In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L 2 and H 1 norm are derived. The theoretical findings are illustrated by the numerical experiments.
In this paper we consider virtual element discretization of the following optimal control problem: find ( y , p , u ) ∈ V × Q × U a d satisfying
min u ∈ U a d J ( y , u ) = 1 2 ‖ y − y d ‖ L 2 ( Ω ) 2 + γ 2 ‖ u ‖ L 2 ( Ω ) 2 (1.1)
subject to
( − Δ y + ∇ p + K − 1 y = f + u in Ω , ∇ ⋅ y = 0 in Ω , y = 0 on Γ , (1.2)
where J ( y , u ) is the objective functional, y d is the desired state, γ > 0 is the regularization parameter, and Ω is a bounded domain in ℝ 2 with the boundary Γ . We suppose that the K is a uniformly symmetric positive definite tensor, i.e. there exist two positive constants λ 1 , λ 2 > 0 such that
λ 1 η T η ≤ η T K − 1 η ≤ λ 2 η T η .
The admissible control set U a d is defined by
U a d = { u ∈ L 2 ( Ω ) : u a ≤ u ( x ) ≤ u b a . e . in Ω } .
The quantities u a , u b ∈ ℝ 2 are constant vectors and the inequality u a ≤ u ( x ) ≤ u b is understood componentwise.
Flow control problems have important applications in aerospace, chemical engineering and so on. The Brinkman equations can be viewed as a parameter-dependent combination of the Darcy and the Stokes equations [
The virtual element method (VEM), first introduced in [
For the literature on the application of virtual element method to optimal control problem, we can refer to [
The structure of this paper is as follows. In Section 2, we give the continuous first order optimality condition of problem (1.1)-(1.2). Then, some basic concepts about VEM are introduced. In Section 3, we derive the virtual element discrete scheme for (1.1)-(1.2) and the discrete first order optimality condition. In Section 4, a priori error estimates of the state, adjoint state and control variables are proved. In Section 5, we show numerical results to verify the theoretical results.
Throughout this paper, for an open bounded domain K, we will denote scale and vector Sobolev space by H s ( K ) and H s ( K ) equipped with seminorm | ⋅ | s , K and norm ‖ ⋅ ‖ s , K , while ( ⋅ , ⋅ ) 0, K will denote the L 2 ( K ) or L 2 ( K ) inner product for scale and vector.
In this section, we firstly recall the continuous first order optimality condition for problem (1.1)-(1.2). Then we introduce the definitions of virtual element space and two projection operators.
We consider the spaces:
V : = H 0 1 ( Ω ) , Q : = L 0 2 ( Ω ) = { q ∈ L 2 ( Ω ) s .t . ∫ Ω q d Ω = 0 } .
We endow the space V with the norm
| | | v | | | V : = ‖ ∇ v ‖ 0 2 + ‖ K − 1 / 2 v ‖ 0 2
and the space Q with L 2 -norm.
Then the weak formulation of the optimal control problem (1.1)-(1.2) is given by seeking ( y , p , u ) ∈ V × Q × U a d satisfying
min u ∈ U a d J ( y , u )
s.t.
( A ( y , v ) − b ( v , p ) = ( f + u , v ) ∀ v ∈ V , b ( y , q ) = 0 ∀ q ∈ Q ,
where
A ( y , v ) = a ( y , v ) + d ( y , v ) , a ( y , v ) = ∫ Ω ∇ y : ∇ v d Ω ,
d ( y , v ) = ∫ Ω K − 1 y ⋅ v d Ω , b ( v , p ) = ∫ Ω p div v d Ω .
Additionally, we introduce the kernel:
Z : = { v ∈ V s .t . b ( v , q ) = 0 ∀ q ∈ Q } .
Following [
· A ( ⋅ , ⋅ ) and b ( ⋅ , ⋅ ) are continuous, i.e.
| A ( u , v ) | ≤ C | | | u | | | V | | | v | | | V ∀ u , v ∈ V ,
| b ( v , q ) | ≤ C | | | v | | | V ‖ q ‖ Q ∀ v ∈ V , q ∈ Q .
· A ( ⋅ , ⋅ ) is coercive on the kernel Z , i.e.
A ( v , v ) ≥ | | | v | | | V 2 ∀ v ∈ Z .
· b ( ⋅ , ⋅ ) satisfies the inf-sup condition, i.e.
∃ β > 0, such that max v ∈ V , v ≠ 0 b ( v , p ) | | | v | | | V ≥ β ‖ p ‖ Q ∀ p ∈ Q .
We introduce the following Lagrangian functional:
L ( y , p , u , z , ϕ ) : = J ( y , u ) − A ( y , z ) + b ( z , p ) − b ( y , ϕ ) + ( f + u , z ) .
Then the following continue first order optimality condition can be obtained by computing the derivatives of L ( ⋅ , ⋅ , ⋅ , ⋅ , ⋅ ) with respect to ( y , p , u , z , ϕ ) :
( A ( y , w ) − b ( w , p ) = ( f + u , w ) ∀ w ∈ V , b ( y , ψ ) = 0 ∀ ψ ∈ Q , (2.1)
( A ( w , z ) + b ( w , ϕ ) = ( y − y d , w ) ∀ w ∈ V , b ( z , ψ ) = 0 ∀ ψ ∈ Q , (2.2)
( γ u + z , v − u ) ≥ 0 ∀ v ∈ U a d , (2.3)
where ( z , ϕ ) is the adjoint state variable. Following [
u = P U a d ( − 1 γ z ) ,
where
P U a d ( u ) = max { u a , min { u , u b } }
denotes the projection onto the admissible set U a d .
Let T h be a sequence of decompositions of Ω into general polygonal elements K with
h K : = diameter ( K ) , h : = max K ∈ T h h K .
Assumption 2.1. We assume that there exists two positive constants c and ρ such that, every K ∈ T h satisfies the following assumptions:
(A1) Each element K is star-shaped with respect to a ball of radius ≥ c h K ,
(A2) The distance D h between any two points of each element K satisfies D h ≥ ρ h K .
