A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems
Hou et al. Journal of Inequalities and Applications
A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems
Chunjuan Hou 0
Yanping Chen 2
Zuliang Lu 1
0 Department of Accounting, Huashang College, Guangdong University of Finance , Guangzhou, 511300 , P.R. China
1 Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University , Chongqing, 404000 , P.R. China
2 School of Mathematical Sciences, South China Normal University , Guangzhou, 510631 , P.R. China
In this paper, a fourth order quadratic parabolic optimal control problem is analyzed. The state and co-state are discretized by the order k Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k (k ≥ 0). At last, the results of a posteriori error estimates are given in Lemma 2.1 by using mixed elliptic reconstruction methods.
optimal control problems; fourth parabolic equation; mixed finite element methods; elliptic reconstruction
-
with a
v pm,p =
a semi-norm | · |m,p given by
|v|pm,p =
v H(div) =
v , + div v ,
subject to the state equations
yt(x, t) +
y(x, t) =
v Ls(,T;Wm,q( )) =
, for s ∈ [, ∞),
(u, v) =
∀(u, v) ∈ W × W .
y + p˜ + y – yd + u dt
˜
subject to
subject to
y + p˜ + y – yd + u dt
˜
(p˜ , v) – (y, div v) = ,
∀v ∈ V , t ∈ J,
(div p˜ , w) = (y˜, w),
∀w ∈ W , t ∈ J,
(p, v) – (y˜, div v) = ,
∀v ∈ V , t ∈ J,
∀w ∈ W , t ∈ J,
(p˜ , v) – (y, div v),
(div p˜ , w) = (y˜, w),
∀v ∈ V , t ∈ J,
∀w ∈ W , t ∈ J,
(p, v) – (y˜, div v) = ,
∀v ∈ V , t ∈ J,
∀w ∈ W , t ∈ J,
(q˜ , v) – (z, div v) = ,
∀v ∈ V , t ∈ J,
∀w ∈ W , t ∈ J,
∀v ∈ V , t ∈ J,
∀w ∈ W , t ∈ J,
(u + z, u˜ – u) dt ≥ ,
∀u˜ ∈ Uad.
u = max{, z¯} – z,
z dx dt denotes the integral average on
dx dt
× J of the function z.
y + p˜ + y – yd + u dt ,
˜
∀vh ∈ Vh, t ∈ J,
∀wh ∈ Wh, t ∈ J,
∀vh ∈ Vh, t ∈ J,
∀wh ∈ Wh, t ∈ J,
∀vh ∈ Vh, t ∈ J,
∀wh ∈ Wh, t ∈ J,
∀vh ∈ Vh, t ∈ J,
∀wh ∈ Wh, t ∈ J,
∀v ∈ V , t ∈ J,
∀wh ∈ Wh, t ∈ J,
∀vh ∈ Vh, t ∈ J,
∀wh ∈ Wh, t ∈ J,
∀u˜ h ∈ Kh.
uh = max{, z¯h} – zh,
∀v ∈ V , t ∈ J,
∀w ∈ W , t ∈ J,
p(u ), v – y(u ),
h ˜ h
v = ,
y (u ), w +
t h
p(u ), w = (f + u , w),
h h
y(u )(x, ) = y (x), x
h
y(u )(x, ) = y (x), x
˜ h
q(u ), v – z(u ),
˜ h h
v = ,
q(u ), w = z(u ), w + y(u ), w ,
˜ h ˜ h ˜ h
q(u ), v – z(u ),
h ˜ h
v = – p(u ), v ,
˜ h
– z (u ), w +
t h
q(u ), w = y(u ) – y , w ,
h h d
z(u )(x, T ) = z(u )(x, T ) = , x
h ˜ h
Now we define the following errors:
e = y(u ) – y ,
y h h
e = q(u ) – q ,
q h h
e = y(u ) – y ,
y ˜ h ˜h
˜
e = q(u ) – q ,
q ˜ h ˜h
˜
e = p(u ) – p ,
p h h
e = z(u ) – z ,
z h h
e = p(u ) – p ,
p ˜ h ˜ h
˜
e = z(u ) – z .
