R ESEARCH I NSTITUTEFOR M ATHEMATICAL S CIENCESKYOTOUNIVERSITY,Kyoto,Japan ByMitsuhiroT.NAKAO,TakehikoKINOSHITAandTakumaKIMURAJuly2011 Onaposterioriestimatesofinverseoperatorsforlinearparabolicinitial-boundaryvalueproblems RIMS-1729

RIMS-1729

On a posteriori estimates of inverse operators for linear parabolic initial-boundary value problems

By

Mitsuhiro T. NAKAO, Takehiko KINOSHITA and Takuma KIMURA

July 2011

R _ESEARCH I NSTITUTE FOR M ATHEMATICAL S _CIENCES

KYOTO UNIVERSITY, Kyoto, Japan

Noname manuscript No.

(will be inserted by the editor)

On a posteriori estimates of inverse operators for linear parabolic initial-boundary value problems

Mitsuhiro T. Nakao, Takehiko Kinoshita and Takuma Kimura

July 5, 2011

Abstract We present constructive a posteriori estimates of inverse operators for linear parabolic differential equations. In general, we can obtain a priori estimates of this inverse operators. However, usually the estimates exponentially increases for time derivative. We propose the technique for obtaining the estimates of the inverse operator by using its Galerkin approximation. It is expected that we obtain the constant that is smaller than a priori estimates by using our technique.

Keywords a posteriori estimates·Galerkin methods·linear parabolic equations PACS 02.60.Ed·02.60.Lj·02.70.Dh·02.70.Hm

Mathematics Subject Classification (2000) 35K10·65G99·65M20·65M60

1 Introduction

We consider the following linear parabolic initial-boundary value problems, 8>

><

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:

∂w

∂t −ν4w+ (b· ∇)w+cw=g, in Ω×J, (1a)

w(x, t) = 0, on ∂Ω×J, (1b)

w(x,0) = 0, inΩ, (1c)

Grants or other notes about the article that should go on the front page should be placed here.

General acknowledgments should be placed at the end of the article.

Mitsuhiro T. Nakao

Sasebo National College of Technology, Nagasaki 857-1193, Japan.

E-mail: [email protected] Takehiko Kinoshita

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

Supported by GCOE ‘Fostering top leaders in mathematics’, Kyoto University.

E-mail: [email protected] Takuma Kimura

Sasebo National College of Technology, Nagasaki 857-1193, Japan.

E-mail: [email protected]

where J := (0, T)⊂ R, (T < ∞) is a bounded interval, Ω ⊂ R^d, (d = 1,2,3) a bounded convex polygonal or polyhedral domain, ν a positive parameter, b ∈ L^{∞`</sup>J;L<sup>∞</sup>(Ω)<sup>´d</sup>,c∈L<sup>∞`}J;L^∞(Ω)^´andg∈L^2`J;L²(Ω)^´.

We denote the operator which gives a solution of the problem (1) by Lt−1

. Therefore, the solution can be written asw=Lt−1g. In this paper, we present a numerical method to compute a positive constantC_L−1

t s.t.

kLt−1k_L(^{L2(J;L2(Ω)),L2(J;H}0¹(Ω)))≤C_Lt−1. (2) In case ofb= 0, following a priori estimates of (2) is known [9],

kLt−1k_L(^{L2(J;L2(Ω)),L2(J;H}0¹(Ω)))≤e^βTCp

ν . (3)

whereCpis a Poincar´e constant,ea Napier number andβa nonnegative parameter such tahtβ+c≥0 inΩ×J. However, in case ofβ >0 andT is large,e^βT can be very large.

Our a posteriori estimates will be expected more accurate than a priori esti- mates of (3). Moreover, our method may remove the exponential dependency of T in some numerical results.

Applying the estimations (2) or (3), we can develop a numerical verification method for a solution of (4) in a similar way in [6].

8>

><

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:

∂u

∂t −ν4u=f(t, x, u,∇u), in Ω×J, (4a)

u(x, t) = 0, on ∂Ω×J, (4b)

u(x,0) =u0, in Ω, (4c)

Estimate by a smaller value is preferable for estimate of (4). Therefore, our nu- merical method will be useful than using a priori estimates.