The bilinear forms A ( ⋅ , ⋅ ) and b ( ⋅ , ⋅ ) , the norms | | | ⋅ | | | V and ‖ ⋅ ‖ Q , can be decomposed into local contributions, i.e.:
A ( y , v ) = : ∑ K ∈ T h A K ( y , v ) = : ∑ K ∈ T h ( a K ( y , v ) + d K ( y , v ) ) ∀ y , v ∈ V ,
b ( v , p ) = : ∑ K ∈ T h b K ( v , p ) ∀ v ∈ V , p ∈ Q ,
and
| | | v | | | V = : ( ∑ K ∈ T h | | | v | | | V , K ) 1 2 ∀ v ∈ V , ‖ q ‖ Q = : ( ∑ K ∈ T h ‖ q ‖ Q , K 2 ) 1 2 ∀ q ∈ Q .
Definition 2.1. For all K ∈ T h , we define the energy projection operator Π 2 ∇ : H 1 ( K ) → [ ℙ 2 ( K ) ] 2 as follows:
( a K ( v h − Π 2 ∇ v h , p ) = 0, ∀ v h ∈ H 1 ( K ) , p ∈ [ ℙ 2 ( K ) ] 2 , ∫ K v h d K = ∫ K Π 2 ∇ v h d K .
It obviously holds Π 2 ∇ p = p for all p ∈ [ ℙ 2 ( K ) ] 2 .
Definition 2.2. For all K ∈ T h , we define the L 2 projection operator Π k 0 : L 2 ( K ) → [ ℙ k ( K ) ] 2 as follows:
( P k 0 v h − v h , p ) = 0, ∀ v h ∈ L 2 ( K ) , p ∈ [ ℙ k ( K ) ] 2 .
For k ∈ ℕ , we define the following spaces:
· ℙ k ( K ) : the set of polynomials on K of degree ≤ k , usually, ℙ − 1 = { 0 } ,
· B k ( K ) : = { v ∈ C 0 ( ∂ K ) s .t v | e ∈ ℙ k ( e ) , ∀ e ⊂ ∂ K } ,
· G k ( K ) : = ∇ ( ℙ k + 1 ( K ) ) ⊆ [ ℙ k ( K ) ] 2 ,
· G k ( K ) ⊥ ⊆ [ ℙ k ( K ) ] 2 is the L 2 -orthogonal complement to G k ( K ) .
In [
V h k : = { v ∈ U h k s .t . ( v − Π 2 ∇ v , g 2 ⊥ ) = 0, ∀ g 2 ⊥ ∈ G 2 ( K ) ⊥ / G 0 ( K ) ⊥ } ,
where
U h K : = { v ∈ H 1 ( K ) s .t v | ∂ K ∈ [ B 2 ( ∂ K ) ] 2 , − Δ v + ∇ s ∈ G 2 ( K ) ⊥ and div v ∈ ℙ 1 ( K ) , for some s ∈ L 2 ( K ) } ,
and G 2 ( K ) ⊥ / G 0 ( K ) ⊥ denotes the polynomials in G 2 ( K ) ⊥ that are L 2 -orthogonal to all polynomials in G 0 ( K ) ⊥ .
For the pressure space we adopt the finite-dimensional space
Q h K : = ℙ 1 ( K ) .
Then we define the global virtual element spaces:
V h : = { v ∈ V s .t v | K ∈ V h K , ∀ K ∈ T h }
and
Q h : = { q ∈ Q s .t q | K ∈ Q h K , ∀ K ∈ T h } .
We remark that the above spaces have the following relation
div V h ⊆ Q h .
This implies an exactly divergence-free discrete velocity.
Lemma 2.3. (See [
‖ φ − P k 0 φ ‖ m , K ≤ C h K s − m | φ | s , K , m = 0,1, s ∈ ℕ , m ≤ s ≤ k + 1.
The virtual element discrete scheme of (1.2) can be defined as follows:
( A h ( y h ( u ) , v h ) − b ( v h , p h ( u ) ) = ∑ K ∈ T h ( f + u , Π 2 0 v h ) 0, K ∀ v h ∈ V h , b ( y h ( u ) , q h ) = 0 ∀ q h ∈ Q h ,
where
A h ( y h ( u ) , v h ) : = ∑ K ∈ T h A h K ( y h ( u ) , v h ) = ∑ K ∈ T h ( a h K ( y h ( u ) , v h ) + d h K ( y h ( u ) , v h ) ) ,
a h K ( y h ( u ) , v h ) : = a K ( Π 2 ∇ y h ( u ) , Π 2 ∇ v h ) + S K ( y h ( u ) − Π 2 ∇ y h ( u ) , v h − Π 2 ∇ v h ) ,
d h K ( y h ( u ) , v h ) : = d K ( Π 2 0 y h ( u ) , Π 2 0 v h ) + R K ( y h ( u ) − Π 2 0 y h ( u ) , v h − Π 2 0 v h ) .
Here, R K ( ⋅ , ⋅ ) and S K ( ⋅ , ⋅ ) are symmetric stabilizing bilinear forms satisfying
c 1 a K ( v h , v h ) ≤ S K ( v h , v h ) ≤ c 2 a K ( v h , v h ) , ∀ v h ∈ V h with Π 2 ∇ v h = 0,
c 3 d K ( v h , v h ) ≤ R K ( v h , v h ) ≤ c 4 d K ( v h , v h ) , ∀ v h ∈ V h with Π 2 0 v h = 0.
where c 1 , c 2 , c 3 and c 4 are positive constants independent of h. One can refer to [
· Consistency:
A h K ( p , v h ) = A K ( p , v h ) , ∀ p ∈ [ ℙ 2 ( K ) ] 2 , v h ∈ V h K .
