R E S E A R C H
Open Access
A posteriori
error estimates of mixed finite
element solutions for fourth order parabolic
control problems
Chunjuan Hou
1*, Yanping Chen
2and Zuliang Lu
3*Correspondence:
1Department of Accounting,
Huashang College, Guangdong University of Finance, Guangzhou, 511300, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, a fourth order quadratic parabolic optimal control problem is analyzed. The state and co-state are discretized by the orderkRaviart-Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of orderk
(k≥0). At last, the results ofa posteriorierror estimates are given in Lemma 2.1 by using mixed elliptic reconstruction methods.
Keywords: optimal control problems; fourth parabolic equation; mixed finite element methods; elliptic reconstruction
1 Introduction
It is known that optimal control problems governed by partial differential equations (PDEs, for short) play a great role in modern science, technology, engineering and so on. There has been extensive theoretical research for finite element approximation of various optimal control problems (see,e.g., [–]), and some scholars have been paying much attention to the mixed finite element methods for PDEs (see,e.g., [–]). As a matter of fact, the fourth PDEs of this method is always a hot special topic. For example, in (see []), Brezzi and Raviart studied fourth order elliptic equations by mixed element meth-ods. In [], Brezzi and Fortin presented some results on the application of the mixed finite element methods to linear elliptic problems. In [], Li developed mixed finite el-ement methods for solving fourth-order elliptic and parabolic problems by using RBFs and gave similar error estimates as classical mixed finite element methods. Several recent works have been devoted to the analysis of this field for the error estimates, for example, Cao and Yang got thea priorierror estimates using Ciarlet-Raviart mixed finite element methods for the fourth order control problems governed by the first bi-harmonic equation (see []). Hou studied a class of fourth order quadratic elliptic optimal control problems, where the state and co-state are approximated by the orderkRaviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of or-derk(k≥), and he deriveda posteriorierror estimates for both the control and the state approximations (see []). Although the error analysis for the finite element discretization of optimal control problems for the fourth order PDEs is discussed in many publications, there are only a few published results on this topic for parabolic problems. Therefore, we
will study the error estimates using mixed finite element for the fourth order parabolic optimal control problems.
This paper is organized as follows. Firstly, we discuss the semi-discrete mixed finite ele-ment approximation for the fourth order parabolic optimal control problem in Section . Next,a posteriorierror estimates of mixed finite element approximation for the control problem are given in Section . Finally, we analyze the conclusion and future work in Sec-tion .
2 Mixed methods for optimal control problem
In the paper, we adopt the standard notation Wm,p() for Sobolev spaces on with a norm · m,pgiven by
vpm,p=
|α|≤m
DαvpLp(),
a semi-norm| · |m,pgiven by
|v|pm,p=
|α|=m
DαvpLp().
Forp= , we denoteHm() =Wm,(), and · m= · m,, · = · ,.
For the sake of simplicity, we takeV=H(div;) ={v∈(L()),divv∈L()}andW=
L(), the Hilbert spaceVis defined by the following norm:
vH(div)=
v,+divv,
.
In this paper, the model problem that we shall investigate is the following two-dimensional optimal control problem:
min u∈Uad
T
y+∇y+y–yd+u
dt
(.)
subject to the state equations
yt(x,t) +y(x,t) =f(x,t) +u(x,t), x∈,t∈(,T], (.)
y(x,t) =y(x,t) = , x∈∂,t∈[,T], (.)
y(x, ) =y(x), x∈, (.)
y(x, ) =y(x), x∈, (.)
where the bounded open set⊂Ris a convex polygon with the bounded∂,J= [,T].
ydis continuously differentiable with respect tot; moreover,f,yd∈L(J;W). We letUad denote the admissible set of the control variable, which is defined by
Uad=
u(x,t)∈L(J;W) :
T
u(x,t)≥,x∈,∀t∈J
We denote byLs(,T;Wm,q()) the Banach space of all Lsintegrable functions from (,T) intoWm,q() with the norm
vLs(,T;Wm,q())=
T
vsWm,q()dt
s
, fors∈[,∞),
and the standard modification for s=∞. Similarly, one can define the spaces Hk(,T;
Wm,q()) andCk(,T;Wm,q()).
Throughout this paper, (·,·) denotes the inner product inL(), the form is as follows:
(u,v) =
uv, ∀(u,v)∈W×W.
Letp˜= –∇yandy˜= –y, then we can rewrite (.)-(.) as
min u∈Uad
T
˜y+˜p+y–yd+u
dt
(.)
subject to
˜
p= –∇y, x∈,t∈J, (.)
divp˜=y˜, x∈,t∈J, (.)
p= –∇˜y, x∈,t∈J, (.)
yt+divp=f +u, x∈,t∈J, (.)
y(x,t) =y˜(x,t) = , x∈∂,t∈J, (.)
y(x, ) =y(x), x∈, (.)
˜
y(x, ) =y(x), x∈. (.)
Then a possible weak formula for the state equation reads: find (p,y˜,p˜,y,u)∈(V×W)×
Uadsuch that
min u∈Uad
T
˜y+˜p+y–yd+u
dt
(.)
subject to
(p˜,v) – (y,divv) = , ∀v∈V,t∈J, (.) (divp˜,w) = (y˜,w), ∀w∈W,t∈J, (.) (p,v) – (y˜,divv) = , ∀v∈V,t∈J, (.) (yt,w) + (divp,w) = (f+u,w), ∀w∈W,t∈J, (.)
y(x, ) =y(x), x∈, (.)
