R E S E A R C H
Open Access
A posteriori
error estimates of the lowest
order Raviart-Thomas mixed finite element
methods for convective diffusion optimal
control problems
Yuchun Hua and Yuelong Tang
**Correspondence:
[email protected] Institute of Computational Mathematics, Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou, Hunan 425100, China
Abstract
In this paper, we consider the mixed finite element methods for quadratic optimal control problems governed by convective diffusion equations. The state and the co-state are discretized by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. Using some proper duality problems, we derivea posteriori L2(0,T;L2()) error estimates for the scalar functions. Such estimates, which are apparently not available in the literature, are an important step toward developing reliable adaptive mixed finite element approximation schemes for the control problem.
MSC: 49J20; 65N30
Keywords: parabolic equations; optimal control problems;a posteriorierror estimates; mixed finite element methods
1 Introduction
As far as we know, optimal control problems [] have been extensively utilized in many aspects of the modern life such as social, economic, scientific, and engineering numeri-cal simulation. Thus, they must be solved successfully with efficient numerinumeri-cal methods. Among these numerical methods, finite element method is a good choice. There have been extensive studies in the convergence of finite element approximation of optimal control problems; see [–]. A systematic introduction to finite element methods for PDEs and optimal control can be found for example in [–].
Recently, the adaptive finite element method has been investigated extensively. It has become one of the most popular methods in the scientific computation and numerical modeling. An adaptive finite element approximation ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate, indicated bya posteriorierror estimators. Hence it is an important approach to boost the accuracy and efficiency of finite element discretizations. There are lots of works concen-trating on the adaptivity of various optimal control problems. See, for example, [–].
In many control problems, the objective functional contains the gradient of the state variables. Thus, the accuracy of the gradient is important in numerical discretization of
the coupled state equations. Mixed finite element methods are appropriate for the state equations in such cases since both the scalar variable and its flux variable can be approxi-mated to the same accuracy by using such methods; see, for example, [–].
We shall use the lowest order Raviart-Thomas mixed finite element to discretize the state and the co-state, and use the piecewise constant space to approximate the control variable. Using some proper duality problems, we derivea posteriori L(,T;L()) error estimates for the scalar functions. The optimal control problems that we are interested in are as follows:
min
u∈K⊂U
T
p– pd+y–yd+u
dt
, (.)
yt+divp+cy=f+u, x∈,t∈J, (.)
p= –a(∇y+ by), x∈,t∈J, (.)
y(x,t) = , x∈∂,t∈J, y(x, ) =y(x), x∈, (.)
where the bounded open set⊂Ris a convex polygon with the boundary∂.J= [,T]. LetKbe a closed convex set in the control spaceU=L(J;L()), p, p
d∈(L(J;H())),
u,y,yd∈L(J;H()),f ∈L(J;L()),y(x)∈H(). Moreover, we assume that <a≤
a≤a,a(x)∈W,∞(),c(x)∈W,∞(), b(x)∈(W,∞()).
We assume that the constraint on the control is an obstacle such that
K=u∈U:u(x,t)≥, a.e. in×J.
In this paper, we adopt the standard notationWm,p() for Sobolev spaces onwith a
norm · m,p given byvmp,p= |α|≤mDαvpLp(), a semi-norm| · |m,p given by|v|pm,p=
|α|=mD
αvp
Lp(). We set Wm,p() = {v∈Wm,p() :v|∂ = }. For p= , we denote
Hm() =Wm,(),Hm
() =Wm,(), and · m= · m,, · = · ,.
We denote byLs(,T;Wm,p()) the Banach space of allLsintegrable functions fromJ
intoWm,p() with normv
Ls(J;Wm,p())= (T
vsWm,p()dt)
s fors∈[,∞), and the
stan-dard modification for s=∞. Similarly, one can define the spaces Hl(J;Wm,p()) and
Ck(J;Wm,p()). In additionCdenotes a general positive constant independent ofhand t, wherehis the spatial mesh-size for the control and state discretization andtis the time increment.