· Stability:
α * A K ( v h , v h ) ≤ A h K ( v h , v h ) ≤ α * A K ( v h , v h ) , ∀ v h ∈ V h K .
Here, α * and α * are two positive constants independent of h. By the stability of A h K ( ⋅ , ⋅ ) and the coercive of A ( ⋅ , ⋅ ) , we obtain that the bilinear form A h ( ⋅ , ⋅ ) is coercive, i.e.:
A h ( v h , v h ) ≥ C | | | v h | | | V 2 ∀ v h ∈ V h . (3.1)
Next, the bilinear b ( ⋅ , ⋅ ) satisfies the inf-sup condition [
Lemma 3.1. Given the discrete spaces V h and Q h , there exists a positive constant β ˜ independent of h with
sup v h ∈ V h , v h ≠ 0 b ( v h , q h ) | | | v h | | | V ≥ β ˜ ‖ q h ‖ Q ∀ q h ∈ Q h .
Then the virtual element approximation of optimal control problem (1.1)-(1.2) is to find ( y h , p h , u h ) ∈ V h × Q h × U a d such that
min u h ∈ U a d J ( y h , u h ) : = 1 2 ∑ K ∈ T h ∫ K ( Π 2 0 y h − y d ) 2 d K + γ 2 ∫ Ω u h 2 d Ω
subject to
( A h ( y h , v h ) − b ( v h , p h ) = ∑ K ∈ T h ( f + u h , Π 2 0 v h ) 0, K ∀ v h ∈ V h , b ( y h , q h ) = 0 ∀ q h ∈ Q h . (3.2)
Here the control variable is implicitly discretized (see [
We introduce the following Lagrangian functional:
L ( y h , p h , u h , z h , ϕ h ) : = J ( y h , u h ) − A h ( y h , z h ) + b ( z h , p h ) − b ( y h , ϕ h ) + ∑ K ∈ T h ( f + u h , Π 2 0 z h ) 0, K .
Then the following discrete first order optimality condition can be obtained by computing the derivatives of L ( ⋅ , ⋅ , ⋅ , ⋅ , ⋅ ) with respect to ( y h , p h , u h , z h , ϕ h ) :
( A h ( y h , w h ) − b ( w h , p h ) = ∑ K ∈ T h ( f + u h , Π 2 0 w h ) 0, K ∀ w h ∈ V h , b ( y h , ψ h ) = 0 ∀ ψ h ∈ Q h , (3.3)
( A h ( w h , z h ) + b ( w h , ϕ h ) = ∑ K ∈ T h ( y h − y d , Π 2 0 w h ) 0, K ∀ w h ∈ V h , b ( z h , ψ h ) = 0 ∀ ψ h ∈ Q h , (3.4)
∑ K ∈ T h ( γ u h + Π 2 0 z h , v h − u h ) 0, K ≥ 0 ∀ v h ∈ U a d . (3.5)
Lemma 4.1. (See [
‖ y ‖ 2 + ‖ p ‖ 1 ≤ C ‖ f ‖ 0 .
To achieve a priori error estimates, we introduce some auxiliary problems: ∀ ( w h , ψ h ) ∈ V h × Q h ,
( A h ( y h ( u ) , w h ) − b ( w h , p h ( u ) ) = ∑ K ∈ T h ( f + u , Π 2 0 w h ) 0, K , b ( y h ( u ) , ψ h ) = 0, (4.1)
( A h ( w h , z h ( y ) ) + b ( w h , ϕ h ( y ) ) = ∑ K ∈ T h ( y − y d , Π 2 0 w h ) 0, K , b ( z h ( y ) , ψ h ) = 0, (4.2)
( A h ( w h , z h ( u ) ) + b ( w h , ϕ h ( u ) ) = ∑ K ∈ T h ( y h ( u ) − y d , Π 2 0 w h ) 0, K , b ( z h ( u ) , ψ h ) = 0. (4.3)
Additionally, we introduce the discrete kernel:
Z h : = { v h ∈ V h s .t . b ( v h , q h ) = 0 ∀ q h ∈ Q h } .
Lemma 4.2. (See [
‖ y − y I ‖ 0, K + h K | | | y − y I | | | V , K ≤ C h K s + 1 | y | s + 1, K ,
‖ z − z I ‖ 0, K + h K | | | z − z I | | | V , K ≤ C h K s + 1 | z | s + 1, K ,
where C is a positive constant independent of h.
Lemma 4.3. Let ( y , p ) and ( y h ( u ) , p h ( u ) ) be the solutions (2.1) and (4.1), respectively. Under the Assumption 2.1, we have the following estimates
| | | y − y h ( u ) | | | V ≤ C h 2 ( | f + u | 1 + | y | 3 ) ,
‖ p − p h ( u ) ‖ 0 ≤ C h 2 ( | f + u | 1 + | y | 3 + | p | 2 ) ,
‖ y − y h ( u ) ‖ 0 ≤ C h 3 ( | f + u | 1 + | y | 3 + | p | 2 ) .
Proof. Note that ( y h ( u ) , p h ( u ) ) is the virtual element approximation of ( y , p ) . We observe that, if y ∈ V is the velocity solution to Equation (2.1), then it is also the solution to the following problem: find y ∈ Z , such that
A ( y , w ) = ( f + u , w ) ∀ w ∈ Z .
Analogously, if y h ( u ) ∈ V h is the velocity solution to Equation (4.1), then it is also the solution to problem: find y h ( u ) ∈ Z h , such that
A h ( y h ( u ) , w h ) = ∑ K ∈ T h ( f + u , Π 2 0 w h ) 0, K ∀ w h ∈ Z h .