˜
y(x, ) =y(x), x∈. (.)
It is well known (see []) that the above control problem has a unique solution (p,y˜,p˜,y,u)∈(V×W)×U
only if there exists a co-state (q,z˜,q˜,z)∈(V×W)such that (p,y˜,p˜,y,q,z˜,q˜,z,u) satisfies the following optimal conditions fort∈J:
(p˜,v) – (y,divv), ∀v∈V,t∈J, (.) (divp˜,w) = (y˜,w), ∀w∈W,t∈J, (.) (p,v) – (y˜,divv) = , ∀v∈V,t∈J, (.) (yt,w) + (divp,w) = (f+u,w), ∀w∈W,t∈J, (.)
y(x, ) =y(x), x∈, (.)
˜
y(x, ) =y(x), x∈, (.)
(q˜,v) – (z,divv) = , ∀v∈V,t∈J, (.) (divq˜,w) = (z˜,w) + (y˜,w), ∀w∈W,t∈J, (.) (q,v) – (˜z,divv) = –(p˜,v), ∀v∈V,t∈J, (.) –(zt,w) + (divq,w) = (y–yd,w), ∀w∈W,t∈J, (.)
z(x,T) =z˜(x,T) = , x∈, (.)
T
(u+z,u˜–u)dt≥, ∀˜u∈Uad. (.)
In order to derive our final aim, we now give the following important result (see []).
Lemma .[] Let(p,y˜,p˜,y,q,z˜,q˜,z,u)be the solution of(.)-(.),then we have the relation
u=max{,z¯}–z,
wherez¯=
T
z dx dt
T
dx dt
denotes the integral average on×J of the function z.
In the following, we will consider the semi-discrete finite element for the problem. LetThdenote a regular triangulation of the polygonal domain,Th={T
i}, herehis the maximum diameter of the elementTiinTh. Moreover, letehdenote the set of element sides of the triangulationThwithE
h=
eh. Furthermore, letVh×Wh⊂V×W be the Raviart-Thomas space (see []) associated with the triangulationsThof.P
kdenotes the space of polynomials of total degree at mostk(k≥). LetV(Ti) ={v∈Pk(Ti) +x·Pk(Ti)},
W(Ti) =Pk(Ti), and we define
Wh:=
wh∈W:∀Ti∈Th,wh|Ti∈W(Ti)
,
Vh:=
vh∈V:∀Ti∈Th,vh|Ti∈V(Ti)
,
Kh:=L(J;Wh)∩Uad.
The mixed finite element discretization of (.)-(.) is rewritten as follows: find (ph,y˜h,p˜h,yh,uh)∈(L(J;Vh)×L(J;Wh))×Khsuch that
min uh∈Kh
T
˜y+˜p+y–yd+u
dt
(p˜h,vh) – (yh,divvh) = , ∀vh∈Vh,t∈J, (.) (divp˜h,wh) = (y˜h,wh), ∀wh∈Wh,t∈J, (.) (ph,vh) – (y˜h,divvh) = , ∀vh∈Vh,t∈J, (.) (yh,t,wh) + (divph,wh) = (f +uh,wh), ∀wh∈Wh,t∈J, (.)
yh(x, ) =yh(x), x∈, (.)
˜
yh(x, ) =yh(x), x∈, (.)
whereyh
(x)∈Whandyh(x)∈Whare two approximations ofyandy. The above optimal control problem again has a unique solution (ph,y˜h,p˜h,yh,uh), and that (ph,y˜h,p˜h,yh,uh) is the solution of (.)-(.) if and only if there is a co-state (qh,˜zh,q˜h,zh)∈(L(J;Vh)×
L(J;W
h)) such that (ph,y˜h,p˜h,yh,qh,z˜h,q˜h,zh) satisfies the following optimality condi-tions:
(p˜h,vh) – (yh,divvh) = , ∀vh∈Vh,t∈J, (.) (divp˜h,wh) = (y˜h,wh), ∀wh∈Wh,t∈J, (.) (ph,vh) – (y˜h,divvh) = , ∀vh∈Vh,t∈J, (.) (yh,t,wh) + (divph,wh) = (f +uh,wh), ∀wh∈Wh,t∈J, (.)
yh(x, ) =yh(x), x∈, (.)
˜
yh(x, ) =yh(x), x∈, (.)
(q˜h,vh) – (zh,divvh) = , ∀v∈V,t∈J, (.) (divq˜h,wh) = (z˜h,wh) + (y˜h,wh), ∀wh∈Wh,t∈J, (.) (qh,vh) – (z˜h,divvh) = –(p˜h,vh), ∀vh∈Vh,t∈J, (.) –(zh,t,wh) + (divqh,wh) = (yh–yd,wh), ∀wh∈Wh,t∈J, (.)
zh(x,T) =z˜h(x,T) = , x∈, (.)
T
(uh+zh,u˜h–uh)dt≥, ∀˜uh∈Kh. (.)
Similar to Lemma ., we can get the relationship between the control approximation
uhand the co-state approximationzh, which satisfies
uh=max{,¯zh}–zh,
wherez¯h=
T
zhdx dt T
dx dt
denotes the integral average on×Jof the functionzh.
In order to continue our analysis, we shall introduce some intermediate variables. For any control functionuh∈Kh, we define the state solutionp(uh),˜y(uh),p˜(uh),y(uh),q(uh),
˜
z(uh),q˜(uh),z(uh) satisfying
˜
p(uh),v
–y(uh),divv
= , ∀v∈V,t∈J, (.)
divp˜(uh),w
=y˜(uh),w
p(uh),v
–y˜(uh),divv
= , ∀v∈V,t∈J, (.)
yt(uh),w
+divp(uh),w
= (f +uh,w), ∀w∈W,t∈J, (.)
y(uh)(x, ) =y(x), x∈, (.)