The plan of this paper is as follows. In next section, we shall give a brief review on the mixed finite element method and the backward Euler discretization, and then we con-struct the approximation for the optimal control problems (.)-(.). Then, using two duality problems, we derivea posteriori L(,T;L()) error estimates for the scalar func-tions in Section . Finally, we give a conclusion and indicate some possible future work.
2 Mixed methods of parabolic optimal control problems
In this section, we shall study the mixed finite element and the backward Euler discretiza-tion approximadiscretiza-tion of convective diffusion optimal control problems (.)-(.). For the
sake of simplicity, we assume that the domainis a convex polygon. Now, we introduce
the co-state parabolic equation
–zt–div
a(∇z+ p – pd)
which can be written in the form of the first order system
–zt+divq–a–b·q+cz=y–yd, q= –a(∇z+ p – pd),x∈,t∈J (.)
and
z(x,t) = , x∈∂,t∈J, z(x,T) = , x∈. (.)
To be definite, we shall take the state spaces L =L(J; V) andQ=H(J;W), where V and
W are defined as follows:
V=H(div;) =v∈L(),divv∈L(), W=L().
The Hilbert space V is equipped with the following norm:
vH(div;)=
v,+divv,/.
Letα=a–andβ=αb. We recast (.)-(.) as the following weak form: find (p,y,u)∈
L×Q×Ksuch that
min
u∈K⊂U
T
p– pd+y–yd+u
dt
, (.)
(αp, v) – (y,divv) + (βy, v) = , ∀v∈V,t∈J, (.)
(yt,w) + (divp,w) + (cy,w) = (f +u,w), ∀w∈W,t∈J, (.)
y(x, ) =y(x), ∀x∈. (.)
It follows from [] and [] that the optimal control problem (.)-(.) has a unique solution (p,y,u), and that a triplet (p,y,u) is the solution of (.)-(.) if and only if there is a co-state (q,z)∈L×Qsuch that (p,y, q,z,u) satisfies the following optimality conditions:
(αp, v) – (y,divv) + (βy, v) = , ∀v∈V,t∈J, (.)
(yt,w) + (divp,w) + (cy,w) = (f +u,w), ∀w∈W,t∈J, (.)
y(x, ) =y(x), ∀x∈, (.)
(αq, v) – (z,divv) = –(p – pd, v), ∀v∈V,t∈J, (.)
–(zt,w) + (divq,w) – (β·q,w) + (cz,w) = (y–yd,w), ∀w∈W,t∈J, (.)
z(x,T) = , ∀x∈, (.)
T
(u+z,u˜–u)dt≥, ∀˜u∈K, (.)
where (·,·) is the inner product ofL().
Let Th be regular triangulations of . hτ is the diameter of τ and h=maxhτ. Let
the triangulationsThof.Pkdenotes the space of polynomials of total degree of at most
k(k≥). Let V(τ) ={v∈P
(τ) +x·P(τ)},W(τ) =P(τ). We define Vh:=
vh∈V:∀τ∈Th, vh|τ∈V(τ)
,
Wh:=
wh∈W:∀τ∈Th,wh|τ∈W(τ)
,
Kh:=K∩Wh.
The mixed finite element discretization of (.)-(.) is as follows: compute (ph,yh,uh)∈
L(J; Vh)×H(J;Wh)×Khsuch that
min
uh(t)∈Kh
T
ph– pd+yh–yd+uh
dt
, (.)
(αph, vh) – (yh,divvh) + (βyh, vh) = , ∀vh∈Vh,t∈J, (.)
(yht,wh) + (divph,wh) + (cyh,wh) = (f+uh,wh), ∀wh∈Wh,t∈J, (.)
yh(x, ) =yh(x), ∀x∈, (.)
whereyh
(x)∈Wh is an approximation ofy. The optimal control problem (.)-(.) again has a unique solution (ph,yh,uh), and that a triplet (ph,yh,uh) is the solution of
(.)-(.) if and only if there is a co-state (qh,zh)∈L(J; Vh)×H(J;Wh) such that
(ph,yh, qh,zh,uh) satisfies the following optimality conditions:
(αph, vh) – (yh,divvh) + (βyh, vh) = , ∀vh∈Vh,t∈J, (.)
