APPLIED MATHEMATICS REPORTAMR02/8

A COMPLETE ANALYSIS FOR SOMEA POSTERIORI ERROR ESTIMATES WITHTHE FINITE ELEMENT METHOD OF LINESFOR A NONLINEAR PARABOLIC EQUATION

T. Tran and T-B. Duong

August, 2002

A COMPLETE ANALYSIS FOR SOME A

POSTERIORI ERROR ESTIMATES WITH THE

FINITE ELEMENT METHOD OF LINES FOR A

NONLINEAR PARABOLIC EQUATION

Thanh Tran ∗ Thanh-Binh Duong ∗

ABSTRACT

A posteriori error estimates for semidiscrete finite element methods for a non-linear parabolic initial-boundary value problem are considered. The error estimatesare obtained by solving local parabolic or elliptic equations for corrections to thesolution on each element. The convergence results improve previous results whereunnecessary assumptions are imposed on the approximate solution and the ellipticprojection of the exact solution.

AMS Subject Classification (2000): 65M15, 65M20, 65M60

Key words: nonlinear parabolic equations, finite element, method of lines, a pos-teriori error estimates

1 INTRODUCTION

In this paper, we give a complete analysis of some a posteriori error estimates for semidis-crete finite element methods for a nonlinear parabolic initial-boundary value problem,which were proposed in [11] and not thoroughly analysed before.

A posteriori error estimates are a fundamental component in the design of reliable andefficient adaptive algorithms for solving parabolic equations [1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13,14]. One of the most common strategies for constructing such estimates is p-refinement,in which error estimates are obtained by using a piecewise polynomial approximationhaving a higher degree than that used for the solution. Superconvergence properties ofparabolic equations permit errors at element vertices or edges to be neglected so that theerror estimates can be computed on each element as a correction to the original solution.Therefore, the error estimates are computed by solving local parabolic or elliptic problems,the solution costs of which are rather low. Moreover, when the method of lines is used tosolve the parabolic equations, the set of local elliptic problems used to compute the error

∗School of Mathematics, The University of New South Wales, Sydney 2052, AustraliaEmail: thanh.tran@unsw.edu.au tbduong@maths.unsw.edu.au

1

estimates need not be solved for each time step, but only when needed. This results in asignificant reduction in computational overhead.

Several experiments have been carried out to show the efficiency of this approach tolinear and nonlinear parabolic equations [1, 2, 3, 4, 11]. Convergence of the error estimatesto the true error of a semidiscrete method has been shown for linear equations in [4, 7, 13],and for semilinear equations in [11]. The analysis has also been done in [11] for some fullydiscrete methods (namely SIRK and BDF methods) for nonlinear equations. The work in[14] extends the result of [11] to the case of a semidiscrete method for nonlinear equations,thus provides a basis for an adaptive numerical procedure that carries out the fully discretecomputation with an arbitrary time discretisation.

The analysis in [14] requires some crucial assumptions which are not only hard toverify in general but also fail to hold in many cases. As an example, let Uh and uh be,respectively, the semidiscrete approximation and elliptic projection of the exact solution(see the definitions in (2.10)–(2.11) and (4.1)–(4.2), respectively), and let θ be their differ-ence, θ := uh − Uh. It is required in [14] that the function t 7→ ‖θ(t)‖0 be nondecreasing,where ‖·‖0 denotes the norm in L2. This assumption does not necessarily hold in gen-eral, even for linear problems; see the paragraph after Lemma 4.2 on page 7. The sameassumption is also imposed on the function t 7→ ‖η(t)‖0, where, roughly speaking, η is ananalogue to θ on the space of functions whose restrictions to each subinterval are multiplesof the antiderivative of the Legendre polynomial of degree p, scaled to that subinterval(see definition (4.37)). It is even harder to check this assumption in the general case.

In the present paper, we improve the analysis in [14] and prove the convergence of theerror estimates to the true error without the monotonicity assumptions mentioned above.This guarantees the applicability of the estimates for adaptive methods for nonlinearequations. The main tool we employ that makes our analysis different to that in [14] is aSobolev imbedding theorem; see Lemma 4.5.

In section 2 we state the model problem to be considered and present the spatialdiscretisation scheme. Strategies to approximate the semidiscrete error are designed inSection 3. The proof of the main result of the paper is carried out in section 4. Someconcluding remarks (section 5) finish the paper. An earlier version of this paper appearedas a research report [16] of the School of Mathematical Sciences, ANU.

In the paper, c denotes a generic constant which may take different values at differentoccurrences.