2 Function spaces and projections

In this section, we introduce the function spaces, projections to finite dimensional subspaces and these error estimates will be used in later.

LetSh(Ω)⊂H₀¹(Ω) be a finite dimensional subspace depend on the parameter h. For example,his the mesh size when we employ the finite element method. The basis functions are{φi}1≤i≤n, i.e.Sh(Ω) = span_1≤i≤n{φi}.

We define theL²-projectionP_h⁰:L²(Ω)→Sh(Ω) by

“

u−Ph⁰u, vh

”

L²(Ω)= 0, ∀vh∈Sh(Ω). (5) As an extension of (5), we defineP_h⁰:L^{2`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>→L<sup>2`}J;Sh(Ω)^´by

“

u(t)−Ph⁰u(t), vh

”

L²(Ω)= 0, ∀vh∈Sh(Ω), a.e. t∈J. (6) It is easy to show _‚

‚‚P_h^0‚‚_‚

L`

L²(J;L²(Ω)), L²(J;L²(Ω))´≤1, (7)

Also we define theH₀¹-projectionP_h¹:H₀¹(Ω)→Sh(Ω) by

“

u−Ph¹u, vh

”

H₀¹(Ω)= 0, ∀vh∈Sh(Ω). (8) As an extension of (8), we defineP_h¹:L^{2`</sup>J;H<sub>0</sub><sup>1</sup>(Ω)<sup>´</sup>→L<sup>2`}J;Sh(Ω)^´by

“

u(t)−P_h¹u(t), vh

”

H¹₀(Ω)= 0, ∀vh∈Sh(Ω), a.e. t∈J. (9) We denote the functional spaceV¹(J)⊂H¹(J) by

V¹(J) :=ⁿu∈H¹(J) ; u(0) = 0^o and the inner product by (u, v)_V1(J):=^`u⁰, v^0´_L2(J).

And denoteV^{1`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>⊂H<sup>1`}J;L²(Ω)^´ by

V^{1`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>:=<sup>n</sup>u∈H<sup>1`}J;L²(Ω)^´; u(0) = 0, in L²(Ω)^o and the inner product

(u, v)

V¹`

J;L²(Ω)´:= (ut, vt)

L²`

J;L²(Ω)´+ (u, v)

L²`

J;L²(Ω)´, where,ut= ^∂u_∂t.

LetV :=V^{1`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>∩L<sup>2`}J;H₀¹(Ω)^´andX(Ω) :={u∈L²(Ω);4u∈L²(Ω)}.

We assume the following estimates about P_h¹ holds.

Assumption 1 There exist a constantC(h)>0 satisfying,

‚‚

‚u−Ph¹u

‚‚

‚H¹₀(Ω)≤C(h)k4uk_L²_(Ω), ∀u∈H0¹(Ω)∩X(Ω), (10)

‚‚

‚u−Ph¹u

‚‚

‚L²(Ω)≤C(h)

‚‚

‚u−Ph¹u

‚‚

‚H₀¹(Ω), ∀u∈H0¹(Ω). (11) In many cases, the values ofC(h) satisfying Assumption 1 are known. For exam- ples, see [5].

3 Constructive a priori error estimates for a simple projection

In this section, we discuss the a priori estimate for linear heat equation. It is important to estimate a more general problem (1).

We consider a linear heat equation, 8>

><

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:

∂u

∂t −ν4u=g, in J×Ω, (12a)

u(x, t) = 0, on J×∂Ω, (12b)

u(x,0) = 0, in Ω, (12c)

for anyν >0 andg∈L^2`J;L²(Ω)^´. We define a weak solution of (12) satisfying

“∂u

∂t(t), v

”

L²(Ω)+ (∇u(t),∇v)_L²_(Ω)^d = (g(t), v)_L2(Ω), ∀v∈H0¹(Ω), a.e. t∈J, (13)

It is known that there exists a unique solution of (13) inL^2`J;H₀¹(Ω)^´ (e.g. [9]).