z ˜ h ˜h
˜
(e , v) – (e ,
p y
˜
e , w) – (e , w) = –r (w),
p y
˜ ˜
(e , w) – (
y,t
e , w) = –r (w),
p
(e , v) – (e ,
q z
˜
e , w) – (e , w) – (e , w) = –r (w),
q z y
˜ ˜ ˜
v) + (e , v) = –r (v),
p
˜
–(e , w) + (
z,t
e , w) = (e , w) – r (w),
q y
where r -r are given as follows:
r (v) := (p , v) – (y ,
˜ h h
r (v) := (p , v) – (y ,
h ˜h
r (v) := (q , v) – (z ,
˜h h
r (v) := (q , v) – (z ,
h ˜h
v) + (p , v),
˜ h
r (w) := (
q , w) – (z , w) – (y – y , w).
h h,t h d
p(t), p(t), p(t), p(t)
ˇ ˆ ˇ ˆ
J ,
rer (w) := (
p , w) – (y , w),
˜ h ˜h
r (w) := (y , w) + (
h,t
p , w) – (f + u , w),
h h
r (w) := (
q , w) – (z , w) – (y , w),
˜h ˜h ˜h
V , which satisfies: for any v
and p(t), p(t), p(t), p(t)
ˇ ˆ ˇ ˆ
V satisfy the following equations:
(p – p , v) – (y – y ,
ˇ ˜ h ˆ h
(p – p ), w – (y – y , w) = –r (w),
ˇ ˜ h ˇ ˜h
(p – p , v) – (y – y ,
ˆ h ˇ ˜h
(p – p ), w = –r (w),
ˆ h
(q – q , v) – (z – z ,
ˇ ˜h ˆ h
(q – q ), w – (z – z , w) – (y – y , w) = –r (w),
ˇ ˜h ˇ ˜h ˇ ˜h
(q – q , v) – (z – z ,
ˆ h ˇ ˜h
v) = –(p – p , v) – r (v),
ˇ ˜ h
(q – q ), w = (y – y , w) – r (w),
ˆ h ˆ h
= ,
Ch v
, /q < r < k + , v
∀ ∈
≤ ≤
k + ,
We have the commuting properties
= P
where I denotes an identity operator.
We can derive r (v ) = r (v ) = r (v ) = r (v ), v
h h h h
∀ h ∈
V , and r (w ) = r (w ) = r (w ) =
h h h h
r (w ), w
h
∀ h ∈
e = (y – y ) – y – y(u ) :=
y ˆ h ˆ h
e = (p – p ) – p – p(u ) :=
p ˆ h ˆ h
e = (q – q ) – q – q(u ) :=
q ˆ h ˆ h
e = (z – z ) – z – z(u ) :=
z ˆ h ˆ h
e = (y – y ) – y – y(u ) :=
y ˇ ˜h ˇ ˜ h
˜
e = (p – p ) – p – p(u ) :=
p ˇ ˜ h ˇ ˜ h
˜
e = (q – q ) – q – q(u ) :=
q ˇ ˜h ˇ ˜ h
˜
e = (z – z ) – z – z(u ) :=
z ˇ ˜h ˇ ˜ h
˜
the standard L -orthogonal projection P : W
h
W which satisfies: for any w
h
(w – P w, w ) = ,
h h
,q ≤
–r ≤
C w
≤ ≤
k + , if w
k + , if w
3 A posteriori error estimates
(rp˜, v) – (ry, div v) = ,
∀v ∈ V ,
(div rp˜, w) = (ry˜, w),
∀w ∈ W ,
(rp, v) – (ry˜, div v) = ,
∀v ∈ V ,
(rq˜ , v) – (rz, div v) = ,
∀v ∈ V ,
∀w ∈ W ,
∀v ∈ V ,
∀w ∈ W .
∀w ∈ W ,
(rp˜,t, rp) = (ry,t, div rp).
Now we set w =
r in (.), we derive that
p
r ) = (u – u ,
p h
From (.)-(.), we have
(r , r ) + (
y y,t
˜ ˜
r ) = (u – u ,
p h
dt
C u – u
We integrate the above inequa (...truncated)