2 MODEL PROBLEM AND SPATIAL DISCRETI-

SATION

Let Ω := (0, 1) and T > 0 be fixed, and consider the nonlinear problem

∂tu(x, t)− ∂x(a(u)∂xu(x, t)) + f(u) = 0, x ∈ Ω, t ∈ (0, T ], (2.1)

u(0, t) = u(1, t) = 0, t ∈ [0, T ], (2.2)

u(x, 0) = u0(x), x ∈ Ω, (2.3)

2

where ∂t := ∂/∂t, ∂x := ∂/∂x, u0 is a given smooth function, and a and f are smoothfunctions from R into R satisfying, with positive constants µ, M , and L,

0 < µ ≤ a(s) ≤M ∀s ∈ R, (2.4)

|a(s)− a(r)| ≤ L|s− r| ∀s, r ∈ R, (2.5)

|f(s)− f(r)| ≤ L|s− r| ∀s, r ∈ R. (2.6)

Denoting by 〈·, ·〉 the inner product in L2(Ω), we define the weak solution of (2.1)–(2.3)as: u ∈ H1(0, T ;H1

0(Ω)) satisfying, for t ∈ (0, T ],

〈∂tu, v〉+ 〈a(u)∂xu, ∂xv〉+ 〈f(u), v〉 = 0 ∀v ∈ H10 (Ω), (2.7)

and, for t = 0,

〈a(u0)∂xu, ∂xv〉 = 〈a(u0)∂xu0, ∂xv〉 ∀v ∈ H10 (Ω). (2.8)

For the definition of H1(0, T ;H10(Ω)) we refer to [9] or [10].

In the sequel, we will denote by ‖·‖k the norm in the Sobolev space Hk(Ω), and by|||·|||∞ the norm in L∞(Ω× (0, T )).

We use the finite element method to approximate the weak solution (2.7)–(2.8) by acontinuous piecewise polynomial function of degree p ≥ 1 as follows. Introduce first apartition of Ω:

0 = x0 < x1 < · · · < xN = 1,

and put hj := xj − xj−1, j = 1, . . . , N , and h := maxj hj. We suppose that there existpositive constants τ1 and τ2 such that

τ1 ≤ hj

hj+1

≤ τ2, j = 1, . . . , N − 1.

There are adaptive procedures where refinement criteria satisfy this condition on themeshes; see [11].

Let φj1 be the hat function on (xj−1, xj+1), and φjk be an antiderivative of the Legendrepolynomial Pk−1 of degree k − 1 scaled to the subinterval [xj−1, xj ], i.e.

φj1(x) =

(x− xj−1)/hj, xj−1 ≤ x < xj ,

(xj+1 − x)/hj+1, xj ≤ x < xj+1,

0, otherwise,

j = 1, . . . , N − 1, and

φjk(x) =

√2(2k−1)

hj

∫ x

xj−1Pk−1(y) dy, xj−1 ≤ x < xj ,

0, otherwise,(2.9)

3

j = 1, . . . , N and k = 2, . . . , p. We construct a finite dimensional subspace SN,p0 of H1

0 (Ω)as the set of functions v having the form

v(x) =N−1∑j=1

vj1φj1(x) +N∑

j=1

p∑k=2

vjkφjk(x).

A function Uh ∈ H1(0, T ;SN,p0 ), which has the form

Uh(x, t) =

N−1∑j=1

Uh,j1(t)φj1(x) +

N∑j=1

p∑k=2

Uh,jk(t)φjk(x),

is called a semidiscrete approximate solution to the exact solution u if, for t ∈ (0, T ],

〈∂tUh, v〉+ 〈a(Uh)∂xUh, ∂xv〉+ 〈f(Uh), v〉 = 0 ∀v ∈ SN,p0 , (2.10)

and, for t = 0,

〈a(u0)∂xUh, ∂xv〉 = 〈a(u0)∂xu0, ∂xv〉 ∀v ∈ SN,p0 . (2.11)

3 APPROXIMATION OF THE SEMIDISCRETE ER-

ROR

Let

e(x, t) := u(x, t)− Uh(x, t) (3.1)

be the error in the semidiscrete approximation. From (2.7), (2.8), and (3.1) we infer thate(x, t) satisfies, for t ∈ (0, T ],

〈∂te, v〉+ 〈a(Uh + e)∂xe, ∂xv〉 = −〈f(Uh + e), v〉 − 〈∂tUh, v〉− 〈a(Uh + e)∂xUh, ∂xv〉 ∀v ∈ H1

0 (Ω), (3.2)

and, for t = 0,

〈a(u0)∂xe, ∂xv〉 = 〈a(u0)∂x(u0 − Uh), ∂xv〉 ∀v ∈ H10 (Ω). (3.3)

As in [11], we will approximate the true error e by Eh having the form

Eh(x, t) :=

N∑j=1

Eh,j(t)φj,p+1(x), (3.4)

where φj,p+1 is defined by (2.9) with k = p + 1. Let

SN,p+10 := span φ1,p+1, . . . , φN,p+1.

4

Since supp φj,p+1 = [xj−1, xj ], if ψ ∈ L2(Ω) then the statement

〈ψ, v〉 = 0 ∀v ∈ SN,p+10

is equivalent to

〈ψ, v〉j = 0 ∀v ∈ SN,p+10 , j = 1, . . . , N,

where the local inner product 〈v, w〉j is defined by

〈v, w〉j :=

∫ xj

xj−1

v(x)w(x) dx.

As in [11] and [14], Eh is defined in several ways, with the help of (3.2) and (3.3), asfollows.