(yht,wh) + (divph,wh) + (cyh,wh) = (f+uh,wh), ∀wh∈Wh,t∈J, (.)
yh(x, ) =yh(x), ∀x∈, (.)
(αqh, vh) – (zh,divvh) = –(ph–pd, vh), ∀vh∈Vh,t∈J, (.)
–(zht,wh) + (divqh,wh) – (β·qh,wh) + (czh,wh)
= (yh–yd,wh), ∀wh∈Wh,t∈J, (.)
zh(x,T) = , ∀x∈, (.)
T
(uh+zh,u˜h–uh)dt≥, ∀˜uh∈Kh. (.)
We now consider the fully discrete approximation for the above semidiscrete problem. Lett> ,N=T/t∈Z, andti=it,i∈Z. Also, let
dtψi=
ψi–ψi– t .
We address the fully discrete approximation scheme to find (pih,yih,uih)∈Vh×Wh×Kh,
i= , , . . . ,N, such that
min
ui h∈Kh
N
i=
tpih– pid+yhi –yid+uih
αpih, vh
–yih,divvh
+βyih, vh
= , ∀vh∈Vh, (.)
dtyih,wh
+divpih,wh
+cyih,wh
=fi+uih,wh
, ∀wh∈Wh, (.)
yh(x) =yh(x), ∀x∈, (.)
wherefi=fi(x) =f(x,t
i),yid=yd(x,ti), and pid= pd(x,ti).
It follows that the control problem (.)-(.) has a unique solution (pi
h,yih,uih),
i= , , . . . ,N, and that a triplet (pi
h,yih,uih)∈Vh ×Wh× Kh, i= , , . . . ,N, is the
so-lution of (.)-(.) if and only if there is a co-state (qi–
h ,zih–)∈Vh×Wh such that
(pih,yih, qih–,zhi–,uih)∈(Vh×Wh)×Khsatisfies the following optimality conditions:
αpih, vh
–yih,divvh
+βyih, vh
= , ∀vh∈Vh, (.)
dtyih,wh
+divpih,wh
+cyih,wh
=fi+uih,wh
, ∀wh∈Wh, (.)
yh(x) =yh(x), ∀x∈, (.)
αqih–, vh
–zih–,divvh
= –pih– pid, vh
, ∀vh∈Vh, (.)
–dtzih,wh
+divqih–,wh
–β·qih–,wh
+czih–,wh
=yih–yid,wh
, ∀wh∈Wh, (.)
zhN(x) = , ∀x∈, (.)
uih+zih–,u˜h–uih
≥, ∀˜uh∈Kh. (.)
Fori= andi=N, we let
αph, vh
–yh,divvh
+βyh, vh
= , ∀vh∈Vh, (.)
αqNh, vh
–zNh,divvh
= –pNh – pNd, vh
, ∀vh∈Vh. (.)
Fori= , , . . . ,N, let
Yh|(ti–,ti]=
(ti–t)yhi–+ (t–ti–)yih
/t,
Zh|(ti–,ti]=
(ti–t)zhi–+ (t–ti–)zih
/t,
Ph|(ti–,ti]=
(ti–t)phi–+ (t–ti–)pih
/t,
Qh|(ti–,ti]=
(ti–t)qhi–+ (t–ti–)qih
/t,
Uh|(ti–,ti]=u
i h.
For any functionw∈C(J;L()), let
ˆ
w(x,t)|t∈(ti–,ti]=w(x,ti), w˜(x,t)|t∈(ti–,ti]=w(x,ti–).