We define a linear differential operator4t by

4t:V ∩L^{2`</sup>J;X(Ω)<sup>´</sup>→L<sup>2`}J;L²(Ω)^´, 4t:= ∂

∂t−ν4 We have a following estimate corresponding to4t [4, Lemma 2].

Lemma 2 We have

‚‚

‚∂t^∂u

‚‚

‚L²`

J;L²(Ω)´ ≤ k4tuk_L2`

J;L²(Ω)´, ∀u∈V ∩L^{2`</sup>J;X(Ω)<sup>´</sup>. (14) We define theP<sub>h</sub><sup>V</sup>-projection P<sub>h</sub><sup>V</sup> :V →V<sup>1`}J;Sh(Ω)^´by

“∂

∂t

`u(t)−Ph^Vu(t)^´, vh

”

L²(Ω)+ν^“∇^`u(t)−Ph^Vu(t)^´,∇vh

”

L²(Ω)^d= 0,

∀vh∈Sh(Ω), a.e. t∈J, (15) P_h^V is an operator gives the semi-discrete approximation of a function in V. Es- pecially, we denote a semi-discrete solution of (13) byP_h^Vu.

Here, we introduce the following estimates corresponding toP_h^V-projection [4].

Lemma 3 We have

‚‚

‚_∂t^∂Ph^Vu

‚‚

‚L²`

J;L²(Ω)´ ≤ k4tuk_L2`

J;L²(Ω)´, ∀u∈V ∩L^2`J;X(Ω)^´, (16) Theorem 4 Under the Assumption 1, following inequality hold.

‚‚

‚u−Ph^Vu

‚‚

‚L²`

J;H₀¹(Ω)´≤ ^2C(h)_ν k4tuk_L2`

J;L²(Ω)´, ∀u∈V ∩L^2`J;X(Ω)^´, (17) Proof . See [4, Lemma 2].

The Aubin-Nitsche trick is a well-known technique to obtain a higher order a priori L² error estimation thanH0¹ estimates by considering the regularly dual problem. We represent a prioriL² error estimates used the Aubin-Nitsche trick by [3] for applied numerical verification methods.

Here, we consider the conjugate problem of (12), 8>

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:

∂w

∂t +ν4w=u_⊥, in Ω×J, (18a)

w(x, t) = 0, on ∂Ω×J, (18b)

w(x, T) = 0, in Ω, (18c)

whereu_⊥:=u−P_h^Vufor anyu∈V. Letwbe the weak solution of (18), satisfying

“∂

∂tw, v^”

L²(Ω)−ν(∇w,∇v)_L²_(Ω)^d= (u_⊥, v)_L2(Ω), ∀v∈H0¹(Ω), a.e. t∈J, (19) (13) is obtained by (19) and the variable transformations:=T−t. Hence putting V_∗^{1`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>:= <sup>˘</sup>w∈H<sup>1`}J;L²(Ω)^´; w(T) = 0, in L²(Ω)^¯, (19) has a unique solutionw∈V_∗^{1`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>∩L<sup>2`}J;H₀¹(Ω)^´. Moreover,w satisfies (14).

Next, letwh∈V_∗^1`J;Sh(Ω)^´be the semi-discrete approximate solution of the (19) which satisfying the following equation,

“∂

∂twh, vh

”

L²(Ω)−ν(∇wh,∇vh)_L2(Ω)^d = (u_⊥, vh)_L2(Ω),

∀vh∈Sh(Ω), a.e. t∈J. (20) We remark that the error estimates (17) is consisted forw andwh.