Nonlinear parabolic error estimate: Let Eh of the form (3.4) be defined in each subin-terval (xj−1, xj), j = 1, . . . , N , by

⟨∂tEh, v

⟩j+

⟨a(Uh + Eh)∂xEh, ∂xv

⟩j

= −⟨f(Uh + Eh), v

⟩j− 〈∂tUh, v〉j −

⟨a(Uh + Eh)∂xUh, ∂xv

⟩j

∀v ∈ SN,p+10 , (3.5)

when t ∈ (0, T ], and⟨a(u0)∂xEh, ∂xv

⟩j

= 〈a(u0)∂x(u0 − Uh), ∂xv〉j ∀v ∈ SN,p+10 , (3.6)

when t = 0. The problem (3.5)–(3.6) is a set of N uncoupled local parabolic problems,therefore the solution costs are rather low.

Nonlinear elliptic error estimate: To further the saving of computational costs, we

neglect the time change in the error estimate, and define Eh by replacing (3.5) by

⟨a(Uh + Eh)∂xEh, ∂xv

⟩j

= −⟨f(Uh + Eh), v

⟩j− 〈∂tUh, v〉j −

⟨a(Uh + Eh)∂xUh, ∂xv

⟩j

∀v ∈ SN,p+10 . (3.7)

The set (3.7)–(3.6) now represents N uncoupled local elliptic problems, which need notbe solved for each t, but only when needed.

Linear parabolic error estimate: Additional savings can be performed by neglecting

the term Eh in a(Uh + Eh) and f(Uh + Eh), thereby reducing the nonlinear equations (3.5)by the linear equation

⟨∂tEh, v

⟩j+

⟨a(Uh)∂xEh, ∂xv

⟩j

= −〈f(Uh), v〉j − 〈∂tUh, v〉j − 〈a(Uh)∂xUh, ∂xv〉j ∀v ∈ SN,p+10 . (3.8)

5

Linear elliptic error estimate: Similar procedure can be applied to (3.7) to obtain

⟨a(Uh)∂xEh, ∂xv

⟩j

= −〈f(Uh), v〉j − 〈∂tUh, v〉j − 〈a(Uh)∂xUh, ∂xv〉j ∀v ∈ SN,p+10 . (3.9)

Note that in all cases the same initial condition (3.6) is imposed.Introduce the effectivity index

Θ(t) :=

∥∥∥Eh(t)∥∥∥

1

‖e(t)‖1

.

In the following section, we will prove the main result of the paper, namely we will provethat

limh→0

Θ(t) = 1.

4 THE MAIN RESULT

In order to state and prove the main result of the paper, we need the following notionsand lemmas.

As in [15], the analysis requires an elliptic projection of the exact solution, which isdefined as: uh ∈ H1(0, T ;SN,p

0 ) satisfying, for t ∈ (0, T ],

〈a(u)∂xuh, ∂xv〉 = 〈a(u)∂xu, ∂xv〉 ∀v ∈ SN,p0 , (4.1)

and, for t = 0,

〈a(u0)∂xuh, ∂xv〉 = 〈a(u0)∂xu0, ∂xv〉 ∀v ∈ SN,p0 . (4.2)

Introducing the notation

ρ(x, t) := u(x, t)− uh(x, t), (4.3)

we have the following lemma:

Lemma 4.1 If u ∈ H1(0, T ;Hp+1(Ω) ∩H10 (Ω)) then

‖ρ(t)‖0 + ‖∂tρ(t)‖0 + h ‖∂xρ(t)‖0 + h ‖∂txρ(t)‖0 ≤ c (u)hp+1,

and

|||∂xuh|||∞ ≤ c (u). (4.4)

Moreover, if ∂ttρ(·, t) and ∂tttρ(·, t) belong to L2(Ω), then

‖∂ttρ(t)‖0 + ‖∂tttρ(t)‖0 ≤ c (u)hp+1.

6

Proof. See [15, Lemmas 13.2 and 13.3]. 2

Recalling the definition of e and ρ (see (3.1) and (4.3)), we can write

e(x, t) = ρ(x, t) + θ(x, t),

where

θ(x, t) := uh(x, t)− Uh(x, t). (4.5)

The following result was proved in [15]:

Lemma 4.2 If u ∈ H1(0, T ;Hp+1(Ω) ∩ H10 (Ω)) and Uh ∈ H1(0, T ;SN,p

0 ), then for t ∈(0, T ] there holds

‖e(t)‖1 ≤ c(u)hp.

As in the case of linear equations, it will be seen that Uh is a better approximation touh than to u, in the sense that

‖θ(t)‖1 ≤ c(u)hp+1. (4.6)

This phenomenon is referred to as superconvergence in [15], and plays a key role in ouranalysis. The proof of (4.6) was first carried out in [14] under the assumption that thefunction t 7→ ‖θ(t)‖0 is nondecreasing. Unfortunately, this assumption fails to be satisfiedin many cases, even in the case of linear problems. In fact, it was proved in [15, page 9]that

‖θ(t)‖0 ≤ e−λ1t ‖u0,h − u0‖0 + ch

(e−λ1t ‖u0‖1 +

∫ t

0

e−λ1(t−s) ‖∂tu(s)‖1 ds

),

where u0,h ∈ SN,p0 is some approximation to u0, and λ1 is the smallest eigenvalue of

