Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions

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UMER. NAL c

Vol. 51, No. 3, pp. 1494–1524

MAXIMUM NORM A POSTERIORI ERROR ESTIMATION FOR

PARABOLIC PROBLEMS USING ELLIPTIC RECONSTRUCTIONS

∗

NATALIA KOPTEVA† AND TORSTEN LINSS‡

Abstract. A semilinear second-order parabolic equation is considered in a regular and a sin-gularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank–Nicolson, and discontinuous Galerkin dG(r) methods are addressed. For their full discretizations, we employ elliptic reconstructions that are, respectively, piecewise-constant, piecewise-linear, and piecewise-quadratic forr= 1 in time. We also use certain bounds for the Green’s function of the parabolic operator.

Key words. a posteriori error estimate, maximum norm, singular perturbation, elliptic recon-struction, backward Euler, Crank–Nicolson, discontinuous Galerkin, parabolic equation, reaction-diﬀusion

AMS subject classifications.65M15, 65M60

DOI.10.1137/110830563

1. Introduction.

Consider a semilinear parabolic equation in the form

(1.1a)

M

u

:=

∂

_t

u

+

L

u

+

f

(

x, t, u

) = 0

for (

x, t

)

∈

Q

:= Ω

×

(0

, T

]

with a second-order linear elliptic operator

L

=

L

(

t

) in a spatial domain Ω

⊂

R

n

_with

Lipschitz boundary, subject to the initial and Dirichlet boundary conditions

(1.1b)

u

(

x,

0) =

ϕ

(

x

)

for

x

∈

Ω

¯

,

u

(

x, t

) = 0

for (

x, t

)

∈

∂

Ω

×

[0

, T

]

.

We assume that

f

is continuous in ¯

Ω

×

[0

, T

]

×

R

, is diﬀerentiable in the third argument,

and, for some nonnegative constants

γ

and ¯

γ

, satisﬁes

(1.2)

0 ≤

γ

2

≤

∂

_z

f

(

x, t, z

)

≤

γ

¯

2

for

(

x, t, z

)

∈

Ω

¯

×

[0

, T

]

×

R

.

The purpose of this paper is to obtain computable a posteriori error estimates for

fully discrete methods applied to problem (1.1). We consider the ﬁrst-order backward

Euler and the second-order Crank–Nicolson discretizations in time. Furthermore, we

study the discontinuous Galerkin method dG(

r

),

r

≥

1, with Radau quadrature.

These results are applied to the model equation with

L

:=

−

ε

2

=

−

ε

2

n_i₌₁

∂

_x2

i

:

(1.3)

M

u

:=

∂

_t

u

−

ε

2

u

+

f

(

x, t, u

) = 0

posed in a bounded polyhedral spatial domain Ω

⊂

R

n

_with

_n

_{= 1}

_,

₂

_,

_{3. This equation}

will be considered in two regimes:

(i)

ε

= 1

, γ

≥

0;

(ii)

ε

1 , γ >

0 .

∗_{Received by the editors April 13, 2011; accepted for publication (in revised form) March 20,} 2013; published electronically May 22, 2013. This research was conducted with the ﬁnancial support of Science Foundation Ireland under Research Frontiers Programme 2008 grant 08/RFP/MTH1536.

http://www.siam.org/journals/sinum/51-3/83056.html

†_{Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland (natalia.} [email protected]).

‡_Fakult¨_{at f¨}_{ur Mathematik und Informatik, FernUniversit¨}_{at in Hagen, 58095 Hagen, Germany} ([email protected]).

1494

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Note that regime (ii) yields a singularly perturbed reaction-diﬀusion equation, whose

solutions may exhibit sharp layer phenomena. It is important in this regime that a

posteriori error estimates are robust in the sense that any dependence on the small

perturbation parameter

ε

should be shown explicitly [21, 25].

We will give error estimates in the

maximum norm

, which is suﬃciently strong to

capture sharp layers and singularities that may occur, in particular, if problem (1.1) is

of singularly perturbed type. Our estimates will be of

interpolation type

in the sense

that they will include certain terms that may be interpreted as approximating

τ

_jp

|

∂

_tp

u

|

,

where

p

and

τ

_j

are the discretization order and local step size in time, respectively.