˜
y(uh)(x, ) =y(x), x∈, (.)
˜
q(uh),v
–z(uh),divv
= , ∀v∈V,t∈J, (.)
divq˜(uh),w
=z˜(uh),w
+y˜(uh),w
, ∀w∈W,t∈J, (.)
q(uh),v
–z˜(uh),divv
= –p˜(uh),v
, ∀v∈V,t∈J, (.) –zt(uh),w
+divq(uh),w
=y(uh) –yd,w
, ∀w∈W,t∈J, (.)
z(uh)(x,T) =z˜(uh)(x,T) = , x∈, (.)
where the exact solutionsy(uh) andz(uh) satisfy the zero boundary condition. Now we define the following errors:
ey=y(uh) –yh, ey˜=y˜(uh) –y˜h, ep=p(uh) –ph, ep˜=p˜(uh) –p˜h,
eq=q(uh) –qh, e˜q=q˜(uh) –q˜h, ez=z(uh) –zh, ez˜=z˜(uh) –˜zh.
Next, from (.)-(.), (.)-(.), we can get the error equations as follows:
(ep˜,v) – (ey,divv) = –r(v), ∀v∈V,t∈J, (.) (divep˜,w) – (ey˜,w) = –r(w), ∀w∈W,t∈J, (.) (ep,v) – (e˜y,divv) = –r(v), ∀v∈V,t∈J, (.) (ey,t,w) – (divep,w) = –r(w), ∀w∈W,t∈J, (.) (eq˜,v) – (ez,divv) = –r(v), ∀v∈V,t∈J, (.) (dive˜q,w) – (ez˜,w) – (e˜y,w) = –r(w), ∀w∈W,t∈J, (.) (eq,v) – (ez˜,divv) + (ep˜,v) = –r(v), ∀v∈V,t∈J, (.) –(ez,t,w) + (diveq,w) = (ey,w) –r(w), ∀w∈W,t∈J, (.)
wherer-rare given as follows:
r(v) := (p˜h,v) – (yh,divv), r(w) := (divp˜h,w) – (y˜h,w),
r(v) := (ph,v) – (y˜h,divv), r(w) := (yh,t,w) + (divph,w) – (f +uh,w),
r(v) := (q˜h,v) – (zh,divv), r(w) := (divq˜h,w) – (z˜h,w) – (y˜h,w),
r(v) := (qh,v) – (z˜h,divv) + (p˜h,v), r(w) := (divqh,w) – (zh,t,w) – (yh–yd,w).
Then, we introduce mixed elliptic reconstructions yˇ(t),yˆ(t),zˇ(t),zˆ(t) ∈ H() and
ˇ
andpˇ(t),pˆ(t),pˇ(t),pˆ(t)∈V satisfy the following equations:
(pˇ–p˜h,v) – (yˆ–yh,divv) = –r(v), ∀v∈V,t∈J, (.)
div(pˇ–p˜h),w
– (yˇ–y˜h,w) = –r(w), ∀w∈W,t∈J, (.) (pˆ–ph,v) – (yˇ–y˜h,divv) = –r(v), ∀v∈V,t∈J, (.)
div(pˆ–ph),w
= –r(w), ∀w∈W,t∈J, (.) (qˇ–q˜h,v) – (ˆz–zh,divv) = –r(v), ∀v∈V,t∈J, (.)
div(qˇ–q˜h),w
– (zˇ–z˜h,w) – (yˇ–y˜h,w) = –r(w), ∀w∈W,t∈J, (.) (qˆ–qh,v) – (zˇ–˜zh,divv) = –(pˇ–p˜h,v) –r(v), ∀v∈V,t∈J, (.)
div(qˆ–qh),w
= (yˆ–yh,w) –r(w), ∀w∈W,t∈J. (.)
We can deriver(vh) =r(vh) =r(vh) =r(vh),∀vh∈Vh, andr(wh) =r(wh) =r(wh) =
r(wh),∀wh∈Wh, we note thatph,y˜h,p˜h,yhare standard mixed elliptic projections ofpˆ,
ˇ
y,pˇ,yˆ, respectively,qh,z˜h,q˜h,zhare nonstandard mixed elliptic projections ofqˆ,ˇz,qˇ,zˆ. We can define as follows by mixed elliptic reconstructions:
ey= (yˆ–yh) –
ˆ
y–y(uh)
:=ηy–ξy, ey˜= (yˇ–y˜h) –
ˇ
y–˜y(uh)
:=η˜y–ξ˜y,
ep= (pˆ–ph) –
ˆ
p–p(uh)
:=ηp–ξp, ep˜= (pˇ–p˜h) –
ˇ
p–p˜(uh)
:=ηp˜–ξp˜,
eq= (qˆ–qh) –
ˆ
q–q(uh)
:=ηq–ξq, eq˜= (qˇ–q˜h) –
ˇ
q–q˜(uh)
:=ηq˜–ξq˜,
ez= (zˆ–zh) –
ˆ
z–z(uh)
:=ηz–ξz, ez˜= (zˇ–z˜h) –
ˇ
z–z˜(uh)
:=η˜z–ξ˜z.
Next, we will give some preliminary results about the intermediate solution. We define the standardL-orthogonal projectionPh:W→Whwhich satisfies: for anyw∈W,
(w–Phw,wh) = , ∀wh∈Wh, (.)