Moreover, we let
¯
pd|(ti–,ti]=
(ti–t)pid+ (t–ti–)pid+
/t, i= , , . . . ,N– , p¯d|(tN–,tN]= p
N d,
¯
Ph|(ti–,ti]=
(ti–t)pih+ (t–ti–)pih+
/t, i= , , . . . ,N– , P¯h|(tN–,tN]= p
Then the optimality conditions (.)-(.) satisfy
(αPˆh, vh) – (Yˆh,divvh) + (βYˆh, vh) = , ∀vh∈Vh, (.)
(Yht,wh) + (divPˆh,wh) + (cYˆh,wh) = (fˆ+Uh,wh), ∀wh∈Wh, (.)
Yh(x, ) =yh(x), ∀x∈, (.)
(αQ˜h, vh) – (Z˜h,divvh) = –(Pˆh–pˆd, vh), ∀vh∈Vh, (.)
–(Zht,wh) + (divQ˜h,wh) – (β· ˜Qh,wh) + (cZ˜h,wh) = (Yˆh–yˆd,wh), ∀wh∈Wh, (.)
Zh(x,T) = , ∀x∈, (.)
(Uh+Z˜h,u˜h–Uh)≥, ∀˜uh∈Kh. (.)
In the rest of the paper, we shall use some intermediate variables. For any control func-tionUh∈Kh, we first define the state solution (p(Uh),y(Uh), q(Uh),z(Uh)) to satisfy
αp(Uh), v
–y(Uh),divv
+βy(Uh), v
= , ∀v∈V, (.)
yt(Uh),w
+divp(Uh),w
+cy(Uh),w
= (f +Uh,w), ∀w∈W, (.)
y(Uh)(x, ) =y(x), ∀x∈, (.)
αq(Uh), v
–z(Uh),divv
= –p(Uh) – pd, v
, ∀v∈V, (.)
–zt(Uh),w
+divq(Uh),w
–β·q(Uh),w
+cz(Uh),w
=y(Uh) –yd,w
, ∀w∈W, (.)
z(Uh)(x,T) = , ∀x∈. (.)
LetRh:W→Whbe the orthogonalL()-projection intoWh[], which satisfies
(Rhw–w,χ) = , w∈W,χ∈Wh, (.)
Rhw–w,q≤Chw,q, ifw∈W∩W,q(). (.)
Let h: V→Vh be the Raviart-Thomas projection operator [], which satisfies: for
any v∈V,
E
wh(v – hv)·νEds= , wh∈Wh,E∈Eh, (.)
τ
(v – hv)·vhdx dy= , vh∈Vh,τ ∈Th, (.)
whereEhdenotes the set of element sides inTh.
We have the commuting diagram property
div◦ h=Rh◦div: V→Wh and div(I– h)V⊥Wh, (.)
whereIdenotes the identity operator.
Further, the interpolation operator hsatisfies a local error estimate:
v– hv,≤Ch|v|,Th, v∈V∩H
(T
3 A posteriori error estimates
In this section we studya posteriorierror estimates for the mixed finite element approxi-mation to the parabolic optimal control problems.
For the following analysis, we divide the domaininto three parts:
–=
x∈:Z˜h(x)≤
,
=
x∈:Z˜h(x) > ,Uh(x) =
,
+=
x∈:Z˜h(x) > ,Uh(x) >
.
It is easy to see that the partition of the above three subsets is dependent ont. For allt, the three subsets are not intersected each other, and
¯
=¯–∪ ¯∪ ¯+.
Firstly, let us derive thea posteriorierror estimates for the controlu.
Theorem . Let(y, p,z, q,u)and(Yh,Ph,Zh,Qh,Uh)be the solutions of(.)-(.)and
(.)-(.),respectively.Then we have
u–UhL(J;L())≤Cη+Z˜h–z(Uh)
L(J;L()), (.)
where
η=Uh+Z˜hL(J;L(–∪+)).