−∆, with Dirichlet boundary data. (This result was proved for linear problems, butsimilar result holds for the nonlinear problem considered in this paper.) Therefore, byusing Lebesgue’s Dominated Convergence Theorem, one can prove that if the mappingt 7→ ‖∂tu(t)‖1 belongs to L1(0,∞), then limt→∞ ‖θ(t)‖0 = 0, and this contradicts theaforementioned assumption. Moreover, if ‖∂tu(t)‖1 approaches zero exponentially as t→∞, then so does ‖θ(t)‖0. An example that ‖∂tu(t)‖1 approaches zero exponentially ast→∞ is when u satisfies the equation

∂tu−∆u = f,

and conditions (2.2) and (2.3), where f is a function depending on x only. By using thesuperposition principle and method of separation of variables, we can prove that

u(x, t) = v(x, t) + w(x),

7

where

w(x) = −∫ x

0

∫ y

0

f(s) ds dy + x

∫ 1

0

∫ y

0

f(s) ds dy,

and

v(x, t) =

∞∑n=1

Ane−n2π2t sinnπx,

with

An = 2

∫ 1

0

[u0(x)− w(x)] sinnπx dx.

It is then clear that

‖∂tu(t)‖1 ≤ ce−αt for some α > 0 and for large t.

In the following, we will prove the superconvergence property (4.6) without the as-sumption on the monotonicity of t 7→ ‖θ(t)‖0. The proof will be carried out by proving thenext three lemmas. The result in the following lemma is weaker than that of [14, Lemma3.3]. This is due to the relief of the assumption on monotonicity mentioned above.

Lemma 4.3 For any t > 0, there holds

‖θ(t)‖20 +

∫ t

0

‖∂xθ(s)‖20 ds ≤ c

(h2p+2 + ‖θ(0)‖2

0

).

Proof. It follows from (2.7), (2.10), and (4.1) that

〈∂tθ, v〉+ 〈a(Uh)∂xθ, ∂xv〉 = −〈∂tρ, v〉+ 〈f(Uh)− f(u), v〉+ 〈(a(Uh)− a(u))∂xuh, ∂xv〉 ∀v ∈ SN,p

0 . (4.7)

Letting v = θ in the above equation, and using the formula

1

2

d

dt‖θ(t)‖2

0 = 〈∂tθ, θ〉 ,

we obtain

1

2

d

dt‖θ(t)‖2

0 + 〈a(Uh)∂xθ, ∂xθ〉 = −〈∂tρ, θ〉+ 〈f(Uh)− f(u), θ〉+ 〈(a(Uh)− a(u))∂xuh, ∂xθ〉 . (4.8)

Assumption (2.4) yields

µ ‖∂xθ(t)‖20 ≤ 〈a(Uh)∂xθ, ∂xθ〉 . (4.9)

8

There also holds

| 〈∂tρ, θ〉 | ≤ 12 ‖∂tρ(t)‖2

0 + 12 ‖θ(t)‖2

0 . (4.10)

On the other hand, (2.5), (2.6), and (4.4) yield, with α being a positive constant to bechosen later,

| 〈f(Uh)− f(u), θ〉 | ≤ | 〈f(Uh)− f(uh), θ〉 |+ | 〈f(uh)− f(u), θ〉 |≤ 1

2 ‖f(Uh)− f(uh)‖20 + 1

2 ‖f(uh)− f(u)‖20 + ‖θ(t)‖2

0

≤ (12L

2 + 1) ‖θ(t)‖20 + 1

2L2 ‖ρ(t)‖2

0 , (4.11)

and

| 〈(a(Uh)− a(u))∂xuh, ∂xθ〉 | ≤ |||∂xuh|||∞| 〈a(Uh)− a(uh), ∂xθ〉 |+ |||∂xuh|||∞| 〈a(uh)− a(u), ∂xθ〉 |

≤ 12α

−1|||∂xuh|||2∞ ‖a(Uh)− a(uh)‖20 + 1

2α ‖∂xθ(t)‖20

+ 12α

−1|||∂xuh|||2∞ ‖a(uh)− a(u)‖20 + 1

2α ‖∂xθ(t)‖20

≤ c(‖θ(t)‖2

0 + ‖ρ(t)‖20

)+ α ‖∂xθ(t)‖2

0 . (4.12)

Inequalities (4.8)–(4.12) and Lemma 4.1 yield

1

2

d

ds‖θ(s)‖2

0 + (µ− α) ‖∂xθ(s)‖20 ≤ c

(‖θ(s)‖20 + ‖ρ(s)‖2

0 + ‖∂sρ(s)‖20

)≤ c

(h2p+2 + ‖θ(s)‖2

0

).

By choosing α < µ, integrating over [0, t], and using Gronwall’s Lemma (see e.g. [14,Lemma 3.4]) we obtain the desired result. 2

The following lemma can be compared to [14, Lemma 3.5]. We note that the presentstronger result is crucial in order to prove the superconvergence property (4.6). We alsonote that we do not assume the boundedness and Lipschitz conditions on a′ and f ′.

Lemma 4.4 Assume that there exists a constant M ′ > 0 such that

|||∂tu|||∞ + |||∂tUh|||∞ + |||∂txuh|||∞ ≤M ′. (4.13)

Then

‖∂tθ(t)‖20 +

∫ t

0

‖∂txθ(s)‖20 ds ≤ c

(h2p+2 + ‖θ(0)‖2

0 + ‖∂tθ(0)‖20

).

