the following.Proof of Lemma 4.4. Let p, q ∈ (1, ∞) such that 1

p + 1q = 1. We initially note that

‖Phv − v‖1,p Ch‖Av‖p (v ∈ D(A) = Wp), (A.2)

where A = Ap. In fact, since Phv − Rhv = Ph(v − Rhv),

‖Phv − v‖1,p ‖Phv − Rhv‖1,p + ‖Rhv − v‖1,p

C‖Rhv − v‖1,p Ch‖v‖2,p

by (4.4) and (4.9), which, together with (2.5), imply (A.2). Let v ∈ D(A) and χh ∈ Xh . Then, accordingto (4.2) and (4.25), we have

|(Ah Phv, χh)h | = |(∇ Phv, ∇χh) + (Phv, χh)h | |(∇(Phv − v), ∇χh) + (Phv − v, χh)|

+ |(∇v, ∇χh) + (v, χh)| + |(Phv, χh)h − (Phv, χh)| ‖Phv − v‖1,p‖χh‖1,q + |(Av, χh)| + Ch2‖Phv‖1,p‖χh‖1,q

Ch‖Av‖p · Ch−1‖χh‖q + ‖Av‖p‖χh‖q + Ch2‖Av‖p · Ch−1‖χh‖q

C‖Av‖p‖χh‖h,q ;hence,

‖Ah Phv‖h,p C‖Av‖p.

We can apply Lemma A.1 and obtain, for 0 α < β 1,

‖Aαh Phv‖h,p C‖Aβv‖p (v ∈ D(Aβ)). (A.3)

Now let vh ∈ Xh and θ ∈ (1/2, 1]. Then, by (A.3) and (4.20), we have

‖A1−θh vh‖h,p C‖A1/2vh‖p C‖vh‖1,p. (A.4)

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On the other hand, from (A.1), (4.7) and (4.11), we have

‖vh‖1,p C supχ∈W 1,q

|(∇vh, ∇χ) + (vh, χ)|‖χ‖1,q

C supχ∈W 1,q

|(∇vh, ∇Rhχ) + (vh, Rhχ)|‖Rhχ‖1,q

C supχh∈Xh

|(∇vh, ∇χh) + (vh, χh)|‖χh‖1,q

.

Therefore, by virtue of (4.35) and (A.4),

‖vh‖1,p C supχh∈Xh

|(∇vh, ∇χh) + (vh, χh)h |‖χh‖1,q

+ |(vh, χh) − (vh, χh)h |‖χh‖1,q

C supχh∈Xh

|(Ahvh, χh)h |

‖χh‖1,q+ Ch2‖∇vh‖p‖∇χh‖q

‖χh‖1,q

C supχh∈Xh

|(Aθhvh, A1−θ

h χh)h |‖χh‖1,q

+ Ch2‖vh‖1,p

C‖Aθhvh‖h,p sup

χh∈Xh

‖A1−θh χh‖h,q

‖χh‖1,q+ Ch2‖vh‖1,p

C‖Aθhvh‖h,p + Ch2‖vh‖1,p.

Consequently, we have as h ↓ 0,

‖vh‖1,p C‖Aθhvh‖h,p,

which completes the proof.

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