Therefore, by using the same techniques of Theorem 4.6 and Theorem 4.7 in [
Let ( r , t ) ∈ V × Q be the solution to the dual problem
( − Δ r − ∇ t + K − 1 r = y − y h ( u ) in Ω , ∇ ⋅ r = 0 in Ω , r = 0 on Γ . (4.4)
From Lemma 4.1 we know that r satisfies the regularity bound
‖ r ‖ 2 ≤ C ‖ y − y h ( u ) ‖ 0
and consequently for any interpolation r I as in Lemma 4.2, we have
| | | r − r I | | | V = ( ∑ K ∈ T h | | | r − r I | | | V , K 2 ) 1 2 ≤ C ( ∑ K ∈ T h h K 2 | r | 2, K 2 ) 1 2 ≤ C h | r | 2 ≤ C h ‖ y − y h ( u ) ‖ 0 .
Because of div V h ⊆ Q h , we obtain that the discrete velocity solution of state equation is divergence-free. Thus, we get ∇ ⋅ ( y − y h ( u ) ) = 0 . Further, multiplying (4.4) by y − y h ( u ) and integrating leads to
‖ y − y h ( u ) ‖ 0 2 = ( y − y h ( u ) , − Δ r − ∇ t + K − 1 r ) = A ( y − y h ( u ) , r ) = A ( y − y h ( u ) , r − r I ) + A ( y − y h ( u ) , r I ) = A ( y − y h ( u ) , r − r I ) + A ( y , r I ) − A ( y h ( u ) , r I )
= A ( y − y h ( u ) , r − r I ) + b ( r I , p ) − b ( r I , p h ( u ) ) + A h ( y h ( u ) , r I ) − A ( y h ( u ) , r I ) + ( f + u , r I ) − ∑ K ∈ T h ( Π 2 0 ( f + u ) , r I ) 0, K . (4.5)
We label these as T 1 , T 2 , T 3 and T f , respectively, and bound them separately.
Firstly, we can bound T 1 as follows
T 1 : = A ( y − y h ( u ) , r − r I ) ≤ C | | | y − y h ( u ) | | | V | | | r − r I | | | V ≤ C h 2 ( | f + u | 1 + | y | 3 ) ⋅ C h ‖ y − y h ( u ) ‖ 0 ≤ C h 3 ( | f + u | 1 + | y | 3 ) ‖ y − y h ( u ) ‖ 0 .
Due to ∇ ⋅ r = 0 the estimate of term T 2 follows
T 2 : = b ( r I , p ) − b ( r I , p h ( u ) ) = b ( r I , p − p h ( u ) ) = b ( r I − r , p − p h ( u ) ) ≤ | | | r I − r | | | V ‖ p − p h ( u ) ‖ 0 ≤ C h ‖ y − y h ( u ) ‖ 0 ⋅ C h 2 ( | f + u | 1 + | y | 3 + | p | 2 ) ≤ C h 3 ( | f + u | 1 + | y | 3 + | p | 2 ) ‖ y − y h ( u ) ‖ 0 .
For the inconsistency term T 3 we have
T 3 : = A h ( y h ( u ) , r I ) − A ( y h ( u ) , r I ) = ∑ K ∈ T h ( A h K ( y h ( u ) , r I ) − A K ( y h ( u ) , r I ) ) = ∑ K ∈ T h ( A h K ( y h ( u ) − Π 2 0 y , r I − Π 1 0 r ) − A K ( y h ( u ) − Π 2 0 y , r I − Π 1 0 r ) ) .
Note that
( ∑ K ∈ T h | | | y h ( u ) − Π 2 0 y | | | V , K 2 ) 1 2 = ( ∑ K ∈ T h | | | y h ( u ) − y + y − Π 2 0 y | | | V , K 2 ) 1 2 ≤ C ( ∑ K ∈ T h | | | y h ( u ) − y | | | V , K 2 + ∑ K ∈ T h | | | y − Π 2 0 y | | | V , K 2 ) 1 2 ≤ C ( ∑ K ∈ T h | | | y h ( u ) − y | | | V , K 2 ) 1 2 + C ( ∑ K ∈ T h | | | y − Π 2 0 y | | | V , K 2 ) 1 2
≤ C | | | y h ( u ) − y | | | V + C ( ∑ K ∈ T h h K 4 | y | 3, K 2 ) 1 2 ≤ C | | | y h ( u ) − y | | | V + C h 2 | y | 3 ≤ C h 2 ( | f + u | 1 + | y | 3 )
and
( ∑ K ∈ T h | | | r I − Π 1 0 r | | | V , K 2 ) 1 2 = ( ∑ K ∈ T h | | | r I − r + r − Π 1 0 r | | | V , K 2 ) 1 2 ≤ C ( ∑ K ∈ T h | | | r − r I | | | V , K 2 + ∑ K ∈ T h | | | r − Π 1 0 r | | | V , K 2 ) 1 2 ≤ C ( ∑ K ∈ T h | | | r − r I | | | V , K 2 ) 1 2 + C ( ∑ K ∈ T h | | | r − Π 1 0 r | | | V , K 2 ) 1 2
≤ C | | | r I − r | | | V + C ( ∑ K ∈ T h h K 2 | r | 2, K 2 ) 1 2 ≤ C | | | r I − r | | | V + C h | r | 2 ≤ C h ‖ y − y h ( u ) ‖ 0 .
Then applying the Cauchy-Schwarz inequality [
∑ K ∈ T h ( A h K ( y h ( u ) − Π 2 0 y , r I − Π 1 0 r ) − A K ( y h ( u ) − Π 2 0 y , r I − Π 1 0 r ) ) ≤ ∑ K ∈ T h ( | | | y h ( u ) − Π 2 0 y | | | V , K | | | r I − Π 1 0 r | | | V , K ) ≤ ( ∑ K ∈ T h | | | y h ( u ) − Π 2 0 y | | | V , K 2 ) 1 2 ( ∑ K ∈ T h | | | r I − Π 1 0 r | | | V , K 2 ) 1 2 ≤ C h 3 ( | f + u | 1 + | y | 3 ) ‖ y − y h ( u ) ‖ 0 .