We employ the

elliptic reconstruction

technique, which was introduced in the

recent papers [22, 19, 6] as a counterpart of the Ritz-projection in the a posteriori

error estimation for parabolic problems. We also use certain bounds for the

Green’s

function

of the continuous parabolic operator in a manner similar to [6], only for a

more general semilinear parabolic operator of (1.3) (compared to

∂

_t

−

in [6]).

One distinctive feature of our analysis in this paper (compared, e.g., to [1, 6])

is that we use computed solutions and elliptic reconstructions that are

piecewise-polynomial of degree

p

−

1 in time, where

p

is the time discretization order. In

partic-ular, they are

piecewise-constant

in time when dealing with the ﬁrst-order backward

Euler method and

piecewise-linear

and

-quadratic

, respectively, when dealing with

the second-order Crank–Nicolson method, and the third-order dG(1) method.

Con-sequently, we allow the residuals of computed solutions, as well as other functions, to

be understood as

distributions

; this inclusion plays a crucial role in our analysis.

Note that earlier pointwise/maximum norm a posteriori error estimates for

parabolic equations either are given for regular linear problems [9, 3, 6, 7] or are

not robust in the sense that they involve negative powers of

ε

[3]. For a more detailed

comparison of our results with various earlier a posteriori error estimates, we refer

the reader to Remarks 5.3, 9.5, 9.9, and 11.7 below.

The paper is organized as follows. In section 2, we introduce the Green’s function

and obtain a certain stability lemma, which is the key ingredient of our a posteriori

error analysis. The contents of sections 3–6 and 8–11 are summarized in the table

below, while section 7 looks into elliptic a posteriori error estimators.

Summary

Backward

Crank–

of results

Euler

Nicolson

dG(

r

)

Semidiscretizations

section 3

section 4

section 5

section 6 Full discretizations

section 8

section 9

section 10

section 11 Notation.

Throughout the paper,

C

, as well as

c

, denotes a generic positive

constant that may take diﬀerent values in diﬀerent formulas but is

independent of

the diﬀusion coeﬃcient

ε

and any mesh sizes

. We use

|

x

|

for the Euclidean norm of

x

∈

R

n

. The usual spaces

C

( ¯

Ω) and

H

₀1

(Ω) are used, as well as the spaces

L

_p

(Ω),

1 ≤

p

≤ ∞

, with the norm

·

_p,_Ω

, while

φ, ψ

=

_Ω

φ

(

x

)

ψ

(

x

) d

x

denotes the inner

product in

L

₂

(Ω).

Distributions and left-continuity convention.

Certain functions will be understood

as distributions [13], which will in most cases be indicated. By contrast, if a certain

function is Lebesgue-integrable in Ω

×

(0

, T

), we shall refer to it as a regular function.

Whenever we deal with a regular function, it will be understood as

left-continuous

for

all

t

∈

(0

, T

]. In particular, this convention will be applied to all piecewise-continuous

temporal derivatives.

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1496

NATALIA KOPTEVA AND TORSTEN LINSS

2. The Green’s function of the parabolic operator.

In this section we

consider the Green’s function

G

associated with the operator

M

of (1.1). Our interest

in the Green’s function is in that it will be used to express the error of a numerical

approximation in terms of its residual.

For deﬁnitions and properties of fundamental solutions and Green’s functions of

parabolic operators with variable coeﬃcients, we refer the reader to [12, Chapter 1

and section 7 of Chapter 3]. For any pair of bounded functions

v

and

w

that vanish

on

∂

Ω, the standard linearization yields

M

v

− M

w

= [

∂

_t

+

L

+

a

(

x, t

)](

v

−

w

), where

a

(

x, t

) :=

₀1

∂

_z

f

(

x, t, w

+

z

[

v

−

w

]) d

z

. Hence, the diﬀerence

v

−

w

is represented as

[

v

−

w

](

x, t

) =

Ω

G

(

x, t

;

ξ,

0) [

v

−

w

](

ξ,

0) d

ξ

+

_t

0

Ω

G

(

x, t

;

ξ, s

) [

M

v

− M

w

](

ξ, s

) d

ξ

d

s

(2.1)

with the help of the Green’s function

G

that we now deﬁne. For ﬁxed (

x, t

)

∈

Q

, the

Green’s function

G

(

x, t

;

ξ, s

) =: Γ(

ξ, s

) solves the adjoint terminal-value problem

[

−

∂

_s

− L

∗

+

a

(

ξ, s

) ] Γ(

ξ, s

) = 0

for (

ξ, s

)

∈

Ω

×

[0

, t

)

,

(2.2a)

Γ(

ξ, t

) =

δ

(

ξ

−

x

) for

ξ

∈

Ω

,

(2.2b)

Γ(

ξ, s

) = 0

for (

ξ, s

)

∈

∂

Ω

×

[0

, t

]

.