Theorem 5 Under the Assumption 1, we have the following inequality,

‚‚

‚u−P_h^Vu^‚‚_‚

L²`

J;L²(Ω)´≤4C(h)^‚‚_‚u−P_h^Vu^‚‚_‚

L²`

J;H₀¹(Ω)´, ∀u∈V. (21) Proof For arbitrary u ∈ V, we put u_⊥ := u−P_h^Vu. Let w ∈ V_∗^{1`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>∩ L<sup>2`}J;H₀¹(Ω)^´ be a solution of (19). For almost everywheret∈J then by taking the test functionv=u_⊥(t)∈H₀¹(Ω) in (19), we have

ku⊥(t)k²_L2(Ω)=^“_∂t^∂w, u_⊥^”

L²(Ω)−ν(∇w,∇u⊥)_L2(Ω)^d

= _dt^d (w, u_⊥)_L2(Ω)−^“w,_∂t^∂u_⊥

”

L²(Ω)−ν(∇w,∇u⊥)_L2(Ω)^d. By the definition ofP_h^V-projection, we get the following equality,

ku⊥(t)k²_L²_(Ω)=_dt^d (w, u_⊥)_L2(Ω)−^“w−wh,_∂t^∂u_⊥^”

L²(Ω)

−ν(∇(w−wh),∇u⊥)_L2(Ω)^d

=_dt^d (w, u_⊥)_L2(Ω)−_dt^d (w−wh, u_⊥)_L2(Ω)

+^“_∂t^∂(w−wh), u_⊥^”

L²(Ω)−ν(∇(w−wh),∇u⊥)_L2(Ω)^d, wherewh(t)∈Sh(Ω) is the solution of (20). SinceP_h¹u(t)∈Sh(Ω), from (19) and (20), we have

ku_⊥(t)k²_L²_(Ω)=_dt^d (wh, u_⊥)_L2(Ω)+^“_∂t^∂(w−wh), u−P_h¹u^”

L²(Ω)

−ν^“∇(w−wh),∇(u−Ph¹u)^”

L²(Ω)^d. We calculate the integration of this equation onJ ast, then we get

ku⊥k²_L2^`_J;L2(Ω)^´

=^“_∂t^∂(w−wh), u−P_h¹u^”

L²`

J;L²(Ω)´−ν^“∇(w−wh),∇(u−P_h¹u)^”

L²`

J;L²(Ω)´d

≤ kw−whk_V1(J;L²)

‚‚

‚u−P_h¹u

‚‚

‚L²(J;L²)+νkw−whk_L2(J;H¹₀)

‚‚

‚u−P_h¹u

‚‚

‚L²(J;H₀¹)

(22)

where we use the propertiesu_⊥(0) = 0 andwh(T) = 0. By the fact that the error ofP_h¹-projection is minimized of theH₀¹ norm and Assumption 1, we have

‚‚

‚u−P_h¹u^‚‚_‚

L²`

J;L²(Ω)´ ≤C(h)^‚‚_‚u−P_h¹u^‚‚_‚

L²`

J;H₀¹(Ω)´

≤C(h)^‚‚_‚u−P_h^Vu^‚‚_‚

L²`

J;H₀¹(Ω)´.

Next, we can obtain the same error estimates of Theorem 4 for the duality problem, then we have

kw−whk_L2`

J;H₀¹(Ω)´≤ ^2C(h)_ν ^‚‚‚∂t^∂w+ν4w^‚‚‚

L²`

J;L²(Ω)´= ^2C(h)_ν ku⊥k_L2`

J;L²(Ω)´. Finally, From Lemma 2 and Lemma 3, we have

kw−whk_V1`

J;L²(Ω)´≤2k4twk_L2`

J;L²(Ω)´= 2ku⊥k_L2`

J;L²(Ω)´. Therefore, these estimates using at (22), we obtain

ku⊥k²_L2`

J;L²(Ω)´≤4C(h)ku⊥k_L2`

J;L²(Ω)´ku⊥k_L2`

J;H₀¹(Ω)´.

4 A numerically verified a priori estimates for solutions of parabolic problems

In this section, we discuss the a priori estimation for (1), especially, computation ofC_L−1

t .

We define a linear parabolic operatorLtby

Lt:V ∩L^{2`</sup>J;X(Ω)<sup>´</sup>→L<sup>2`}J;L²(Ω)^´, Lt:= _∂t^∂ −ν4+ (b· ∇) +c. (23) Generally, it is known that Lt has the inverse operator(e.g. [9]). We denote the inverse operator ofLtbyL⁻t¹.