Proof It follows from (.) that
u–UhL(J;L())
=
T
(u–Uh,u–Uh)dt
=
T
(u+z,u–Uh)dt+
T
(Uh+Z˜h,Uh–u)dt
+
T
˜
Zh–z(Uh),u–Uh
dt+
T
z(Uh) –z,u–Uh
dt
≤
T
(Uh+Z˜h,Uh–u)dt+
T
˜
Zh–z(Uh),u–Uh
dt
+
T
z(Uh) –z,u–Uh
dt
=:I+I+I. (.)
We first estimateI. Note that
I=
T
(Uh+Z˜h,Uh–u)dt
=
T
–∪+(Uh+
˜
Zh)(Uh–u)dx dt+
T
It is easy to see that
T
–∪+(Uh+
˜
Zh)(Uh–u)dx dt
≤C(δ)Uh+Z˜hL(J;L(–∪+))+δu–UhL(J;L(–∪+))
=C(δ)η+δu–UhL(J;L()), (.)
whereδis an arbitrary small positive number,C(δ) is dependent onδ–. Furthermore, we have
Uh+Z˜h≥ ˜Zh> , Uh–u= –u≤ on.
It yields
T
(Uh+Z˜h)(Uh–u)dx dt≤. (.)
Then (.)-(.) imply that
I≤C(δ)η+δu–UhL(J;L()). (.)
Moreover, it is clear that
I=
T
˜
Zh–z(Uh),u–Uh
dt
≤C(δ)Z˜h–z(Uh)
L(J;L())+δu–UhL(J;L()). (.)
Now we turn toI. Note that
y(x, ) =y(Uh)(x, ) =y(x) and z(x,T) =z(Uh)(x,T) = .
Then from (.)-(.) and (.)-(.), we have
I=
T
z(Uh) –z,u–Uh
dt=
T
u–Uh,z(Uh) –z
dt
=
T
y–y(Uh)
t,z(Uh) –z
+divp– p(Uh)
,z(Uh) –z
dt
+
T
cy–y(Uh)
,z(Uh) –z
–βy–y(Uh)
, q(Uh) – q
dt
–
T
αp– p(Uh)
, q(Uh) – q
–y–y(Uh),div
q(Uh) – q
dt
=
T
–z(Uh) –z
t,y–y(Uh)
+divq(Uh) – q
,y–y(Uh)
dt
+
T
cz(Uh) –z
,y–y(Uh)
–β·q(Uh) – q
,y–y(Uh)
–
T
αq(Uh) – q
, p – p(Uh)
–z(Uh) –z,div
p– p(Uh)
dt
=
T
y(Uh) –y,y–y(Uh)
+p(Uh) – p, p – p(Uh)
dt≤. (.)
Thus, we obtain from (.) and (.)-(.)
u–UhL(J;L())≤Cη+Z˜h–z(Uh)
L(J;L()), (.)
which proves (.).
In order to estimate the error ˜Zh–z(Uh)L(J;L()), we need the following well-known
stability results (see [, ] for the details) for the following dual equations:
⎧ ⎪ ⎨ ⎪ ⎩
φt–div(a∇φ+ bφ) +cφ=F, x∈,t∈J, φ|∂= , t∈J,
φ(x, ) = , x∈
(.)
and
⎧ ⎪ ⎨ ⎪ ⎩
–ψt–div(a∇ψ) + b· ∇ψ+cψ=F, x∈,t∈J, ψ|∂= , t∈J,
ψ(x,T) = , x∈.
(.)
Lemma .[] Letφandψbe the solutions of(.)and(.),respectively.Letbe a convex domain.Then,forϕ=φorϕ=ψ,
ϕ(x,t)dx≤CFL(J;L()), ∀t∈J, T
|∇ϕ|dx dt≤CFL(J;L()), T
Dϕdx dt≤CFL(J;L()), T
|ϕt|dx dt≤CFL(J;L()),
where|Dϕ|=max{|∂ϕ/∂x
i∂xj|, ≤i,j≤}.
We also need the following Gronwall lemma.