(2.2c)

Here

δ

(

· ) is the Dirac

δ

-distribution in

R

n

_{[13], and}

_L

∗

_{is the adjoint operator to the}

linear operator

L

.

The analysis in this paper will be carried out under the following condition.

Condition 2.1.

There are constants

κ

0

, κ

1

>

0 and

κ

2

≥

0 such that the Green’s

function

G

of

(2.2)

,

(1.2)

satisﬁes

G

(

x, t

;

· , s

)

₁_,_Ω

≤

κ

₀

e

−γ2(t−s)

,

_t₋_τ

0

∂

_s

G

(

x, t

;

· , s

)

₁_,_Ω

d

s

≤

κ

₁

(

τ, t

) +

κ

₂

,

where

x

∈

Ω

,

τ

∈

(0

, t

]

,

t

∈

(0

, T

]

, and

(

τ, t

) :=

_τt

s

−1

e

−12γ2s

d

s

≤

ln(

t/τ

)

.

Note that our model problem satisﬁes this condition as follows.

Lemma 2.2.

Let

ε

∈

(0

,

1]

and

γ

≥

0 . Under assumption

(1.2)

, the model

problem

(1.3)

satisﬁes Condition

2.1 with

κ

₀

:= 1

,

κ

₁

:=

₂_n/3n₂₊₁

, and an

ε

-independent

constant

κ

₂

≥

0 . If

f

(

x, t, z

) =

a

(

x

)

z

+

b

(

x, t

)

, then

κ

₂

= 0

. In general,

κ

₂

=

(¯

γ

2

−

γ

2

) ˆ

κ

₂

, where

κ

ˆ

₂

= ˆ

κ

₂

(

γ

)

if

γ >

0 and

κ

ˆ

₂

= ˆ

κ

₂

(

T

)

if

γ

= 0

.

1

Proof

. We defer the proof to section 12.

Condition 2.1 will be employed by means of the following lemma, which plays a

crucial role in our analysis. The lemma is formulated in the context of an arbitrary

nonuniform mesh in the time direction

(2.3)

0 =

t

₀

< t

₁

< t

₂

<

· · ·

< t

_M

=

T

with

τ

_j

=

t

_j

−

t

_j₋₁

for

j

= 1

, . . . , M.

Lemma 2.3.

Suppose the parabolic operator

M

of

(1.1)

satisﬁes

(1.2)

and

Con-dition

2.1 , and

v, w

are bounded in

Ω

¯

×

[0

, T

]

. Furthermore, let

v

(

· , t

)

, w

(

· , t

)

∈

H

₀1

(Ω)

∩

C

( ¯

Ω)

for

t

∈

[0

, T

]

, and

1_{The constants} _κ

0 and κ1 given by Lemma 2.2 are reasonably sharp. For example, for the constant-coeﬃcient version ∂tu − ε2∂2xu + γ2u = b(x, t) of (1.3) in the spa-tial domain Ω :=R, a calculation [16] yields G(x, t;·, s)1,Ω= e−γ

2₍_t₋_s₎

and ∂sG(x, t;·, s)1,Ω≤

2 (πe)−1(t−s)−1+γ2_e−γ2(t−s)_{, so Condition 2.1 is satisﬁed with}_κ 0= 1 (as in Lemma 2.2),κ1=

2 (πe)−1≈0.48,κ2= 1, while Lemma 2.2 givesκ1= 3·2−3/2≈1.06.