We define then×nmatricesLφ,Dφ andQφ by

Lφ,i,j := (φj, φi)_L2(Ω), Dφ,i,j:= (∇φj,∇φi)_L2(Ω)^d (24) Qφ,i,j :=ν(∇φj,∇φi)_L2(Ω)^d+ ((b· ∇)φj, φi)_L2(Ω)+ (cφj, φi)_L2(Ω). (25) Since Sh(Ω)⊂H₀¹(Ω),Dφ andLφ are symmetric positive definite matrices. Let D^1/2_φ andL^1/2_φ be the Cholesky factors ofDφ andLφ respectively, i.e.

Dφ =D^1/2_φ D^{T /2}_φ and Lφ=L^1/2_φ L^{T /2}_φ .

Here, we consider the semi-discrete finite element solution of (1). Denote a semi-discrete finite element solution bywh satisfying

“∂

∂twh, vh

”

L²(Ω)+ν(∇wh,∇vh)_L2(Ω)^d+ ((b· ∇)wh+cwh, vh)_L2(Ω)

= (g, vh)_L2(Ω), ∀v∈Sh(Ω), a.e. t∈J.(26)

There existαandβ s.t.wh=^Pn_i=1αi(t)φi(x),βi:= (g, φi)_L2(Ω), (i= 1,· · ·, n).

Then, (26) can be rewritten as

“ Lφ d

dt+Qφ

”

α=β, a.e. t∈J.

Therefore, if we can estimate^“Lφ d dt +Qφ

”₋1

then we can estimatewh. We define a positive constant

M_φ¹⁰(h) =

‚‚

‚‚D^{T /2}_φ ^“Lφ d dt +Qφ

”₋₁ L^1/2_φ

‚‚

‚‚_L`

L²(J)ⁿ, L²(J)ⁿ´. (27) We can compute an upper bound ofM_φ¹⁰(h) by applying the numerical method in- troduced in [2]. Also we define positive constantsCb:=

‚‚

‚ q

b²₁+· · ·+b²_d

‚‚

‚L^∞`

J;L^∞(Ω)´, C1:=Cb+Cpkck_L∞^{`</sup><sub>J;L∞(Ω)</sub><sup>´</sup> andC2:=Cb+ 4C(h)kck<sub>L∞</sub><sup>`}_J;L∞(Ω)^´.

Theorem 6 Under the Assumption 1, Letκφ>0 be satisfied κφ:= 2C(h)C2

`1 +C1M_φ¹⁰(h)^´< ν. (28) Then, we have following estimates,

‚‚

‚L⁻t¹

‚‚

‚L`

L²(J;L²(Ω)), L²(J;H¹₀(Ω))´≤ νM_φ¹⁰(h) + 2C(h) + 2C(h)C1M_φ¹⁰(h) ν−κφ

. (29) Proof For arbitraryg∈L^{2`</sup>J;L<sup>2</sup>(Ω)<sup>´</sup>, we putu:=L<sup>−</sup>t<sup>1</sup>g∈V ∩L<sup>2`}J;X(Ω)^´. We separate the differential equation that is satisfied ofu byP_h^V-projection as finite and infinite part,

∂u

∂t −ν4u+ (b· ∇)u+cu=g ⇐⇒ u=4⁻¹t

`−(b· ∇)u−cu+g^´

⇐⇒

( Ph^Vu=Ph^V4⁻¹t

`−(b· ∇)u−cu+g^´, (30a) (I−P_h^V)u= (I−P_h^V)4⁻t¹

`−(b· ∇)u−cu+g^´. (30b) In shortly, we denoteu_⊥:=u−P_h^Vuand∂t:= _∂t^∂. From (30a), almost everywhere t∈J and for arbitraryvh∈Sh(Ω), we have

“

∂tP_h^Vu(t), vh

”

L²(Ω)+ν^“∇Ph^Vu(t),∇vh

”

L²(Ω)^d

=

“

∂tPh^V4⁻¹t

`−(b(t)· ∇)u(t)−c(t)u(t) +g(t)^´, vh

”

L²(Ω)