Finally, the definition and estimate of the L 2 projection operator leads to the estimate of T f
T f : = ( f + u , r I ) − ∑ K ∈ T h ( Π 2 0 ( f + u ) , r I ) 0, K = ∑ K ∈ T h ( f + u − Π 2 0 ( f + u ) , r I ) 0, K = ∑ K ∈ T h ( f + u − Π 2 0 ( f + u ) , r I − Π 1 0 r I ) 0, K ≤ ∑ K ∈ T h ‖ f + u − Π 2 0 ( f + u ) ‖ 0, K ‖ r I − Π 1 0 r I ‖ 0, K ≤ ( ∑ K ∈ T h ‖ f + u − Π 2 0 ( f + u ) ‖ 0, K 2 ) 1 2 ( ∑ K ∈ T h ‖ r I − Π 1 0 r I ‖ 0, K 2 ) 1 2
≤ C ( ∑ K ∈ T h h K 2 | f + u | 1, K 2 ) 1 2 ⋅ C h 2 | r | 2 ≤ C h | f + u | 1 ⋅ C h 2 | r | 2 ≤ C h 3 | f + u | 1 ‖ y − y h ( u ) ‖ 0 .
Here, the estimate of ( ∑ K ∈ T h ‖ r I − Π 1 0 r I ‖ 0, K 2 ) 1 2 is derived as follows:
( ∑ K ∈ T h ‖ r I − Π 1 0 r I ‖ 0, K 2 ) 1 2 = ( ∑ K ∈ T h ‖ r I − r + r − Π 1 0 r + Π 1 0 r − Π 1 0 r I ‖ 0, K 2 ) 1 2 ≤ C ( ∑ K ∈ T h ‖ r I − r ‖ 0, K 2 + ∑ K ∈ T h ‖ r − Π 1 0 r ‖ 0, K 2 + ∑ K ∈ T h ‖ Π 1 0 r − Π 1 0 r I ‖ 0, K 2 ) 1 2 ≤ C ( 2 ∑ K ∈ T h ‖ r I − r ‖ 0, K 2 + ∑ K ∈ T h ‖ r − Π 1 0 r ‖ 0, K 2 ) 1 2
≤ C ( ∑ K ∈ T h ‖ r I − r ‖ 0, K 2 ) 1 2 + C ( ∑ K ∈ T h ‖ r − Π 1 0 r ‖ 0, K 2 ) 1 2 ≤ C ( ∑ K ∈ T h h K 4 | r | 2, D ( K ) 2 ) 1 2 + C ( ∑ K ∈ T h h K 4 | r | 2, K 2 ) 1 2 ≤ C h 2 | r | 2 .
Inserting above bounds into (4.5) yields the third estimate.
For the adjoint state variables, we have the following results.
Lemma 4.4. Let ( z , ϕ ) and ( z h ( y ) , ϕ h ( y ) ) be the solutions (2.2) and (4.2), respectively. Under the Assumption 2.1, we have the following estimates
| | | z − z h ( y ) | | | V ≤ C h 2 ( | y − y d | 1 + | z | 3 ) ,
‖ ϕ − ϕ h ( y ) ‖ 0 ≤ C h 2 ( | y − y d | 1 + | z | 3 + | ϕ | 2 ) ,
‖ z − z h ( y ) ‖ 0 ≤ C h 3 ( | y − y d | 1 + | z | 3 + | ϕ | 2 ) .
Proof. Note that ( z h ( y ) , ϕ h ( y ) ) is the virtual element approximation of ( z , ϕ ) . In a similar way to state variables, the Equation (2.2) can be rewritten as: finding z ∈ Z , such that
A ( w , z ) = ( y − y d , w ) ∀ w ∈ Z ,
while, the Equation (4.2) can be rewritten as: finding z h ( y ) ∈ Z h , such that
A h ( w h , z h ( y ) ) = ∑ K ∈ T h ( y − y d , Π 2 0 w h ) 0, K ∀ w h ∈ Z h .
Then by using the same techniques of Theorem 4.6 and Theorem 4.7 in [
Theorem 4.5. (A priori error estimate)Suppose that ( y , p , u , z , ϕ ) is the solution of (2.1)-(2.3), and ( y h , p h , u h , z h , ϕ h ) is the solution of (3.3)-(3.5). Under the Assumption 2.1, we derive
‖ y − y h ‖ 0 + h | | | y − y h | | | V ≤ C h 3 , ‖ p − p h ‖ 0 ≤ C h 2 ,
‖ z − z h ‖ 0 + h | | | z − z h | | | V ≤ C h 3 , ‖ ϕ − ϕ h ‖ 0 ≤ C h 2 ,
‖ u − u h ‖ 0 ≤ C h 3 ,
where C is a positive constant independent of h.
Proof. We decompose the errors y − y h , p − p h , z − z h and ϕ − ϕ h into
y − y h = y − y h ( u ) + y h ( u ) − y h , p − p h = p − p h ( u ) + p h ( u ) − p h ,
z − z h = z − z h ( y ) + z h ( y ) − z h , ϕ − ϕ h = ϕ − ϕ h ( y ) + ϕ h ( y ) − ϕ h .
Recalling Lemma 4.3, we know
| | | y − y h ( u ) | | | 0 ≤ C h 3 , | | | y − y h ( u ) | | | V ≤ C h 2 , ‖ p − p h ( u ) ‖ 0 ≤ C h 2 .
Moreover, by the governing equations of y h ( u ) and y h we have: ∀ ( w h , ψ h ) ∈ V h × Q h ,
( A h ( y h ( u ) − y h , w h ) − b ( w h , p h ( u ) − p h ) = ∑ K ∈ T h ( u − u h , Π 2 0 w h ) 0, K , b ( y h ( u ) − y h , ψ h ) = 0.
Setting w h = y h ( u ) − y h and ψ h = p h ( u ) − p h gives
A h ( y h ( u ) − y h , y h ( u ) − y h ) = ∑ K ∈ T h ( u − u h , Π 2 0 ( y h ( u ) − y h ) ) 0, K .
