A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems

Journal of Inequalities and Applications, Jul 2015

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.

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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)


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Chunjuan Hou, Yanping Chen, Zuliang Lu. A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems, Journal of Inequalities and Applications, 2015, pp. 240, 2015, DOI: 10.1186/s13660-015-0762-9