Phw–w,q≤Cwt,qht, ≤t≤k+ , ifw∈W∩Wt,q(), (.)
Phw–w–r≤Cwthr+t, ≤r,t≤k+ , ifw∈Ht(). (.)
Recall the Fortin projection (see [] and [])h:V→Vh, which satisfies: for anyv∈V,
div(v–hv),wh
= , ∀wh∈Wh, (.)
v–hv,q≤Chrvr,q, /q<r<k+ ,∀v∈V∩
Wr,q(), (.)
div(v–hv)≤Chrdivvr, ≤r≤k+ ,∀divv∈Hr(). (.)
We have the commuting properties
div◦h=Ph◦divV→Wh and div(I–h)V⊥Wh, (.)
3 A posteriori error estimates
In this section, we give some lemmas to prepare for our results, and then we givea pos-terioriestimates for the mixed finite element approximation to the fourth order parabolic optimal control problems. Let (p,y˜,p˜,y,q,˜z,q˜,z,u) and (ph,y˜h,p˜h,yh,qh,˜zh,q˜h,zh,uh) be the solutions of (.)-(.) and (.)-(.), respectively. Now we decompose the errors as the following forms:
p–ph=p–p(uh) +p(uh) –ph:=rp+ep,
˜
y–y˜h=y˜–y˜(uh) +y˜(uh) –y˜h:=ry˜+e˜y,
˜
p–p˜h=p˜–p˜(uh) +p˜(uh) –p˜h:=rp˜+ep˜,
y–yh=y–y(uh) +y(uh) –yh:=ry+ey,
q–qh=q–q(uh) +q(uh) –qh:=rq+eq,
˜
z–z˜h=z˜–z˜(uh) +z˜(uh) –z˜h:=rz˜+e˜z,
˜
q–q˜h=q˜–q˜(uh) +q˜(uh) –q˜h:=rq˜+eq˜,
z–zh=z–z(uh) +z(uh) –zh:=rz+ez.
From (.)-(.), (.)-(.) and (.)-(.), we can get the error equations as follows:
(rp˜,v) – (ry,divv) = , ∀v∈V, (.) (divrp˜,w) = (ry˜,w), ∀w∈W, (.) (rp,v) – (ry˜,divv) = , ∀v∈V, (.) (ry,t,w) + (divrp,w) = (u–uh,w), ∀w∈W, (.) (rq˜,v) – (rz,divv) = , ∀v∈V, (.) (divrq˜,w) = (rz˜,w) + (ry˜,w), ∀w∈W, (.) (rq,v) – (rz˜,divv) = –(rp˜,v), ∀v∈V, (.) –(rz,t,w) + (divrq,w) = (ry,w), ∀w∈W. (.)
Lemma . Let rp,ry˜,rp˜,ry,rq,rz˜,rq˜,rzsatisfy(.)-(.),then there exists a constant C>
independent of h such that
ryL∞(J;W)+ry˜L∞(J;W)+rpL(J;W)+rp˜L(J;W)≤Cu–uhL(J;W), (.)
rzL∞(J;W)+rz˜L(J;W)+rqL(J;W)+r˜qL(J;W)≤Cu–uhL(J;W). (.)
Proof PartI. Lett= andv=rp˜() in (.), sincery() = , so we find thatrp˜= .
Differ-entiate (.) with respect tot, and setv=rpas the test function, then we have
(rp˜,t,rp) = (ry,t,divrp). (.) Then, letv=rp˜,tin (.), and fromdivrp˜=r˜y, we get that
Now we setw=divrpin (.), we derive that
(ry,t,divrp) + (divrp,divrp) = (u–uh,divrp). (.)
From (.)-(.), we have
(r˜y,r˜y,t) + (divrp,divrp) = (u–uh,divrp),
the above equation can be rewritten as follows:
d dtry˜
+divr p≤C
u–uh+divrp
.
We integrate the above inequality from toTand notery˜() = , then we get
ry˜L∞(J;W)+divrpL(J;W)≤Cu–uhL(J;W). (.)
Setv=rpandv=rp˜ as the test functions in (.) and (.), respectively, we have
(ry,divrp) = (ry˜,divrp˜) = (ry˜,ry˜). (.)
Then, letw=ryin (.) and combine with (.), we derive that
(ry,t,ry) + (r˜y,ry˜) = (u–uh,ry),
which leads to
d dtry
+r
˜
y≤C
u–uh+ry
.
On integrating the above inequality from tot, using Gronwall’s lemma and notingry() = , we can get
ryL∞(J;W)+ry˜L(J;W)≤Cu–uhL(J;W). (.)
In (.), letv=rp, we have
(rp,rp) = (r˜y,divrp).
Integrating the above equation with respect to time from toT, combining with (.) and (.), we arrive at
rpL(J;W)≤ ry˜L(J;W)+divrpL(J;W)≤Cu–uhL(J;W). (.)
Letv=rp˜, so we have
we can get the following inequality from the above equation:
rp˜≤C
ry+r˜y
.
Integrating the above inequality again from toTand noticing (.), we can obtain
rp˜L(J;W)≤Cu–uhL(J;W). (.)
By (.), (.)-(.), we derive (.).
PartII. Choosingv=rqin (.) andv=rq˜in (.), respectively, we obtain
(divrq,rz) = (rz˜,divrq˜) – (rp˜,rq˜). (.)
Letv=rq˜in (.), we have
(rp˜,rq˜) = (ry,divrq˜). (.)