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(2.4)

M

v

− M

w

=

∂

_t

μ

+

ϑ

in

Q,

where the function

μ

is continuous and bounded on

[

t

₀

, t

₁

]

and each

(

t

_j₋₁

, t

_j

]

, while

∂

_t

μ

is continuous and bounded on

(

t

_m₋₁

, t

_m

]

for some

1 ≤

m

≤

M

, and

ϑ

(

· , s

)

_∞_,_Ω

is integrable w.r.t.

s

in

(0

, t

_m

)

(possibly, in the sense of distributions). Then

[

v

−

w

](

· , t

_m

)

∞,Ω

≤

κ

₀

e

−γ2tm

_[

_v

₋

_w

₋

_μ

_](

_·

_,

₀₎

∞,Ω

+ (

κ

1

m

+

κ

2

)

sup

s∈[0,tm−1]

μ

(

· , s

)

_∞

,Ω

+

κ

₀

lim

s→t+_m₋₁

μ

(

· , s

)

_∞

,Ω

+

κ

0

τ

m

sup

s∈(tm−1,tm]

∂

_s

μ

(

· , s

)

_∞

,Ω

+

κ

₀

_t_m

0

e

−γ2(tm−s)

_ϑ

₍

_·

_{, s}

₎

∞,Ω

d

s,

(2.5)

where

_m

=

_m

(

γ

) :=

_τtm m

s

−1

_e

−1

2γ2s

d

s

≤

ln(

t

_m

/τ

_m

)

.

Remark

_2.4.

The term

∂

_t

μ

in the right-hand side of (2.4) is understood in the

sense of distributions.

A typical

μ

is continuously diﬀerentiable in time on each

(

t

_j₋₁

, t

_j

] and has jumps at

t

∈ {

t

_j

}

m_j₌₁−1

, but our left-continuity convention allows us

to avoid ambiguity when integrating by parts. It may help the reader to consider

an equivalent interpretation of such evaluations. For some small positive

λ

, one can

replace

t

+_j

by

t

_j

+

λ

and

μ

by

μ

_λ

such that

μ

_λ

=

μ

for

t

∈

[

t

_j₋₁

+

λ, t

_j

], and it

is continuous and linear in time on each [

t

_j

, t

_j

+

λ

]. Then one deals with a regular

function

∂

_t

μ

_λ

, while the ﬁnal result is obtained by taking the limit as

λ

→

0

+

.

Similarly, in all calculations involving Γ, one can initially replace it by a regular

function Γ

_λ

obtained using a regular approximation

δ

_λ

of

δ

in (2.2b), and then let

λ

→

0

+

. With regard to the regularity of Γ, Condition 2.1 implies for any

τ

∈

(0

, t

)

that

∂

_s

Γ

∈

L

₁

(Ω

×

[0

, t

−

τ

]), while an inspection of the proof of Lemma 2.2 yields a

stronger regularity with

∂

_s

Γ

∈

L

₂

(Ω

×

[0

, t

−

τ

]).

Remark

_2.5.

One can easily check that if

γ

= 0, then

_m

= ln(

t

_m

/τ

_m

). Otherwise,

if

γ >

0, one has

_m

(

γ

) =

E

₁

(

1₂

γ

2

τ

_m

)

−

E

₁

(

1₂

γ

2

t

_m

), where

E

₁

(

t

) =

∞

t

s

−1

e

−s

d

s

; so

_m

(

γ

)

≤ |

ln(

1₂

γ

2

τ

_m

)

|

provided that

1₂

γ

2

τ

_m

≤

0 .

67. (This is easily checked by ﬁnding

the only root

≈

0 .

67 of the equation

E

₁

(

s

) =

|

ln

s

|

on (0

,

1).) Note also that

₁

= 0

for any

γ

≥

0. Proof of Lemma

2.3 .

Combining representation (2.1) with the notation Γ(

ξ, s

) :=

G

(

x, t

_m

;

ξ, s

) for the Green’s function of (2.2), one gets

[

v

−

w

](

x, t

_m

) =

[

v

−

w

](

· ,

0)

,

Γ(

· ,

0)

+

_t_m

0

[

M

v

− M

w

](

· , s

)

,

Γ(

· , s

)

d

s.

Here, in view of (2.4), the integral on the right-hand side involves

μ

and

ϑ

and so can

be represented as a sum

J

_μ

+

J

_ϑ

of the corresponding integrals, which we consider

separately. We use the notation

b+

:= lim

_β_→0+

b+β

and so split

J

_μ

as

J

_μ

=

J

_μ(1)

+

J

_μ(2)

:=

_t+

m−1

0

∂

_s

μ,

Γ(

· , s

)

d

s

+

_t_m

t+_m₋₁

∂

_s

μ,

Γ(

· , s

)

d

s.

Here, for

J

_μ(1)

, an integration by parts yields

J

_μ(1)

=

μ

(

· , t

+_m₋₁

)

,

Γ(

· , t

_m₋₁

)

−

μ

(

· ,

0)

,

Γ(

· ,

0)

−

tm−1

0

μ

(

· , s

)

, ∂

_s

Γ(

· , s

)

d

s.