Lemma .[] Let f and g be piecewise continuous nonnegative functions defined on
≤t≤T,g being non-decreasing.If,for each t∈J,
f(t)≤g(t) +
t
f(s)ds, (.)
then f(t)≤etg(t).
Theorem . Let(Yh,Ph,Zh,Qh,Uh)and(y(Uh), p(Uh),z(Uh), q(Uh),Uh)be the solutions
of(.)-(.)and(.)-(.),respectively.Then we have
Yh–y(Uh)
L(J;L())≤C
i=
ηi, (.)
where
η=
T
τ
hτ
τ
(Yht+divPˆh+cYˆh–fˆ–Uh)dx dt;
η=
T
τ
hτ
τ
(αPh+βYh)dx dt; η=ˆPh–PhL(J;L());
η=ˆf –fL(J;L()); η= ˆYh–YhL(J;L()); η=yh(x) –y(x)
L().
Proof From (.) and (.), we get the equality
(αPh, vh) – (Yh,divvh) + (βYh, vh) = , ∀vh∈Vh. (.)
Letψ be the solution of (.) withF=Yh–y(Uh), using (.)-(.), (.)-(.), and
(.)-(.), we infer that
Yh–y(Uh)
L(J;L())
=
T
Yh–y(Uh),F
dt = T
Yh–y(Uh), –ψt–div(a∇ψ) + b· ∇ψ+cψ
dt = T
Yh–y(Uh)
t,ψ
–Yh,div
h(a∇ψ)
+p(Uh),∇ψ
dt + T
(bYh,∇ψ) +
cYh–y(Uh)
,ψdt+Yh–y(Uh)
(x, ),ψ(x, )
=
T
Yh–y(Uh)
t,ψ
–αPh, h(a∇ψ)
–βYh, h(a∇ψ)
–divp(Uh),ψ
dt + T
(βYh,a∇ψ) +
cYh–y(Uh)
,ψdt+yh(x) –y(x),ψ(x, )
=
T
(Yht,ψ) +
αPh,a∇ψ– h(a∇ψ)
– (Pˆh–Ph,∇ψ) – (divPˆh,ψ)
dt + T
βYh,a∇ψ– h(a∇ψ)
+ (cYh–f–Uh,ψ)
dt+yh(x) –y(x),ψ(x, )
=
T
(Yht+divPˆh+cYˆh–fˆ–Uh,ψ)dt+
T
αPh+βYh,a∇ψ– h(a∇ψ)
dt + T
(fˆ–f,ψ) +c(Yh–Yˆh),ψ
+ (Pˆh–Ph,∇ψ)
dt+yh(x) –y(x),ψ(x, )
Using (.), (.), the Cauchy inequality, and Lemma ., we have
L=
T
(Yht+divPˆh+cYˆh–fˆ–Uh,ψ–Phψ)dt
≤C(δ)η+δψL(J;H())
≤Cη+
Yh–y(Uh)
L(J;L()). (.)
Similarly, using the Cauchy inequality and Lemma ., we have
L≤Cη+
Yh–y(Uh)
L(J;L()), (.)
L≤C
η+η+η+
Yh–y(Uh)
L(J;L()), (.)
L≤Cη+
Yh–y(Uh)
L(J;L()). (.)
Hence, using (.)-(.), we get
Yh–y(Uh)
L(J;L())≤C
i=
ηi. (.)
This proves (.).
Theorem . Let(y, p,z, q,u)and(Yh,Ph,Zh,Qh,Uh)be the solutions of(.)-(.)and
(.)-(.),respectively.Let(y(Uh), p(Uh),z(Uh), q(Uh),Uh)be defined as in(.)-(.).
Then we have the following error estimate:
Z˜h–z(Uh)L(J;L())≤C
i=,,–
ηi +CYh–y(Uh)L(J;L()), (.)
where
η=
T
τ
hτ
τ
(–Zht+divQ˜h–β· ˜Qh+cZ˜h–Yˆh+yˆd)dx dt;
η=
T
τ
hτ
τ
(αQh+P¯h–p¯d)dx dt; η= ˜Qh–QhL(J;L());
η=¯Ph–PhL(J;L()); η= ˜Zh–ZhL(J;L());
η=¯pd– pdL(J;L()); η =ˆyd–ydL(J;L()),
ηandηare defined in Theorem..