Setw=rzin (.), we arrive at
–(rz,t,rz) + (divrq,rz) = (ry,rz). (.)
Now from (.)-(.) we can get that
–(rz,t,rz) + (rz˜,divrq˜) – (ry,divrq˜) = (ry,rz). (.)
Then, setw=divrq˜ in (.) and combine with (.), we obtain
–(rz,t,rz) + (divrq˜,divrq˜) = (ry,divrq˜) + (ry˜,divrq˜) + (ry,rz),
which leads to
–
d dtrz
+divr
˜
q≤C
ry+r˜y+divrq˜+rz
.
Integrating the above equation with respect to time fromttoT, using Gronwall’s lemma and (.), notingrz(T) = , we can derive that
rzL∞(J;W)+divrq˜L(J;W)≤C
u–uhL(J;W)
. (.)
Letv=rq˜in (.), we get
(rq˜,r˜q) = (rz,divrq˜).
We integrate the above equation from toT and notice (.), then we can obtain that
rq˜L(J;W)≤C
u–uhL(J;W)
Letw=rz˜in (.), so we arrive at
(r˜z,r˜z) = (divrq˜,rz˜) – (ry˜,rz˜),
the above equation can be rewritten as follows:
rz˜≤C
divrq˜+r˜z+r˜y
.
We integrate the above equation from toT, and from (.), (.), we have
rz˜L(J;W)≤C
u–uhL(J;W)
. (.)
Setv=rqin (.), it yields that (rq,rq) = (rz˜,divrq) – (rp˜,rq), then we have
rq≤C
r˜z+rp˜+divrq
. (.)
Letw=divrqin (.), we have
(divrq,divrq) = (rz,t,divrq) + (ry,divrq), it also can be restated as
divrq≤C
rz,t+divrq+ry
,
where it leads to
divrq≤C
rz,t+u–uh
. (.)
Now setw=rz,tin (.), we get that –(rz,t,rz,t) + (divrq,rz,t) = (ry,rz,t), we can rewrite the above equation as follows:
rz,tL(J;W)≤C
ryL∞(J;W)+divrqL(J;W)
≤Cu–uhL(J;W)+divrqL(J;W)
. (.)
From (.)-(.), we can obtain that
rqL(J;W)≤Cu–uhL(J;W). (.)
Lemma . Let(p,y˜,p˜,y,q,z˜,q˜,z,u)and(ph,y˜h,p˜h,yh,qh,˜zh,q˜h,zh,uh)be the solutions of (.)-(.)and(.)-(.),respectively.Suppose(uh+zh)|Ti∈H
(T
i)and that there
exists w∈Khsuch that
T
(uh+zh,w–u)dt
≤C T Ti
hTi|uh+zh|H(T
i)u–uhL(Ti)dt.
Then we have
u–uhL(J;W)≤Cηu+Czh–z(uh)
L(J;W), (.)
whereηu= (
T
Tih
Ti|uh+zh|
H(T
i)dt)
.
Proof From (.), (.) and (.), we derive that
u–uhL(J;W)=
T
(u–uh,u–uh)dt
=
T
(u+z,u–uh)dt+
T
(zh+uh,uh–u)dt
+
T
zh–z(uh),u–uh
dt+
T
z(uh) –z,u–uh
dt
≤
T
(zh+uh,w–u)dt+
T
zh–z(uh),u–uh
dt + T
z(uh) –z,u–uh
dt
≤C(δ)ηu+δu–uhL(J;W)+Czh–z(uh) +rz
L(J;W)
≤C(δ)ηu+δu–uhL(J;W)+Czh–z(uh), (.)
whereδdenotes an arbitrary small positive number,C(δ) is dependent onδ–. By using
(.), we can easily obtain (.).
Lemma . Letyˇ(t),yˆ(t),zˇ(t),zˆ(t),pˇ(t),pˆ(t),qˇ(t),qˆ(t)satisfy(.)-(.).Then we can derive the following properties:
ˇ
p= –∇ˆy, pˇ=ˇy, pˆ= –∇ˇy, qˇ= –∇ˆz, qˇ=zˇ+yˇ, qˆ+pˇ= –∇ˇz.
Using (.)-(.) in (.)-(.), we obtain the following error equations:
(ξp˜,v) – (ξy,divv) = , ∀v∈V,t∈J, (.) (divξp˜,w) – (ξy˜,w) = , ∀w∈W,t∈J, (.)
(divξq˜,w) – (ξz˜,w) – (ξy˜,w) = , ∀w∈W,t∈J, (.)
(ξq,v) – (ξ˜z,divv) = –(ξp˜,v), ∀v∈V,t∈J, (.) –(ξz,t,w) + (divξq,w) = (ξy,w) + (ηz,t,w), ∀w∈W,t∈J. (.)
Lemma . Letξy, ξ˜y, ξp,ξp˜,ξz,ξ˜z, ξq, ξq˜ satisfy (.)-(.).Then we have the error
estimates as follows:
ξyL∞(J;W)+ξy˜L(J;W)+ξp˜L(J;W)
≤Cηy()+y–yh+ηy,tL(J;W)
, (.)
ξpL(J;W)≤Cηy()+y–yh+η˜y()+y–yh+ηy,tL(J;W)
, (.)
ξ˜yL∞(J;W)≤Cηy˜()+y–yh+ηy,tL(J;W)
, (.)
ξzL∞(J;W)+ξz˜L(J;W)+ξqL(J;W)+ξq˜L(J;W)
≤Cηy()+y–yh+ηy,tL(J;W)+ηz(T). (.)