+ν^“∇Ph^V4⁻t¹

`−(b(t)· ∇)u(t)−c(t)u(t) +g(t)^´,∇vh

”

L²(Ω)^d

= (−(b· ∇)u−cu+g, vh)_L2(Ω)

(31)

“

∂tPh^Vu, vh

”

L²(Ω)+ν^“∇Ph^Vu,∇vh

”

L²(Ω)^d+^“(b· ∇)Ph^Vu+cPh^Vu, vh

”

L²(Ω)

= (−(b· ∇)u_⊥−cu_⊥+g, vh)_L2(Ω)

=^“Ph⁰

`−(b· ∇)u⊥−cu_⊥+g^´, vh

”

L²(Ω). (32)

From P_h^Vu and P_h^{0`</sup>−(b· ∇)u<sub>⊥</sub>−cu<sub>⊥</sub>+g<sup>´</sup> are elements of V<sup>1`}J;Sh(Ω)^´ and L^2`J;Sh(Ω)^´, respectively. Therefore, these are expressible by linear combination of the basis of Sh(Ω). Namely, there exists α := (α1,· · ·, αn) ∈ V¹(J)ⁿ and β:= (β1,· · ·, βn)∈L²(J)ⁿsuch that

Ph^Vu(x, t) = Xn i=1

αi(t)φi(x), Ph⁰

`−(b· ∇)u⊥−cu_⊥+g^´(x, t) = Xn i=1

βi(t)φi(x).

(32) is rewritten by usingαandβ, then we have

Lφα⁰+Qφα=Lφβ, (33) where the matricesLφ andQφare defined by (24) and (25), respectively. By (33), theH₀¹ norm ofP_h^Vuattis satisfying that

‚‚

‚P_h^Vu(t)

‚‚

‚²

H¹₀(Ω)=α(t)^TDφα(t)

=^“D^{T /2}_φ α(t)^”TDφ

“ Lφd

dt+Qφ(t)^”−1L^1/2_φ ^“L^{T /2}_φ β(t)^”. We integrate this equation aboutton J, then we get

‚‚

‚Ph^Vu

‚‚

‚²

L²`

J;H¹₀(Ω)´

= Z

J

“

D_φ^{T /2}α(t)^”TDφ

“ Lφd

dt +Qφ(t)^”−1L^1/2_φ ^“L^{T /2}_φ β(t)^”dt

≤^‚‚‚D_φ^{T /2}α

‚‚

‚L²(J)ⁿ

‚‚

‚‚Dφ

“ Lφ d

dt+Qφ

”₋1

L^1/2_φ

‚‚

‚‚_L`

L²(J)ⁿ, L²(J)ⁿ´

‚‚

‚L^{T /2}_φ β

‚‚

‚L²(J)ⁿ

≤^‚‚‚Ph^Vu

‚‚

‚L²`

J;H¹₀(Ω)´Mφ¹⁰(h)

‚‚

‚Ph⁰

`−(b· ∇)u⊥−cu_⊥+g^´‚‚_‚

L²`

J;L²(Ω)´, where we are using (27). Therefore, by (7), we have

‚‚

‚P_h^Vu

‚‚

‚L²`

J;H¹₀(Ω)´≤M_φ¹⁰(h)k−(b· ∇)u⊥−cu_⊥+gk_L2`

J;L²(Ω)´. Moreover, from (21), we have

‚‚

‚Ph^Vu

‚‚

‚L²`

J;H₀¹(Ω)´=C2Mφ¹⁰(h)ku⊥k_L2`

J;H₀¹(Ω)´+Mφ¹⁰(h)kgk_L2`

J;L²(Ω)´. (34) Next, we calculateL^2`J;H₀¹(Ω)^´norm of (30b). By (17), we have

ku⊥k_L2`

J;H₀¹(Ω)´=

‚‚

‚(I−Ph^V)4⁻¹t

`−(b· ∇)u−cu+g^´‚‚_‚

L²`

J;H₀¹(Ω)´

≤ ^2C(h)_ν k−(b· ∇)u−cu+gk_L2`

J;H₀¹(Ω)´

≤ ^2C(h)_ν Cb

„‚‚‚P_h^Vu^‚‚_‚

L²`

J;H₀¹(Ω)´+ku_⊥k_L2`

J;H₀¹(Ω)´«

+^2C(h)_ν kgk_L2`

J;L²(Ω)´ +^2C(h)_ν kck_L∞^`_J;L∞(Ω)^´

„ Cp

‚‚

‚P_h^Vu^‚‚_‚

L²`

J;H¹₀(Ω)´+ku_⊥k_L2`

J;L²(Ω)´« .