Proof. By differentiating (4.7) with respect to t we obtain

〈∂ttθ, v〉+ 〈a(Uh)∂txθ, ∂xv〉 = −〈∂ttρ, v〉 − 〈∂ta(Uh)∂xθ, ∂xv〉+ 〈∂tf(Uh)− ∂tf(u), v〉+ 〈(∂ta(Uh)− ∂ta(u))∂xuh, ∂xv〉+ 〈(a(Uh)− a(u))∂txuh, ∂xv〉 ∀v ∈ SN,p

0 . (4.14)

9

By choosing v = ∂tθ and using (2.4) we obtain

1

2

d

dt‖∂tθ(t)‖2

0 + µ ‖∂txθ(t)‖20 ≤ −〈∂ttρ, ∂tθ〉+ 〈∂ta(Uh)∂xθ, ∂txθ〉

+ 〈∂tf(Uh)− ∂tf(u), ∂tθ〉+ 〈(∂ta(Uh)− ∂ta(u))∂xuh, ∂txθ〉+ 〈(a(Uh)− a(u))∂txuh, ∂txθ〉

=: T1 + T2 + T3 + T4 + T5. (4.15)

The term T1 can be easily estimated as

|T1| ≤ 12 ‖∂ttρ(t)‖2

0 + 12 ‖∂tθ(t)‖2

0 . (4.16)

To estimate the other terms on the right-hand side of (4.15), we first note that fromLemma 4.2 there holds

|||Uh|||∞ ≤ ess supt∈(0,T ]

‖Uh(t)‖1 ≤ ess supt∈(0,T ]

‖e(t)‖1 + ess supt∈(0,T ]

‖u(t)‖1 ≤ c 0,

which implies, for x ∈ Ω and t ∈ [0, T ],

|a′(Uh)| ≤ sup|z|≤c 0

|a′(z)| ≤ c 1, (4.17)

|f ′(Uh)| ≤ sup|z|≤c 0

|f ′(z)| ≤ c 2, (4.18)

|a′(Uh)− a′(u)| ≤ sup|z|≤c 0+|||u|||∞

|a′′(z)||Uh − u| ≤ c 3(|ρ|+ |θ|), (4.19)

|f ′(Uh)− f ′(u)| ≤ sup|z|≤c 0+|||u|||∞

|f ′′(z)||Uh − u| ≤ c 4(|ρ|+ |θ|). (4.20)

Let α be some positive constant to be chosen later. Inequalities (4.13) and (4.17) imply

|T2| ≤ |||a′(Uh)|||∞|||∂tUh|||∞| 〈∂xθ, ∂txθ〉 |≤ c 1M

′ ‖∂xθ(t)‖0 ‖∂txθ(t)‖0

≤ 12α

−1c 21M

′2 ‖∂xθ(t)‖20 + 1

2α ‖∂txθ(t)‖20 . (4.21)

Inequalities (4.13), (4.18), and (4.20) yield

|T3| ≤ ‖f ′(Uh)∂tUh − f ′(u)∂tu‖0 ‖∂tθ(t)‖0

≤ (‖f ′(Uh)∂tθ‖0 + ‖f ′(Uh)∂tρ‖0 + ‖(f ′(Uh)− f ′(u))∂tu‖0) ‖∂tθ(t)‖0

≤ (c 2 ‖∂tθ(t)‖0 + c 2 ‖∂tρ(t)‖0 + c 4M′ ‖ρ(t)‖0 + c 4M

′ ‖θ(t)‖0) ‖∂tθ(t)‖0

≤ c(‖ρ(t)‖2

0 + ‖∂tρ(t)‖20 + ‖θ(t)‖2

0 + ‖∂tθ(t)‖20

). (4.22)

Similarly, (4.4), (4.13), (4.17), and (4.19) imply

|T4| ≤ |||∂xuh|||∞ ‖a′(Uh)∂tUh − a′(u)∂tu‖0 ‖∂txθ(t)‖0

≤ c ‖∂txθ(t)‖0 (‖a′(Uh)∂tθ‖0 + ‖a′(Uh)∂tρ‖0 + ‖(a′(Uh)− a′(u))∂tu‖0)

≤ c ‖∂txθ(t)‖0 (c 1 ‖∂tθ(t)‖0 + c 1 ‖∂tρ(t)‖0 + c 3M′ ‖ρ(t)‖0 + c 3M

′ ‖θ(t)‖0)

≤ 12α ‖∂txθ(t)‖2

0 + c(‖ρ(t)‖2

0 + ‖∂tρ(t)‖20 + ‖θ(t)‖2

0 + ‖∂tθ(t)‖20

). (4.23)

10

Finally, from (4.13) and (2.5) we deduce

|T5| ≤ |||∂txuh|||∞ ‖a(Uh)− a(u)‖0 ‖∂txθ(t)‖0

≤ α−1L2M ′2 (‖ρ(t)‖20 + ‖θ(t)‖2

0

)+ 1

2α ‖∂txθ(t)‖20 . (4.24)

Inequalities (4.15), (4.16), (4.21)–(4.24), and Lemmas 4.1 and 4.3 imply

1

2

d

dt‖∂tθ(t)‖2

0 + (µ− 32 α) ‖∂txθ(t)‖2

0 ≤ c(‖ρ(t)‖2

0 + ‖∂tρ(t)‖20 + ‖∂ttρ(t)‖2

0

+ ‖θ(t)‖20 + ‖∂xθ(t)‖2

0 + ‖∂tθ(t)‖20

)≤ c

(h2p+2 + ‖θ(0)‖2

0 + ‖∂xθ(t)‖20 + ‖∂tθ(t)‖2

0

).