Proof First of all, we differentiate equation (.) with respect totand derive
(ξp˜,t,v) – (ξy,t,divv) = , ∀v∈V,t∈J. (.)
Setv=ξp˜,tin (.) as the test function, and fromdivξp˜ =ξ˜y, we obtain
(ξp,ξp˜,t) = (ξ˜y,divξp˜,t) = (ξ˜y,ξ˜y,t). (.)
Choosew=divξpin (.) as the test function, we have
(ξy,t,divξp) + (divξp,divξp) = (ηy,t,divξp). (.)
From (.)-(.), we derive
(ξy˜,ξy˜,t) + (divξp,divξp) = (ηy,t,divξp),
it can also be read as
d dtξ˜y
+divξ p≤C
ηy,t+divξp
.
Integrating the above equation with respect to time from tot, we have
ξ˜yL∞(J;W)+divξpL(J;W)≤Cξy˜()+ηy,tL(J;W). (.)
Setv=ξpandv=ξp˜ as the test functions in (.) and (.), respectively, and note that divξp˜=ξy˜, we have
Choosew=ξyin (.), by using (.), we obtain (ξy,t,ξy) + (ξ˜y,ξ˜y) = (ηy,t,ξy),
it can be rewritten as
d dtξy
+ξ
˜
y≤C
ηy,t+ξy
.
On integrating the above inequality with respect from totand using Gronwall’s lemma, it reduces to
ξyL∞(J;W)+ξy˜L(J;W)≤C
ηy,tL(J;W)+ξy(). (.)
Letv=ξpin (.), we have (ξp,ξp) = (ξ˜y,divξp),
integrate it from toT, and from (.)-(.), we get
ξpL(J;W)≤ ξ˜yL(J;W)+divξpL(J;W)
≤Cξ˜y()+ξy()+ηy,tL(J;W)
,
it also means that
ξpL(J;W)≤Cξy˜()+ξy()+ηy,tL(J;W)
. (.)
Choosev=ξp˜ in (.), we derive
(ξp˜,ξp˜) = (ξy,divξp˜) = (ξy,ξ˜y),
which leads to
ξp˜≤C
ξy+ξ˜y
.
Integrate the above inequality from toT, using (.), we can see that
ξp˜L(J;W)≤Cξy()+ηy,tL(J;W), (.)
and notice that
ξy()≤Cηy()+y–yh, ξ˜y()≤Cηy˜()+y–yh. (.)
From (.) and (.)-(.), then (.)-(.) is proved.
Choosev=ξqandv=ξq˜ as the test functions in (.) and (.), respectively, we get
Letv=ξq˜ in (.), we have
(ξp˜,ξq˜) = (ξy,divξq˜). (.)
Setw=ξzin (.), we obtain
–(ξz,t,ξz) + (divξq,ξz) = (ξy,ξz) + (ηz,t,ξz). (.)
From (.)-(.), we derive
–(ξz,t,ξz) + (ξz˜,divξ˜q) – (ξy,divξq˜) = (ξy,ξz) + (ηz,t,ξz). (.)
Setw=divξ˜qin (.) and combine with (.), we can find that
–(ξz,t,ξz) + (divξq˜,divξq˜) = (ξy,divξ˜q) + (ξ˜y,divξq˜) + (ξy,ξz) + (ηz,t,divξq˜),
the above equality is equivalent to
–
d dtξz
+divξ
˜
q≤C
ξy+ξy˜+divξq˜+ξz+ηy,t
.
Integrating this inequality fromttoT and using Gronwall’s lemma, we have
ξzL∞(J;W)+divξq˜L(J;W)
≤CξyL(J;W)+ξy˜L(J;W)+ηy,tL(J;W)+ξz(T)
≤Cηy()+y–yh
+ηy,tL(J;W)+ξz(T)
. (.)
Choosev=ξq˜in (.), we get
(ξq˜,ξq˜) = (ξz,divξ˜q),
integrating the two sides from toT and using (.), we obtain
ξq˜L(J;W)≤Cηy()
+y–yh
+ηy,tL(J;W)+ξz(T)
. (.)
Letw=ξ˜zin (.), we derive (ξz˜,ξz˜) = (divξq˜,˜z) – (ξy˜,ξz˜),
namely,
ξ˜z≤C
divξq˜+ξ˜z+ξy˜
.
Integrating the two sides from toT again and using (.) and (.), we get
Letv=ξqin (.), we get (ξq,ξq) = (ξz˜,divξq) – (ξp˜,ξq),
it can be read as
ξq≤C
ξz˜+ξp˜+divξq
. (.)
Setw=divξqin (.), we can see that
(divξq,divξq) = (ξz,t,divξq) + (ξy,divξq) + (ηz,t,divξq),
which equals
divξq≤C
ξz,t+ξy+ηy,t
. (.)
Choosew=ξz,tin (.), we have
–(ξz,t,ξz,t) + (divξq,ξz,t) = (ξy,ξz,t) + (ηz,t,ξz,t).
From inequality (.), we deduce that
ξz,tL(J;W)≤C
ξz,tL(J;W)+ξyL∞(J;W)+ηy,tL(J;W)
. (.)
Due to (.), (.)-(.), we can give that
ξqL(J;W)≤Cηy()+y–yh+ηy,tL(J;W)+ξz(T). (.)
Note that ez+ξz=ηz, from (.)-(.) and (.), we obtain the results (.) and
(.).