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1498

Consequently, we arrive at

[

v

−

w

](

x, t

_m

) =

[

v

−

w

−

μ

](

· ,

0)

,

Γ(

· ,

0)

−

_t_m₋₁

0

μ

(

· , s

)

, ∂

_s

Γ(

· , s

)

d

s

+

μ

(

· , t

+_m₋₁

)

,

Γ(

· , t

_m₋₁

)

+

_t_m

t+_m₋₁

∂

_s

μ,

Γ(

· , s

)

d

s

+

_t_m

0

ϑ

(

· , s

)

,

Γ(

· , s

)

d

s,

(2.6)

where the last term represents

J

_ϑ

. Finally, Condition 2.1 implies that

Γ(

· , s

)

₁_,_Ω

≤

κ

₀

e

−γ2(tm−s)

_≤

_κ

0

,

_t_m₋₁

0

∂

_s

Γ(

· , s

)

₁_,_Ω

d

s

≤

κ

₁

_m

+

κ

₂

,

so we get the desired result.

The following version of Lemma 2.3 involves certain approximations Γ

j_h

of Γ(

· , t

_j

).

Lemma 2.3

∗

.

Under conditions of Lemma

2.3 , suppose that instead of

(2.4)

one

has

M

v

− M

w

=

∂

_t

μ

+

ϑ

+

ϑ

_∗

, where

ϑ

_∗

(

· , t

) =

m_j₌₁−1

ϑ

j

_δ

₍

_t

₋

_t

j

)

for

t

∈

[0

, t

m

]

. If

there exist some functions

{

Γ

j_h

}

m_j₌₁−1

such that

ϑ

j

,

Γ

j_h

= 0

for

j

= 1

, . . . , m

−

1 , and

_m₋₁

j=1

τ

j

H

j−2

{

Γ(

· , t

j

)

−

Γ

hj

}

1,Ω

≤

κ

3

(

τ, t

)

for some positive weight functions

{H

j

}

and some constant

κ

₃

, then the statement of Lemma

2.3 remains valid, only with an

additional term

κ

₃

(

τ, t

) max

_j₌₁_,...,m₋₁

τ

_j−1

H

2_j

ϑ

j

∞,Ω

in the ﬁnal line of

(2.5)

.

Proof

. Imitate the proof of Lemma 2.3, and note that now we have (2.6) with an

additional term

m_j₌₁−1

ϑ

j

,

Γ(

· , t

_j

)

=

_jm₌₁−1

ϑ

j

,

Γ(

· , t

_j

)

−

Γ

j_h

.

3. Summary of results for semidiscrete methods (no spatial

discretiza-tion).

In this section we describe our results for the abstract parabolic problem (1.1)

discretized in time on an arbitrary nonuniform mesh (2.3) using semidiscrete backward

Euler, Crank–Nicolson, and discontinuous Galerkin methods.

Let

u

solve problem (1.1) with the parabolic operator

M

satisfying (1.2) and let

Condition 2.1, and let

U

j

_∈

_H

1

0

(Ω)

∩

C

( ¯

Ω), associated with the time level

t

j

, solve a

corresponding semidiscrete problem with

U

0

=

ϕ

. Then, for

m

= 1

, . . . , M

, we give

a

posteriori error estimates

of the type

U

m

−

u

(

· , t

_m

)

_∞

,Ω

≤

C

1

(

κ

1

m

+

κ

2

)

_j₌₁

max

_,...,m₋₁

χ

j

∞,Ω

+

C

2

κ

0

χ

m

∞,Ω

+

κ

₀

m

j=1

_t_j

tj−1

e

−γ2(tm−s)

_ϑ

₍

_·

_{, s}

₎

∞,Ω

d

s .