By choosing α > 0 such that µ − 32 α > 0, changing the variable from t to s, integrating

over (0, t), and using Lemma 4.3 and Gronwall’s Lemma we obtain the desired result. 2

In the following lemma, by using a Sobolev imbedding theorem, we deduce from Lem-mas 4.3 and 4.4 the superconvergence property (4.6). We also prove an estimate for‖∂ttθ(t)‖0, which was not proved in [14] and is required in the proof of Lemma 4.6.

Lemma 4.5 Under the assumption of Lemma 4.4, there holds, for any t ∈ [0, T ],

‖θ(t)‖21 ≤ c

(h2p+2 + ‖θ(0)‖2

0 + ‖∂tθ(0)‖20

). (4.25)

If we further assume that

|||∂ttu|||∞ + |||∂ttUh|||∞ + |||∂ttxuh|||∞ ≤ M ′′, (4.26)

then

‖∂ttθ(t)‖20 ≤ c

(h2p+2 + ‖θ(0)‖2

0 + ‖∂tθ(0)‖20 + ‖∂ttθ(0)‖2

0

). (4.27)

Proof. Lemmas 4.3 and 4.4 imply that θ ∈ L2(0, T ;H10(Ω)) and ∂tθ ∈ L2(0, T ;H1

0(Ω)),and that ∫ T

0

(‖θ(t)‖21 + ‖∂tθ(t)‖2

1

)dt ≤ c (T )

(h2p+2 + ‖θ(0)‖2

0 + ‖∂tθ(0)‖20

).

Therefore, after a possible modification on a set of measure zero (see [9, Lemma 1.2] or[10, Theorem 3.1]), θ is a continuous mapping from [0, T ] into H1

0 (Ω) satisfying (4.25) forall t ∈ [0, T ].

To prove (4.27) we differentiate (4.14) with respect to t to obtain

〈∂tttθ, v〉+ 〈a(Uh)∂ttxθ, ∂xv〉= −〈∂tttρ, v〉+ 〈∂ttf(Uh)− ∂ttf(u), v〉− 〈∂tta(Uh)∂xθ, ∂xv〉 − 2 〈∂ta(Uh)∂txθ, ∂xv〉+ 〈(∂tta(Uh)− ∂tta(u))∂xuh, ∂xv〉+ 2 〈(∂ta(Uh)− ∂ta(u))∂txuh, ∂xv〉+ 〈(a(Uh)− a(u))∂ttxuh, ∂xv〉 .

11

By substituting v by ∂ttθ we infer

1

2

d

dt‖∂ttθ(t)‖2

0 + µ ‖∂ttxθ(t)‖20

≤ ‖∂tttρ(t)‖0 ‖∂ttθ(t)‖0 + ‖∂ttf(Uh)− ∂ttf(u)‖0 ‖∂ttθ(t)‖0

+ ‖∂tta(Uh)∂xθ‖0 ‖∂ttxθ(t)‖0 + 2 ‖∂ta(Uh)∂txθ‖0 ‖∂ttxθ(t)‖0

+ ‖(∂tta(Uh)− ∂tta(u))∂xuh‖0 ‖∂ttxθ(t)‖0

+ 2 ‖(∂ta(Uh)− ∂ta(u))∂txuh‖0 ‖∂ttxθ(t)‖0

+ ‖(a(Uh)− a(u))∂ttxuh‖0 ‖∂ttxθ(t)‖0

=: T1 + · · ·+ T7. (4.28)

Proceeding as in the proof of Lemma 4.4 we obtain

T1 ≤ 12 ‖∂tttρ(t)‖2

0 + 12 ‖∂ttθ(t)‖2

0 , (4.29)

T2 ≤ c(‖ρ(t)‖2

0 + ‖∂tρ(t)‖20 + ‖∂ttρ(t)‖2

0 + ‖θ(t)‖20 + ‖∂tθ(t)‖2

0 + ‖∂ttθ(t)‖20

), (4.30)

T3 ≤ c ‖∂xθ(t)‖20 + 1

2α ‖∂ttxθ(t)‖20 , (4.31)

T4 ≤ c ‖∂txθ(t)‖20 + 1

2α ‖∂ttxθ(t)‖20 , (4.32)

T5 ≤ c(‖ρ(t)‖2

0 + ‖∂tρ(t)‖20 + ‖∂ttρ(t)‖2

0 + ‖θ(t)‖20 + ‖∂tθ(t)‖2

0 + ‖∂ttθ(t)‖20

)+ 1

2α ‖∂ttxθ(t)‖20 , (4.33)

T6 ≤ c(‖ρ(t)‖2

0 + ‖∂tρ(t)‖20 + ‖θ(t)‖2

0 + ‖∂tθ(t)‖20

)+ 1

2α ‖∂ttxθ(t)‖20 , (4.34)