It follows from (3.1) that
C | | | y h ( u ) − y h | | | V 2 ≤ A h ( y h ( u ) − y h , y h ( u ) − y h ) = ∑ K ∈ T h ( u − u h , Π 2 0 ( y h ( u ) − y h ) ) 0, K ≤ ∑ K ∈ T h ‖ u − u h ‖ 0, K ‖ y h ( u ) − y h ‖ 0, K ≤ ‖ u − u h ‖ 0 | | | y h ( u ) − y h | | | V .
We can deduce
| | | y h ( u ) − y h | | | V ≤ C ‖ u − u h ‖ 0 .
Combining above inequalities leads to
‖ y − y h ‖ 0 ≤ C ( h 3 + ‖ u − u h ‖ 0 ) (4.6)
and
| | | y − y h | | | V ≤ C ( h 2 + ‖ u − u h ‖ 0 ) . (4.7)
Next we estimate ‖ p h ( u ) − p h ‖ 0 . Based on Lemma 3.1, we get
β ˜ ‖ p h ( u ) − p h ‖ 0 ≤ sup w h ∈ V h , w h ≠ 0 b ( w h , p h ( u ) − p h ) | | | w h | | | V = sup w h ∈ V h , w h ≠ 0 A h ( y h ( u ) − y h , w h ) + ∑ K ∈ T h ( u h − u , Π 2 0 w h ) 0, K | | | w h | | | V ≤ C ( | | | y h ( u ) − y h | | | V + ‖ u h − u ‖ 0 ) ≤ C ‖ u − u h ‖ 0 .
By the triangle inequality, it holds
‖ p − p h ‖ 0 ≤ C ( h 2 + ‖ u − u h ‖ 0 ) . (4.8)
In a similar way, from Lemma 4.4, we have
‖ z − z h ( y ) ‖ 0 ≤ C h 3 , | | | z − z h ( y ) | | | V ≤ C h 2 , ‖ ϕ − ϕ h ( y ) ‖ 0 ≤ C h 2 .
By the governing equations of z h ( y ) and z h we have: ∀ ( w h , ψ h ) ∈ V h × Q h ,
( A h ( w h , z h ( y ) − z h ) + b ( w h , ϕ h ( y ) − ϕ h ) = ∑ K ∈ T h ( y − y h , Π 2 0 w h ) 0, K , b ( z h ( y ) − z h , ψ h ) = 0.
Choosing w h = z h ( y ) − z h and ψ h = ϕ h ( y ) − ϕ h , we obtain
A h ( z h ( y ) − z h , z h ( y ) − z h ) = ∑ K ∈ T h ( y − y h , Π 2 0 ( z h ( y ) − z h ) ) 0, K .
Further, it follows that
C | | | z h ( y ) − z h | | | V 2 ≤ A h ( z h ( y ) − z h , z h ( y ) − z h ) = ∑ K ∈ T h ( y − y h , Π 2 0 ( z h ( y ) − z h ) ) 0, K ≤ ∑ K ∈ T h ‖ y − y h ‖ 0, K ‖ z h ( y ) − z h ‖ 0, K ≤ ‖ y − y h ‖ 0 | | | z h ( y ) − z h | | | V .
This implies
| | | z h ( y ) − z h | | | V ≤ C ‖ y − y h ‖ 0 ≤ C ( h 3 + ‖ u − u h ‖ 0 ) .
Combining above inequalities gives
‖ z − z h ‖ 0 ≤ C ( h 3 + ‖ u − u h ‖ 0 ) (4.9)
and
| | | z − z h | | | V ≤ C ( h 2 + ‖ u − u h ‖ 0 ) . (4.10)
Using Lemma 3.1, we can derive
β ˜ ‖ ϕ h ( y ) − ϕ h ‖ 0 ≤ sup w h ∈ V h , w h ≠ 0 b ( w h , ϕ h ( y ) − ϕ h ) | | | w h | | | V = sup w h ∈ V h , w h ≠ 0 A h ( w h , z h − z h ( y ) ) + ∑ K ∈ T h ( y − y h , Π 2 0 w h ) 0, K | | | w h | | | V ≤ C ( | | | z h − z h ( y ) | | | V + ‖ y − y h ‖ 0 ) ≤ C ( h 3 + ‖ u − u h ‖ 0 ) .
By the triangle inequality, we have
‖ ϕ − ϕ h ‖ 0 ≤ C ( h 2 + ‖ u − u h ‖ 0 ) . (4.11)
Since the estimates of state and adjoint state variables both depend on the estimate of control variable, now it remains to estimate ‖ u − u h ‖ 0 . Define
J ^ ′ h ( u ) ( v − u ) : = ∑ K ∈ T h ∫ K ( γ u + Π 2 0 z h ( u ) ) ( v − u ) d K .
We can prove that
J ^ ′ h ( v ) ( v − u ) − J ^ ′ h ( u ) ( v − u ) ≥ γ ‖ v − u ‖ 0 2 . (4.12)
Note that
J ^ ′ h ( v ) ( v − u ) − J ^ ′ h ( u ) ( v − u ) = ∑ K ∈ T h ∫ K ( γ v + Π 2 0 z h ( v ) − γ u − Π 2 0 z h ( u ) ) ( v − u ) d K = ∑ K ∈ T h ∫ K γ ( v − u ) 2 d K + ∑ K ∈ T h ∫ K ( Π 2 0 ( z h ( v ) − z h ( u ) ) ) ( v − u ) d K = γ ∫ Ω ( v − u ) 2 d Ω + ∑ K ∈ T h ∫ K ( Π 2 0 ( z h ( v ) − z h ( u ) ) ) ( v − u ) d K .
Using (4.1), we can derive: ∀ ( w h , ψ h ) ∈ V h × Q h ,
( A h ( y h ( v ) − y h ( u ) , w h ) − b ( w h , p h ( v ) − p h ( u ) ) = ∑ K ∈ T h ( v − u , Π 2 0 w h ) 0, K , b ( y h ( v ) − y h ( u ) , ψ h ) = 0.