Proof Similar to (.), using (.), (.), and the definitions ofZh,Qh,P¯h, andp¯d, we
get
Letφ be the solution of (.) with F=Zh–z(Uh). Then it follows from (.)-(.),
(.)-(.), and (.)-(.) that
Zh–z(Uh)
L(J;L())
=
T
Zh–z(Uh),F
dt = T
Zh–z(Uh),φt–div(a∇φ+ bφ) +cφ
dt = T
–Zh–z(Uh)
t,φ
–Zh,div
h(a∇φ+ bφ)
dt + T
αq(Uh) + p(Uh) – pd,a∇φ+ bφ
dt+
T
cZh–z(Uh)
,φdt
=
T
–Zh–z(Uh)
t,φ
–αQh+P¯h–p¯d, h(a∇φ+ bφ)
dt + T
p(Uh) – pd,a∇φ+ bφ
–divq(Uh),φ
+β·q(Uh),φ
dt + T
cZh–z(Uh)
,φdt
=
T
–Zh–z(Uh)
t,φ
+αQh+P¯h–p¯d,a∇φ+ bφ– h(a∇φ+ bφ)
dt + T
α(Q˜h–Qh) –αQ˜h,a∇φ+ bφ
–divq(Uh),φ
+β·q(Uh),φ
dt + T
cZh–z(Uh),φ
dt+
T
p(Uh) –P¯h+p¯d– pd,a∇φ+ bφ
dt
=
T
(–Zht+divQ˜h–β· ˜Qh+cZ˜h–Yˆh+yˆd,φ)dt+
T
c(Zh–Z˜h),φ
dt + T
αQh+P¯h–p¯d,a∇φ+ bφ– h(a∇φ+ bφ)
dt + T
α(Q˜h–Qh),a∇φ+ bφ
dt+
T
yd–ˆyd+Yˆh–y(Uh),φ
dt + T
p(Uh) –P¯h+p¯d– pd,a∇φ+ bφ
dt
=:J+J+· · ·+J. (.)
First, using the same estimates as (.)-(.), we have
J≤Cη+
Zh–z(Uh)
L(J;L()), (.)
J≤Cη+
Zh–z(Uh)
L(J;L()), (.)
J≤Cη+
Zh–z(Uh)
L(J;L()), (.)
J≤Cη +
Zh–z(Uh)
ForJ, using the Cauchy inequality and Lemma ., we have
J=
T
ˆ
Yh–Yh+Yh–y(Uh) +yd–yˆd,φ
dt
≤Cη+η+CYh–y(Uh)L(J;L())+
Zh–z(Uh)
L(J;L()). (.)
Finally, forJ, using (.), (.), the Cauchy inequality, and Lemma ., we derive
J=
T
p(Uh) –Ph+Ph–P¯h+p¯d– pd,a∇φ+ bφ
dt
=
T
αp(Uh) –Ph
,a∇φ+abφdt
+
T
(Ph–P¯h+p¯d– pd,a∇φ+ bφ)dt
=
T
y(Uh),div
a∇φ+abφ–βy(Uh),a∇φ+abφ
dt
+
T
αPh, h
a∇φ+abφ–a∇φ–abφdt
+
T
βYh, h
a∇φ+abφ–Yh,div
h
a∇φ+abφdt
+
T
(Ph–P¯h+p¯d– pd,a∇φ+ bφ)dt
=
T
y(Uh) –Yh,div
a∇φ+abφ+βYh–y(Uh)
,a∇φ+abφdt
+
T
αPh+βYh, h
a∇φ+abφ–a∇φ–abφdt
+
T
(Ph–P¯h+p¯d– pd,a∇φ+ bφ)dt
≤Cη+η +η+CYh–y(Uh)L(J;L())+
Zh–z(Uh)
L(J;L()). (.)