(3.1)

The quantities that appear in this estimate are speciﬁed by Theorems 4.1 and 5.1

and Corollary 6.3 below, and can be summarized as follows:

p

χ

j+1

ϑ

C

₁

C

₂

Backward Euler 1

U

j+1

−

U

j

ψ

˜

−

ψ

j

on (

t

_j₋₁

, t

_j

]

1

2 Crank–Nicolson 2

τ

_j₊₁

(

ψ

j+1

−

ψ

j

)

ψ

˜

−

I

₁_,t

ψ

˜

1₈ 1₂

dG(1)-Radau 3

3 τ

_j₊₁

(2

ψ

j

₋

₃

_ψ

j+1/3

₊

_ψ

j+1

₎

_ψ

˜

₋

_I

2,t

ψ

˜

₈₁2 1₆

Here for the evaluation of

χ

j+1

_and

_ϑ

_{we use}

ψ

j+α

:=

L

(

t

_j₊_α

)

U

j+α

+

f

(

· , t

_j₊_α

, U

j+α

)

,

ψ

˜

:=

L

(

t

) ˜

U

+

f

(

· , t,

U

˜

)

,

(3.2)

(6)

where

α

∈

(0

,

1] is any value for which the approximate solution

U

j+α

_{at time}

_t

j+α

:=

t

_j

+

ατ

_j₊₁

is available from the deﬁnition of the semidiscrete method. Also, ˜

U

is

a piecewise-polynomial interpolant of the computed solution of degree

p

−

1, while

I

_p₋₁_,t

ψ

˜

is a piecewise-polynomial interpolant of ˜

ψ

of the same degree using the same

interpolation points.

Remark

_{3.1 (}

interpolation-type estimates

_).

The quantity

|

χ

j

_|

_{in (3.1)}

approxi-mates

τ

_jp

|

∂

_tp

u

(

· , t

_j

)

|

. This immediately follows from

χ

j

₌

_U

j

₋

_U

j−1

_{for the backward}

Euler method. For the Crank–Nicolson and dG(1) methods, note that

ψ

j+α

approxi-mates

L

u

+

f

(

· , t, u

) at

t

=

t

_j₊_α

, so

χ

j

_approximates

_τ

p j

|

∂

p−1

t

(

L

u

+

f

(

· , t, u

))

|

(in fact,

χ

j

₌

_τ

p

j

∂

tp−1

(

I

p−1,t

ψ

˜

)), while (1.1) gives

|

∂

tp−1

(

L

u

+

f

(

· , t, u

))

|

=

|

∂

tp

u

|

.

Remark

3.2 (

p

th-order estimates

).

Remark 3.1 and the deﬁnitions of

ϑ

for the

backward Euler, Crank–Nicolson, and dG(1) methods imply that (3.1) gives an a

pos-teriori error estimate of order

p

with

p

= 1

,

2, and 3, respectively.

4. Semidiscrete backward Euler method (no spatial discretization).

Consider an arbitrary nonuniform mesh (2.3) in the time direction and discretize

the abstract parabolic problem (1.1) in time using the ﬁrst-order backward Euler

method as follows. We associate an approximate solution

U

j

∈

H

₀1

(Ω)

∩

C

( ¯

Ω) with

the time level

t

_j

and require it to satisfy

δ

_t

U

j

+

L

j

U

j

+

f

j

= 0

in Ω

,

j

= 1

, . . . , M,

U

0

=

ϕ,

(4.1a)

where

δ

_t

U

j

:=

U

j

₋

_U

j−1

τ

_j

,

L

j

_:=

_L

₍

_t

j

)

,

and

f

j

:=

f

(

· , t

j

, U

j

)

.

(4.1b)

For this discretization, we give the following a posteriori error estimate.

Theorem 4.1.

Let

u

solve problem

(1.1)

with the parabolic operator

M

satisfying

(1.2)

and Condition

2.1 and

U

j

_{solve the corresponding semidiscrete problem}

_(4.1)

_.

Then, for

m

= 1

, . . . , M

, one has

(3.1)

with

χ

j

₌

_U

j

₋

_U

j−1

_,

_C

1

= 1

,

C

2

= 2

, and

ϑ

deﬁned by

ϑ

(

· , t

) = ˜

ψ

(

· , t

)

−

ψ

˜

(

· , t

_j

)

,

ψ

˜

(

· , t

) =

L

(

t

)

U

j

+

f

(

· , t, U

j

)

for

t

∈

(

t

_j₋₁

, t

_j

]

.

(4.2)

Proof

. Let

I

₁_,t

U

be the standard piecewise-linear interpolant of

U

j

_{in time:}

(4.3)

I

₁_,t

U

(

· , t

) :=

tj_τ−t j

U

j−1

₊

t−tj−1

τj

U

j

_for

_t

_∈

_[

_t

j−1

, t

j

]

,

j

= 1

, . . . , M.