Let w h = z h ( v ) − z h ( u ) and ψ h = ϕ h ( v ) − ϕ h ( u ) , then we obtain
( A h ( y h ( v ) − y h ( u ) , z h ( v ) − z h ( u ) ) − b ( z h ( v ) − z h ( u ) , p h ( v ) − p h ( u ) ) = ∑ K ∈ T h ( v − u , Π 2 0 ( z h ( v ) − z h ( u ) ) ) 0, K , b ( y h ( v ) − y h ( u ) , ϕ h ( v ) − ϕ h ( u ) ) = 0. (4.13)
Using (4.3), we can obtain: ∀ ( w h , ψ h ) ∈ V h × Q h ,
( A h ( w h , z h ( v ) − z h ( u ) ) + b ( w h , ϕ h ( v ) − ϕ h ( u ) ) = ∑ K ∈ T h ( y h ( v ) − y h ( u ) , Π 2 0 w h ) 0, K , b ( z h ( v ) − z h ( u ) , ψ h ) = 0.
Taking w h = y h ( v ) − y h ( u ) and ψ h = p h ( v ) − p h ( u ) yields
( A h ( y h ( v ) − y h ( u ) , z h ( v ) − z h ( u ) ) + b ( y h ( v ) − y h ( u ) , ϕ h ( v ) − ϕ h ( u ) ) = ∑ K ∈ T h ( y h ( v ) − y h ( u ) , Π 2 0 ( y h ( v ) − y h ( u ) ) ) 0, K , b ( z h ( v ) − z h ( u ) , p h ( v ) − p h ( u ) ) = 0. (4.14)
According to (4.13)-(4.14) and the property of the L 2 projection, we deduce
∑ K ∈ T h ( v − u , Π 2 0 ( z h ( v ) − z h ( u ) ) ) 0, K = A h ( y h ( v ) − y h ( u ) , z h ( v ) − z h ( u ) )
= ∑ K ∈ T h ( y h ( v ) − y h ( u ) , Π 2 0 ( y h ( v ) − y h ( u ) ) ) 0, K = ∑ K ∈ T h ( Π 2 0 ( y h ( v ) − y h ( u ) ) , Π 2 0 ( y h ( v ) − y h ( u ) ) ) 0, K ≥ 0.
Therefore, from (4.12), it follows
γ ‖ u − u h ‖ 0 2 ≤ J ^ ′ h ( u ) ( u − u h ) − J ^ ′ h ( u h ) ( u − u h ) = ∑ K ∈ T h ∫ K ( γ u + Π 2 0 z h ( u ) − γ u h − Π 2 0 z h ( u h ) ) ( u − u h ) d K = ∑ K ∈ T h ∫ K ( γ u + z + Π 2 0 z h ( u ) − z ) ( u − u h ) d K + ∑ K ∈ T h ∫ K ( γ u h + Π 2 0 z h ( u h ) ) ( u h − u ) d K
= ( γ u + z , u − u h ) + ∑ K ∈ T h ( Π 2 0 z h ( u ) − z , u − u h ) 0, K + ∑ K ∈ T h ( γ u h + Π 2 0 z h ( u h ) , u h − u ) 0, K ≤ 0 + ∑ K ∈ T h ( Π 2 0 z h ( u ) − z , u − u h ) 0, K + 0.
This shows
‖ u − u h ‖ 0 ≤ C ( ∑ K ∈ T h ‖ Π 2 0 z h ( u ) − z ‖ 0, K 2 ) 1 2 . (4.15)
Note that
‖ Π 2 0 z h ( u ) − z ‖ 0, K ≤ ‖ Π 2 0 z h ( u ) − Π 2 0 z ‖ 0, K + ‖ Π 2 0 z − z ‖ 0, K ≤ ‖ z h ( u ) − z ‖ 0, K + ‖ Π 2 0 z − z ‖ 0, K . (4.16)
By Lemma 2.3, we have ‖ Π 2 0 z − z ‖ 0, K ≤ C h 3 . We decompose the error z h ( u ) − z into
z h ( u ) − z = z h ( u ) − z h ( y ) + z h ( y ) − z .
Applying Lemma 4.4 yields
‖ z − z h ( y ) ‖ 0 ≤ C h 3 .
By governing equations of z h ( u ) and z h ( y ) we have: ∀ ( w h , ψ h ) ∈ V h × Q h
( A h ( w h , z h ( u ) − z h ( y ) ) + b ( w h , ϕ h ( u ) − ϕ h ( y ) ) = ∑ K ∈ T h ( y h ( u ) − y , Π 2 0 w h ) 0, K , b ( z h ( u ) − z h ( y ) , ψ h ) = 0.
Setting w h = z h ( u ) − z h ( y ) and ψ h = ϕ h ( u ) − ϕ h ( y ) , we obtain
( A h ( z h ( u ) − z h ( y ) , z h ( u ) − z h ( y ) ) + b ( z h ( u ) − z h ( y ) , ϕ h ( u ) − ϕ h ( y ) ) = ∑ K ∈ T h ( y h ( u ) − y , Π 2 0 ( z h ( u ) − z h ( y ) ) ) 0, K , b ( z h ( u ) − z h ( y ) , ϕ h ( u ) − ϕ h ( y ) ) = 0.
It can be deduced immediately by (3.1)
C | | | z h ( u ) − z h ( y ) | | | V 2 ≤ A h ( z h ( u ) − z h ( y ) , z h ( u ) − z h ( y ) ) = ∑ K ∈ T h ( y h ( u ) − y , Π 2 0 ( z h ( u ) − z h ( y ) ) ) 0, K ≤ ∑ K ∈ T h ‖ y h ( u ) − y ‖ 0, K ‖ z h ( u ) − z h ( y ) ‖ 0, K ≤ ‖ y h ( u ) − y ‖ 0 ‖ z h ( u ) − z h ( y ) ‖ 1 .