T7 ≤ c(‖ρ(t)‖2

0 + ‖θ(t)‖20

)+ 1

2α ‖∂ttxθ(t)‖20 , (4.35)

where α is a positive constant to be chosen later. Inequalities (4.25), (4.28)–(4.35), andLemmas 4.1 and 4.4 imply

1

2

d

dt‖∂ttθ(t)‖2

0 + (µ− 52α) ‖∂ttxθ(t)‖2

0

≤ c(‖ρ(t)‖2

0 + ‖∂tρ(t)‖20 + ‖∂ttρ(t)‖2

0 + ‖∂tttρ(t)‖20

+ ‖θ(t)‖21 + ‖∂tθ(t)‖2

0 + ‖∂txθ(t)‖20 + ‖∂ttθ(t)‖2

0

)≤ c

(h2p+2 + ‖θ(0)‖2

0 + ‖∂tθ(0)‖20 + ‖∂txθ(t)‖2

0 + ‖∂ttθ(t)‖20

).

By choosing α > 0 such that µ − 52α > 0, integrating over (0, t), and using Gronwall’s

Lemma and Lemma 4.4 we obtain (4.27), thus prove the lemma. 2

Analogous to Eh, we define eh having the form

eh(x, t) :=

N∑j=1

eh,j(t)φj,p+1(x)

such that uh := uh + eh is an elliptic projection of u into SN,p+10 , i.e. we define eh by

〈a(u)∂x(uh + eh), ∂xv〉j = 〈a(u)∂xu, ∂xv〉j ∀v ∈ SN,p+10 , (4.36)

12

for t ∈ (0, T ] and j = 1, . . . , N .Let

η(x, t) := eh(x, t)− Eh(x, t), (4.37)

and let

ρ(x, t) := u(x, t)− uh(x, t). (4.38)

Since we can write e as

e = u− (uh + eh) + (uh − Uh) + (eh − Eh) + Eh

= ρ+ θ + η + Eh,

we have

Eh = e− ρ− θ − η. (4.39)

The estimates for ρ and θ are given by Lemma 4.1 (with p replaced by p + 1) andLemma 4.5, respectively. The following lemma yields the corresponding estimate for η.We note that similar results were proved in [14] with the assumption on monotonicityof both functions t 7→ ‖θ(t)‖0 and t 7→ ‖η(t)‖0. Such assumptions will not be requiredhere. What makes our analysis different is the use of a Sobolev imbedding theorem andestimate (4.27).

Lemma 4.6 Let eh ∈ H1(0, T ; SN,p+10 ) be defined by (4.36), and the assumptions of

Lemma 4.5 be satisfied. If Eh ∈ H1(0, T ; SN,p+10 ) is defined by either of the equations

(3.5), (3.7), (3.8), or (3.9), and the condition (3.6), and if uh, Uh, eh, and Eh are chosensuch that

‖θ(0)‖0 + ‖∂tθ(0)‖0 + ‖∂ttθ(0)‖0 + ‖η(0)‖0 ≤ c hp+1, (4.40)

then

‖η(t)‖1 ≤ c hp+1.

Proof. We prove the lemma for the case of nonlinear parabolic error estimate, i.e.whenEh satisfies (3.5) and (3.6). The other cases can be proved similarly. From (2.7), (3.5),and (4.36) we deduce

〈∂tη, v〉j +⟨a(Uh + Eh)∂xη, ∂xv

⟩j=

⟨(a(Uh + Eh)− a(u))∂x(uh + eh), ∂xv

⟩j

−⟨a(Uh + Eh)∂xθ, ∂xv

⟩j− 〈∂tθ, v〉j

− 〈∂tρ, v〉j +⟨f(Uh + Eh)− f(u), v

⟩j, (4.41)

13

for v ∈ SN,p+10 and j = 1, . . . , N . By substituting v by η in the above equation (noting

η ∈ SN,p+10 ), and using (2.4) we obtain

1

2

d

dt‖η(t)‖2

0,j + µ ‖∂xη(t)‖20,j ≤ ‖∂tρ(t)‖0,j ‖η(t)‖0,j + ‖∂tθ(t)‖0,j ‖η(t)‖0,j

+∥∥∥(a(Uh + Eh)− a(u))∂x(uh + eh)

∥∥∥0,j‖∂xη(t)‖0,j

+∥∥∥a(Uh + Eh)∂xθ

∥∥∥0,j‖∂xη(t)‖0,j

+∥∥∥f(Uh + Eh)− f(u)

∥∥∥0,j‖η(t)‖0,j

=: T1 + T2 + T3 + T4 + T5, (4.42)

for j = 1, . . . , N . There hold

T1 ≤ 12 ‖∂tθ(t)‖2

0,j + 12 ‖η(t)‖2

0,j , (4.43)

and

T2 ≤ 12 ‖∂tρ(t)‖2

0,j + 12 ‖η(t)‖2

0,j . (4.44)

Due to (2.5), (4.5), (4.37), and (4.38) there holds∥∥∥a(Uh + Eh)− a(u)∥∥∥

0,j≤ L

(‖θ‖0,j + ‖ρ‖0,j + ‖η‖0,j

).