Furthermore, we deﬁne a

piecewise-constant

interpolant ˜

U

of

U

j

_by

(4.4)

U

˜

(

· , t

) :=

U

j

for

t

∈

(

t

_j₋₁

, t

_j

]

,

j

= 1

. . . , M,

U

˜

(

· ,

0) :=

U

1

(so ˜

U

is continuous on [

t

₀

, t

₁

]). Note that the temporal derivative

∂

_t

U

˜

is understood

as a distribution, while

∂

_t

(

I

₁_,t

U

) is a regular function, equal to

δ

_t

U

j

for

t

∈

(

t

_j₋₁

, t

_j

]

(in agreement with our left-continuity convention). Consequently, (4.1a) implies that

(4.5)

∂

_t

(

I

₁_,t

U

) + ˜

ψ

=

ϑ

for (

x, t

)

∈

Q. Here we also used the observation that by (4.4), the regular function

ϑ

of (4.2) can

be rewritten as

ϑ

= ˜

ψ

−

[

L

j

_U

j

₊

_f

j

_{] for}

_t

_∈

₍

_t

j−1

, t

j

].

As

M

U

˜

=

∂

_t

U

˜

+ ˜

ψ

and

M

u

= 0, so (4.5) implies that

M

_U

˜

_{− M}

_u

₌

_∂

t

[ ˜

U

−

I

1,t

U

] +

ϑ

in

Q.

(7)

1500

Now the desired bound for

U

m

₋

_u

₍

_·

_{, t}

m

) = [ ˜

U

−

u

](

· , t

m

) is obtained by an application

of Lemma 2.3 with

μ

:= ˜

U

−

I

₁_,t

U

and

ϑ

of (4.2), using the following two observations.

First, we note that [ ˜

U

−

u

−

μ

](

· ,

0) =

U

1

−

ϕ

−

(

U

1

−

ϕ

) = 0. Second, for

t

∈

(

t

_j₋₁

, t

_j

],

one has

μ

=

tj_τ−t j

(

U

j

₋

_U

j−1

_{) =}

tj−t τj

χ

j

₌

_⇒

_|

_μ

_{| ≤}

_χ

j

_,

_τ

j

|

∂

t

μ

|

=

χ

j

.

This completes the proof.

Corollary 4.2.

Under assumption

(1.2)

, the a posteriori error estimate of

Theorem

4.1 applies to the model problem

(1.3)

with

ϑ

=

f

(

· , t, U

j

)

−

f

(

· , t

_j

, U

j

)

and

the constants

κ

₀

,

κ

₁

,

κ

₂

from Lemma

2.2 .

5. Semidiscrete Crank–Nicolson method (no spatial discretization).

Con-sider an arbitrary nonuniform mesh (2.3) in the time direction and discretize the

ab-stract parabolic problem (1.1) in time using the second-order Crank–Nicolson method

as follows. We associate an approximate solution

U

j

_∈

_H

1

0

(Ω)

∩

C

( ¯

Ω) with the time

level

t

_j

and require it to satisfy

(5.1a)

δ

_t

U

j

+

1₂

L

j−1

U

j−1

+

L

j

U

j

+

1₂

(

f

j−1

+

f

j

) = 0

in Ω

,

j

= 1

, . . . , M,

where we again let

(5.1b)

U

0

=

ϕ,

δ

_t

U

j

:=

U

j

₋

_U

j−1

τ

_j

,

L

j

_:=

_L

₍

_t

j

)

,

and

f

j

:=

f

(

· , t

j

, U

j

)

.

To give an a posteriori error estimate for this discretization, we will use the

stan-dard piecewise linear interpolation

I

₁_,t

, which, for any continuous function

w

=

w

(

t

),

is deﬁned by

(5.2)

I

₁_,t

w

(

t

) :=

tj_τ−t

j

w

(

t

j−1

) +

t−tj−1

τj

w

(

t

j

)

for

t

∈

[

t

j−1

, t

j

]

,

j

= 1

, . . . , M.

Recall an almost identical deﬁnition (4.3) for the

piecewise-linear

interpolant

I

₁_,t

U

of

the computed solution; the latter plays a crucial role in our analysis of this section.

Theorem 5.1.