Lemma . Considering Raviart-Thomas elements, there exists a positive constant C,
which is in relation to the domain,polynomial degree k and the shape regularity of the elements,such that
ηy≤C
h+min{,k}(divp˜h+y˜h)
+η˜y+ min wh∈Wh
h(p˜h–∇hwh)
, (.)
ηy,t≤C
h+min{,k}(divp˜h+y˜h)t+ηy˜+ min
wh∈Wh
h(p˜h–∇hwh)
, (.)
ηy˜≤C
h+min{,k}(yh,t+divph–f –uh)
+ min wh∈Wh
h(p˜h–∇hwh)
, (.)
ηp≤Ch(divp˜h+y˜h)
+η˜y+h
J(p˜
h·t)
+h·curlh(p˜h)
, (.)
ηp˜≤Ch(yh,t+divph–f–uh)
+hJ(p
h·t)
+h·curlh(ph)
, (.)
ηq≤C
η˜z+η˜y+h
J(q˜h·t)+h·curlh(q˜h), (.)
ηq˜≤Ch(zh,t+divqh+yh–yd)
+hJ(p˜h+qh)·t
,Eh+h·curlh(p˜h+qh)
, (.)
ηz≤C
ηz˜+η˜y+ min wh∈Wh
h(q˜h–∇hwh)
, (.)
ηz˜≤C
h+min{,k}(zh,t+divqh+yh–yd)+ηy+ηp˜
+ min wh∈Wh
h(p˜h+qh–∇hwh)
, (.)
where J(v·t)expresses the jump of v·t across the element edgewith the time t being the tangential unit vector along the edge∈Ehfor all v∈V.
Proof First of all, we must refer to [] and [], based on which we can obtaina posteriori
error estimates forηy,ηy,t,ηy˜,ηp,ηp˜,ηq,ηq˜,ηz,η˜z. We only give the proof ofL-norm estimate ofηz˜for simplicity. Now, with the help of Aubin-Nitsche duality arguments, we
think about∈H
()∩H() as the following elliptic problem:
–div(A∇) =, in, (.)
which satisfies the elliptic regularity result as follows:
≤C. (.)
Exploiting (.), (.), (.) and the definition ofh, furthermore, noting thatqˆ+pˇ= –∇ˇzin Lemma . and the equation of (∇hwh, (I–h)()) = , we gain
(η˜z,) =
η˜z, –div(∇)
=zˇ, –div(∇)+˜zh, –div(∇)
= (∇ˇz,∇) +z˜h,div(∇)
= (–qˆ–pˇ,∇) +˜zh,div
h(∇)
= –(ηq,∇) – (ηp˜,∇) – (p˜h+qh,∇) +
˜
ph+qh,h(∇)
= (divηq,–Ph) – (ηy,–Ph) + (ηy,) – (ηp˜,∇) –
˜
ph+qh, (I–h)∇
=div(qˆ–qh) – (yˆ–yh),–Ph
+ (ηy,)
– (ηp˜,∇) –
˜
ph+qh, (I–h)∇
=div(qˆ–qh) – (yˆ–yh),–Ph
+ (ηy,)
– (ηp˜,∇) –
˜
ph+qh, (I–h)∇
= (zh,t–divqh+yh–yd,–Ph) + (ηy,) – (ηp˜,∇) –
˜
ph+qh–∇hwh, (I–h)∇
≤Ch+min{,k}(z
h,t–divqh+yh–yd)· +ηy · +ηp˜ · ∇+h(p˜h+qh–∇hwh)· ∇
≤Ch+min{,k}(zh,t–divqh+yh–yd)+ηy+ηp˜
Combining (.) with (.), we can derive that
(ηz˜,)
≤C
h+min{,k}(zh,t–divqh+yh–yd) +ηy+ηp˜+ min
wh∈Wh
h(p˜h+qh–∇hwh). (.)
Next, taking supremum over, we should get estimate (.). Using a similar method, we can obtain the other estimates of Lemma . at last.
Now, by the aid of Lemmas .-., we can obtain the final result.
Theorem . Let(p,y˜,p˜,y,q,˜z,q˜,z,u)and(ph,y˜h,p˜h,yh,qh,˜zh,q˜h,zh,uh)be the solutions
of(.)-(.)and(.)-(.),respectively.Then,for∀t∈J,the following a posteriori estimates hold true:
u–uhL(J;W)≤C
ηu+ηy()+y–yh+ηy,tL(J;W)
+ηz(T)+ηzL(J;W)
, (.)
y–yhL∞(J;W)≤C
u–uhL(J;W)+ηyL(J;W)
, (.)
˜y–˜yhL∞(J;W)≤C
u–uhL(J;W)+η˜yL(J;W)
, (.)
p–phL∞(J;W)≤C
u–uhL(J;W)+ηpL(J;W)
, (.)
˜p–p˜hL∞(J;W)≤C
u–uhL(J;W)+ηp˜L(J;W), (.)
z–zhL∞(J;W)≤C
u–uhL(J;W)+ηzL(J;W)
, (.)
˜z–z˜hL∞(J;W)≤C
u–uhL(J;W)+η˜zL(J;W), (.)
q–qhL∞(J;W)≤C
u–uhL(J;W)+ηqL(J;W)
, (.)
˜q–q˜hL∞(J;W)≤C
u–uhL(J;W)+ηq˜L(J;W)
, (.)
whereηuis introduced in Lemma.,andηy,ηy,t, ηy˜,ηp,ηp˜, ηq, ηq˜, ηz,η˜z are given in
Lemma..