Therefore, it follows from the above estimates that
Zh–z(Uh)
L(J;L())≤C
i=,,–
ηi +CYh–y(Uh)
L(J;L()). (.)
The triangle inequality and (.) yield (.).
Remark . If we use the higher order RT mixed finite elements to approximate the state variables and the co-state variables, then the estimatorsη,η,η, andηin Theorem . and Theorem . can be improved by
η=
T
τ
hτ
τ
(Yht+divPˆh+cYˆh–fˆ–Uh)dx dt;
η=
T
τ
hτ
τ
η=
T
τ
hτ
τ
(–Zht+divQ˜h–β· ˜Qh+cZ˜h–Yˆh+yˆd)dx dt;
η=
T
τ
hτ
τ
(αQh+∇hZh+P¯h–p¯d)dx dt,
where∇hχ|τ=∇(χ|τ).
Let (p,y, q,z,u) and (Ph,Yh,Qh,Zh,Uh) be the solutions of (.)-(.) and (.)-(.),
respectively. We decompose the errors as follows:
p–Ph= p – p(Uh) + p(Uh) –Ph:=+ε,
y–Yh=y–y(Uh) +y(Uh) –Yh:=r+e, q–Qh= q – q(Uh) + q(Uh) –Qh:=+ε,
z–Zh=z–z(Uh) +z(Uh) –Zh:=r+e.
From (.)-(.) and (.)-(.), we derive the error equations:
(α, v) – (r,divv) + (βr, v) = , ∀v∈V, (.)
(rt,w) + (div,w) + (cr,w) = (u–Uh,w), ∀w∈W, (.)
(α, v) – (r,divv) = –(, v), ∀v∈V, (.)
–(rt,w) + (div,w) – (β·,w) + (cr,w) = (r,w), ∀w∈W. (.)
Theorem . There is a constant C> ,independent of h,such that
L(J;L())+rL(J;L())≤Cu–UhL(J;L()), (.)
L(J;L())+rL(J;L())≤Cu–UhL(J;L()). (.)
Proof Choosing v =andw=ras the test functions and add the two relations of (.)-(.), we have
(α,) + (rt,r) = (u–Uh,r) – (βr,) – (cr,r). (.)
Then, using the-Cauchy inequality, we can find an estimate as follows:
(a,) + (rt,r)≤C
rL()+u–UhL()
+
(a,). (.)
Note that
(rt,r) =
∂ ∂tr
L(),
then, using the assumption ona, we can obtain
a
L()+
∂ ∂tr
L()≤C
rL()+u–UhL()
Integrating (.) in time, and sincer() = , using Lemma . to get
L(J;L())+rL∞(J;L())≤Cu–UhL(J;L()), (.)
implies (.).
Similarly, we can obtain
L(J;L())+rL∞(J;L())≤C
L(J;L())+rL(J;L())
. (.)
Using (.) and (.), we complete the proof of Theorem ..
Collecting Theorems .-., we can derive the following results.
Theorem . Let(p,y, q,z,u)and(Ph,Yh,Qh,Zh,Uh)be the solutions of(.)-(.)and
(.)-(.),respectively.Then we have
u–UhL(J;L())+y–YhL(J;L())+z–ZhL(J;L())≤C
i=
ηi, (.)
whereηis defined in Theorem.,η, . . . ,ηare defined in Theorem.,andη, . . . ,ηare
defined in Theorems.,respectively.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The first author carried out the molecular genetic studies, participated in the sequence alignment, and drafted the manuscript. The second author conceived of the study, and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The first author is supported by the scientific research program in Hunan University of Science and Engineering (2015). The second author is supported by the National Natural Science Foundation of China (11401201), the Foundation of Hunan Educational Committee (13C338), and the construct program of the key discipline in Hunan University of Science and Engineering.
Received: 7 February 2015 Accepted: 13 August 2015 References
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