Similarly, (2.6), (4.5), (4.37), and (4.38) imply∥∥∥f(Uh + Eh)− f(u)∥∥∥

0,j≤ L

(‖θ‖0,j + ‖ρ‖0,j + ‖η‖0,j

).

Also, due to (4.36) and Lemma 4.1 there holds

|||∂x(uh + eh)|||∞ ≤ c .

Therefore, with α being a positive constant to be chosen later, we have

T3 ≤ c(‖θ(t)‖2

0,j + ‖ρ(t)‖20,j + ‖η(t)‖2

0,j

)+ 1

2α ‖∂xη(t)‖20,j , (4.45)

T4 ≤ c ‖∂xθ(t)‖20,j + 1

2α ‖∂xη(t)‖20,j , (4.46)

T5 ≤ c(‖θ(t)‖2

0,j + ‖ρ(t)‖20,j + ‖η(t)‖2

0,j

). (4.47)

Inequalities (4.42)–(4.47), (4.25), and (4.40) imply

1

2

d

dt‖η(t)‖2

0,j + (µ− α) ‖∂xη(t)‖20,j ≤ c

(h2p+2 + ‖η(t)‖2

0,j

).

14

By summing over j, choosing α > 0 such that α < µ, integrating over (0, t), and usingGronwall’s Lemma we obtain

‖η(t)‖20 +

∫ t

0

‖∂xη(s)‖20 ds ≤ c h2p+2. (4.48)

We next differentiate (4.41) with respect to t, and substitute v by ∂tη to have

1

2

d

dt‖∂tη(t)‖2

0,j + µ ‖∂txη(t)‖20,j ≤ ‖∂ttθ(t)‖0,j ‖∂tη(t)‖0,j + ‖∂ttρ(t)‖0,j ‖∂tη(t)‖0,j

+∥∥∥a(Uh + Eh)∂txθ

∥∥∥0,j‖∂txη(t)‖0,j

+∥∥∥(a(Uh + Eh)− a(u))∂txuh

∥∥∥0,j‖∂txη(t)‖0,j

+∥∥∥∂ta(Uh + Eh)(∂xη + ∂xθ)

∥∥∥0,j‖∂txη(t)‖0,j

+∥∥∥(∂ta(Uh + Eh)− ∂ta(u))∂xuh

∥∥∥0,j‖∂txη(t)‖0,j

+∥∥∥∂tf(Uh + Eh)− ∂tf(u)

∥∥∥0,j‖∂txη(t)‖0,j .

By using Lemmas 4.1, 4.4 and 4.5, and proceeding as before we obtain

1

2

d

dt‖∂tη(t)‖2

0,j + (µ− α) ‖∂txη(t)‖20,j ≤ c

(‖ρ(t)‖2

0,j + ‖∂tρ(t)‖20,j + ‖∂ttρ(t)‖2

0,j

+ ‖θ(t)‖20,j + ‖∂tθ(t)‖2

0,j + ‖∂ttθ(t)‖20,j

+ ‖∂xθ(t)‖20,j + ‖∂txθ(t)‖2

0,j + ‖∂tη(t)‖20,j

)≤ c

(h2p+2 + ‖∂txθ(t)‖2

0,j + ‖∂tη(t)‖20,j

).

Summing over j, choosing α < µ, integrating over (0, t) and using Lemma 4.4 and Gron-wall’s Lemma we obtain

‖∂tη(t)‖20 +

∫ t

0

‖∂txη(s)‖20 ds ≤ c h2p+2. (4.49)

The desired result now follows from (4.48), (4.49), and the argument in the first part ofthe proof of Lemma 4.5. 2

We are now ready to state and prove the main result of the paper.

Theorem 4.7 Let the assumptions of Lemma 4.6 hold. If the exact solution satisfiesu ∈ H1(0, T ;Hp+1(Ω) ∩H1

0 (Ω)), and if

‖e(t)‖1 ≥ c hp, (4.50)

then, for t ∈ [0, T ],

limh→0

Θ(t) = 1.

15

Proof. From equation (4.39) we deduce, using the triangular inequalities,

|Θ(t)− 1| =

∣∣∣‖Eh(t)‖1 − ‖e(t)‖1

∣∣∣‖e(t)‖1

≤ ‖ρ(t)‖1 + ‖θ(t)‖1 + ‖η(t)‖1

‖e(t)‖1

.

By using Lemma 4.1 (with p+ 1 instead of p), Lemmas 4.5 and 4.6, and (4.50) we infer

|Θ(t)− 1| ≤ c h,

thus prove the theorem. 2

5 CONCLUDING REMARKS

Linear and nonlinear parabolic and elliptic a posteriori error estimates for semidiscretemethods for nonlinear parabolic equations are shown to converge to the true error. Ouranalysis relieves the assumptions on functions involving the elliptic projection and semidis-crete approximation of the exact solution, which are not simple to justify; therefore assurethe applicability of the error estimates in the general nonlinear case. These error esti-mates can be used as a basis for an adaptive numerical procedure that uses any timediscretisation when the method of lines is used for space discretisation. Many numericalexperiments have been carried out in [1, 2, 4, 11] that justify the theory.

ACKNOWLEDGMENTS

The authors wish to thank Prof. Vidar Thomee for helpful discussions.

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