Let

u

solve the problem

(1.1)

with the parabolic operator

M

satisfying

(1.2)

and Condition

2.1 , and let

U

j

_{solve the corresponding semidiscrete}

problem

(5.1)

. Then for

m

= 1

, . . . , M

, one has

(3.1)

with

χ

j

₌

_τ

j

ψ

j

₋

_ψ

j−1

_using

ψ

j

₌

_L

j

_U

j

₊

_f

j

_,

_C

₁

₌

1

8

,

C

2

=

12

, and

ϑ

deﬁned by

ϑ

= ˜

ψ

−

I

₁_,t

ψ,

˜

ψ

˜

=

L

(

t

) ˜

U

+

f

(

· , t,

U

˜

)

,

U

˜

(

· , t

) =

I

₁_,t

U

(

· , t

)

(5.3)

for

t

∈

[0

, T

]

, where we use

I

₁_,t

U

(

· , t

)

of

(4.3)

and

I

₁_,t

of

(5.2)

.

Proof

. Let

t

∈

[

t

_j₋₁

, t

_j

]. First, note that ˜

ψ

=

I

₁_,t

ψ

˜

+

ϑ

=

1₂

ψ

j−1

+

ψ

j

+

∂

_t

μ

+

ϑ

,

where

μ

:=

_tt

j

I

₁_,t

ψ

˜

−

1₂

ψ

j−1

₊

_ψ

j

_d

_t

_{, so}

(5.4)

μ

=

τ

_j−1

χ

j

_t

tj

(

t

−

t

_j₋₁_/₂

) d

t

=

−

1₂

(

t

_j

−

t

)(

t

−

t

_j₋₁

)

· τ

_j−2

χ

j

for

t

∈

[

t

_j₋₁

, t

_j

]

.

Next, note that ˜

U

(

· , t

) =

I

₁_,t

U

(

· , t

) implies that

∂

_t

U

˜

=

δ

_t

U

j

₌

₋

1

2

(

ψ

j−1

+

ψ

j

) for

t

∈

(

t

_j₋₁

, t

_j

] (where we also invoked (5.1a)). Combining these two observations, one

deduces that

∂

_t

U

˜

+ ˜

ψ

=

∂

_t

μ

+

ϑ

. As

M

U

˜

=

∂

_t

U

˜

+ ˜

ψ

and

M

u

= 0, so

(5.5)

M

U

˜

− M

u

=

∂

_t

μ

+

ϑ

in

Q.

(8)

Both sides in this relation are regular functions; it is valid for

t

∈

(0

, T

] as

μ

of (5.4)

is continuous for

t

∈

[0

, T

].

Now the desired bound for

U

m

₋

_u

₍

_·

_{, t}

m

) = [ ˜

U

−

u

](

· , t

m

) is obtained by an

application of Lemma 2.3 to (5.5) with

μ

given by (5.4) and

ϑ

by (5.3), using the

following two observations. First, note that [ ˜

U

−

u

−

μ

](

· ,

0) =

U

0

−

ϕ

−

0 = 0.

Second, for

t

∈

(

t

_j₋₁

, t

_j

], one has

(5.6)

|

μ

| ≤

1₈

|

χ

j

|

and

τ

_j

|

∂

_t

μ

| ≤

1₂

|

χ

j

|

,

while

μ

(

· , t

+_m₋₁

) = 0. This completes the proof.

Corollary 5.2.

Under assumption

(1.2)

, the a posteriori error estimate of

Theorem

5.1 applies to the model problem

(1.3)

with

ϑ

=

f

(

· , t, I

₁_,t

U

)

−

I

₁_,t

[

f

(

· , t, I

_t

U

)]

and the constants

κ

₀

,

κ

₁

,

κ

₂

from Lemma

2.2 .

Remark

_5.3.

The a posteriori error estimate given by Theorem 5.1 resembles

(but is not identical to) error estimates of [1]. Our analysis of the semidiscrete Crank–

Nicolson method seems more straightforward as we work with the standard piecewise

linear interpolant of the computed solution, while the analysis in [1] involves a

con-struction of a certain piecewise-quadratic polynomial of the computed solution in

time. Furthermore, in section 10, we derive a posteriori error estimates for fully

discrete Crank–Nicolson methods, which were not considered in [1].

6. Semidiscrete discontinuous Galerkin method dG(r) with Radau

quadrature (no spatial discretization).

Consider an arbitrary nonuniform mesh

(2.3) in the time direction and discretize the abstract parabolic problem (1.1) in time