From (21) and (34), we have ku_⊥k_L2`

J;H₀¹(Ω)´

≤^2C(h)_ν ^“Cb+Cpkck_L∞(J;L^∞)

” “

C2Mφ¹⁰(h)ku⊥k_L²_(J;H¹

0)+Mφ¹⁰(h)kgk_L²_(J;L²₎^” +^2C(h)_ν ^“Cb+ 4C(h)kck_L∞(J;L^∞)

”ku_⊥k_L2(J;H₀¹)+ ^2C(h)_ν kgk_L2(J;L²)

=^2C(h)_ν C2

`1 +C1Mφ¹⁰(h)^´ku⊥k_L²_(J;H¹

0)+ ^2C(h)_ν ^`1 +C1Mφ¹⁰(h)^´kgk_L²_(J;L²₎. By the assumptionκφ:= 2C(h)C2

`1 +C1M_φ¹⁰(h)^´< ν, we have

ku_⊥k_L2`

J;H¹₀(Ω)´ ≤2C(h)1 +C1M_φ¹⁰(h) ν−κφ kgk_L2`

J;L²(Ω)´. (35) We use (35) at (34), then we have

‚‚

‚Ph^Vu

‚‚

‚L²`

J;H₀¹(Ω)´≤ νM_φ¹⁰(h) ν−κφ kgk_L2`

J;L²(Ω)´. (36) Finally, we obtain the following estimates by using (35) and (36),

kuk_L2`

J;H₀¹(Ω)´≤^‚‚‚P_h^Vu

‚‚

‚L²`

J;H₀¹(Ω)´+ku_⊥k_L2`

J;H¹₀(Ω)´

≤ νM_φ¹⁰(h) + 2C(h) + 2C(h)C1M_φ¹⁰(h) ν−κφ kgk_L2`

J;L²(Ω)´. Therefore, this proof is completed.

5 Numerical examples

We use the following nonlinear problem as the example:

8>

<

>:

Ltw≡ _∂t^∂w−ν∆w−2u^k_hw=g(w), inΩ×J, (37a)

w(x, t) = 0, on∂Ω×J, (37b)

w(x,0) = 0, inΩ, (37c)

whereu^k_h is an approximate solution of following nonlinear problem, 8>

<

>:

∂

∂tu−ν∆u=u²+f(x, t), inΩ×J, (38a)

u(x, t) = 0, on∂Ω×J, (38b)

u(x,0) = 0, inΩ, (38c)

(37) is the linearized problem of (38).

Let Ω= (0,1). We perform the numerical experiments for two test problems:

• u(x, t) = 0.5tsin(πx), ν= 0.1,(Example 1);

• u(x, t) = sin(πt) sin(πx), ν= 0.1,(Example 2);

Fig. 1 Example 1. Fig. 2 Example 2.

We compare the values of e^{βT C}_ν^p in (3) and C_L−1

t by (29). We used Sh(Ω) be spanned by piecewise linear functions withn= 5.u^k_h is the linear interpolation of u.

Numerical results of Example 1 and 2 are given in Fig.1 and 2, respectively.

All computations in the tables are carried out on a Dell Precision T7500 (Intel Xeon x5680, 72GB of memory) with MATLAB R2010b. The computation errors has been taken into account by using INTLAB [7], a Toolbox of MATLAB.

Remark 7 In the tables, the values computed by our numerical method are al- ways smaller than the apriori estimates. Moreover, the exponential dependency on T were not observed in results of the our method.

6 Conclusions

We presented a posteriori estimates of inverse operators for linear parabolic dif- ferential equations (1).

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