4 Conclusion and future works
In this paper we discuss the semi-discrete mixed finite element methods of the fourth order quadratic parabolic optimal control problems. We have establisheda posteriorierror estimates for both the state, the co-state and the control variables. Thea posteriorierror estimates for those problems by finite element methods seem to be new.
In our future work, we shall use the mixed finite element method to deal with fourth order hyperbolic optimal control problems. Furthermore, we shall considera posteriori
error estimates and superconvergence of mixed finite element solution for fourth order hyperbolic optimal control problems.
Competing interests
Authors’ contributions
CH, YC and ZL participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
Author details
1Department of Accounting, Huashang College, Guangdong University of Finance, Guangzhou, 511300, P.R. China. 2School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China.3Key Laboratory for
Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, P.R. China.
Acknowledgements
The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation. This work is supported by the Foundation for Talent Introduction of Guangdong Provincial University, Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2008), National Science Foundation of China (10971074, 11201510), Chongqing Research Program of Basic Research and Frontier Technology
(cstc2015jcyjA1193), Scientific and Technological Research Program of Chongqing Municipal Education Commission and Science and Technology Project of Wanzhou District of Chongqing (2013030050).
Received: 19 October 2014 Accepted: 16 July 2015
References
1. Chen, C, Huang, Y: High Accuracy Theory of Finite Element Methods. Hunan Science and Technology Press, Changsha (1995)
2. Chen, C, Thomee, V: The lumped mass finite element method for a parabolic problem. J. Aust. Math. Soc. Ser. B26, 329-354 (1995)
3. Chen, Y, Dai, Y: Superconvergence for optimal control problems governed by semi-linear elliptic equations. J. Sci. Comput.39, 206-221 (2009)
4. Hou, C, Chen, Y, Lu, Z: Superconvergence property of finite element methods for parabolic optimal control problems. J. Ind. Manag. Optim.7, 927-945 (2011)
5. Knowles, G: Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim.20, 414-427 (1982)
6. Kwak, D, Lee, S, Li, Q: Superconvergence of finite element method for parabolic problem. Int. J. Math. Sci.23, 567-578 (2000)
7. Li, R, Ma, H, Liu, W, Tang, T: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim.41, 1321-1349 (2002)
8. Meyer, C, Rösch, A: Superconvergence properties of optimal control problems. SIAM J. Control Optim.43, 970-985 (2004)
9. Neittaanmaki, P, Tiba, D: Optimal Control of Nonlinear Parabolic Systems: Theory: Algorithms and Applications. Dekker, New York (1994)
10. Nie, YY, Thomee, V: A lumped mass finite element method with quadrature for a nonlinear parabolic problem. SIAM J. Numer. Anal.5, 371-396 (1985)
11. Thomee, V: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer, Berlin (2006)
12. Thomee, V, Xu, J, Zhang, N: Superconvergence of the gradient in piecewise linear finite element approximation to a parabolic problem. SIAM J. Numer. Anal.26, 553-573 (2006)
13. Brezzi, F, Fortin, M: Mixed and Hybrid Finite Element Methods, pp. 15-32. Springer, New York (1991)
14. Chen, Y: Superconvergence of quadratic optimal control problems by triangular mixed finite elements. Int. J. Numer. Methods Eng.75, 881-898 (2008)
15. Chen, Y: Superconvergence of optimal control problems by rectangular mixed finite element methods. Math. Comput.77, 1269-1291 (2008)
16. Chen, Y, Liu, W: Error estimates and superconvergence of mixed finite element for quadratic optimal control. Int. J. Numer. Anal. Model.3, 311-321 (2006)
17. Hou, T:A posteriori-error estimates of semidiscrete mixed finite element methods for hyperbolic optimal control problems. Bull. Korean Math. Soc.50, 321-341 (2013)
18. Kwon, Y, Milner, FA:L∞-Error estimates for mixed methods for semilinear second-order elliptic equations. SIAM J. Numer. Anal.25, 46-53 (1988)
19. Lu, Z, Chen, Y:A posteriorierror estimates of triangular mixed finite element methods for semilinear optimal control problems. Adv. Appl. Math. Mech.1, 242-256 (2009)
20. Lu, Z, Chen, Y:L∞-Error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation. Numer. Anal. Appl.12, 74-86 (2009)
21. Xing, X, Chen, Y: Error estimates of mixed methods for optimal control problems governed by parabolic equations. Int. J. Numer. Methods Eng.75, 735-754 (2008)
22. Brezzi, F, Raviart, A: Mixed finite element methods for 4th order elliptic equations. In: Topics in Numerical Analysis, vol. 3, pp. 33-56 (1978)
23. Li, J: Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions. Adv. Comput. Math. 23, 21-30 (2005)
24. Cao, W, Yang, D: Ciarlet-Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation. J. Comput. Appl. Math.233, 372-388 (2009)
25. Hou, T: Error estimates of mixed finite element approximations for a class of fourth order elliptic control problems. Bull. Korean Math. Soc.50, 1127-1144 (2013)
26. Lions, JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971) 27. Chen, Y, Yi, N, Liu, W: A Legendre-Galerkin spectral method for optimal control problems governed by elliptic
28. Douglas, J Jr., Roberts, JE: Global estimates for mixed methods for second order elliptic equations. Math. Comput. 169, 39-52 (1985)
29. Carstensen, C:A posteriorierror estimate for the mixed finite element method. Math. Comput.218, 465-476 (1997) 30. Chen, Y, Liu, W: Superconvergence property of finite element methods for parabolic optimal control problems.