Then we can derive
‖ z h ( u ) − z h ( y ) ‖ 0 ≤ | | | z h ( u ) − z h ( y ) | | | V ≤ C ‖ y h ( u ) − y ‖ 0 ≤ C h 3 .
Using the triangle inequality leads to
‖ z − z h ( u ) ‖ 0 ≤ C h 3 . (4.17)
Combining (4.15), (4.16) and (4.17) results in
‖ u − u h ‖ 0 ≤ C h 3 .
Inserting above estimate into the estimates of state and adjoint state yields the final results.
In this section, we present an example on domain Ω : = [ 0 , 1 ] × [ 0 , 1 ] to validate the performance of our error analysis presented in this paper.
For the convergence test we consider the following two sequences of meshes that are shown in
Example 5.1. Consider the optimal control problem (1.1)-(1.2) on the square domain Ω . Let u a = ( − 0.015, − 0.015 ) T , u b = ( 0.015,0.015 ) T , K = I , γ = 1 . The exact solutions are chosen to be
y = ( y 1 y 2 ) = ( 10 x 1 2 ( 1 − x 1 ) 2 x 2 ( 1 − x 2 ) ( 1 − 2 x 2 ) − 10 x 1 ( 1 − x 1 ) ( 1 − 2 x 1 ) x 2 2 ( 1 − x 2 ) 2 ) ,
p = 10 ( 2 x 1 − 1 ) ( 2 x 2 − 1 ) ,
z = 0.5 y , ϕ = − 0.5 p .
The control variable is given u = P U a d ( − 1 γ z ) . f and y d can be determined from the exact solutions y , p , z , ϕ , u .
In Tables 1-3, we show the numerical results about the state variables y , p , the adjoint state variables z , ϕ and the control variable u on three different meshes. We can observe that the convergence rate is consistent with the previous theoretical analysis. In the following tables, NE is the number of mesh elements. We observe that both errors have optimal convergence rates, which satisfies the
| Distorted square mesh | ||||||||
|---|---|---|---|---|---|---|---|---|
| NE | h | ‖ y − y h ‖ 0 | Rate | | | | y − y h | | | V | Rate | ‖ p − p h ‖ 0 | Rate | |
| 100 | 1.000E−01 | 9.31208E−04 | 3.55874E−02 | 5.30223E−03 | ||||
| 400 | 5.000E−02 | 1.06371E−04 | 3.13 | 8.65357E−03 | 2.04 | 8.65357E−03 | 2.07 | |
| 900 | 3.333E−02 | 3.20232E−05 | 2.96 | 3.89233E−03 | 1.97 | 3.89233E−03 | 1.97 | |
| 1600 | 2.500E−02 | 1.33977E−05 | 3.03 | 2.18359E−03 | 2.01 | 2.18359E−03 | 2.02 | |
| Lloyd mesh | ||||||||
| NE | h | ‖ y − y h ‖ 0 | Rate | | | | y − y h | | | V | Rate | ‖ p − p h ‖ 0 | Rate | |
| 100 | 1.000E−01 | 8.13421E−04 | 8.51484E−02 | 2.51368E−03 | ||||
| 400 | 5.000E−02 | 9.54153E−05 | 3.09 | 2.05802E−02 | 2.05 | 6.45030E−04 | 1.96 | |
| 900 | 3.333E−02 | 2.89133E−05 | 2.94 | 8.99788E−03 | 2.04 | 2.80872E−04 | 2.05 | |
| 1600 | 2.500E−02 | 1.22585E−05 | 2.98 | 5.04953E−03 | 2.01 | 1.58005E−04 | 2.00 | |
| Distorted square mesh | ||||||||
|---|---|---|---|---|---|---|---|---|
| NE | h | ‖ z − z h ‖ 0 | Rate | | | | z − z h | | | V | Rate | ‖ ψ − ψ h ‖ 0 | Rate | |
| 100 | 1.000E−01 | 2.53147E−03 | 1.23694E−01 | 5.24139E−03 | ||||
| 400 | 5.000E−02 | 2.97297E−04 | 3.09 | 3.04978E−02 | 2.02 | 1.27452E−03 | 2.04 | |
| 900 | 3.333E−02 | 8.95017E−05 | 2.96 | 1.36622E−02 | 1.98 | 5.73271E−04 | 1.97 | |
| 1600 | 2.500E−02 | 3.74454E−05 | 3.03 | 7.66446E−03 | 2.01 | 3.21603E−04 | 2.01 | |
| Lloyd mesh | ||||||||
| NE | h | ‖ z − z h ‖ 0 | Rate | | | | z − z h | | | V | Rate | ‖ ψ − ψ h ‖ 0 | Rate | |
| 100 | 1.000E−01 | 6.51423E−04 | 2.54301E−02 | 1.48532E−03 | ||||
| 400 | 5.000E−02 | 7.69442E−05 | 3.08 | 6.14640E−03 | 2.05 | 3.81145E−04 | 1.96 | |
| 900 | 3.333E−02 | 2.32217E−05 | 2.95 | 2.69819E−03 | 2.03 | 1.66370E−04 | 2.04 | |
| 1600 | 2.500E−02 | 9.84543E−06 | 2.98 | 1.51420E−03 | 2.01 | 9.33767E−05 | 2.01 | |
| Distorted square mesh | |||
|---|---|---|---|
| NE | h | ‖ u − u h ‖ 0 | Rate |
| 100 | 1.000E−01 | 5.82184E−03 | |
| 400 | 5.000E−02 | 6.98085E−04 | 3.06 |
| 900 | 3.333E−02 | 2.03451E−04 | 3.04 |
| 1600 | 2.500E−02 | 8.66004E−05 | 2.97 |
| Lloyd mesh | |||
| NE | h | ‖ u − u h ‖ 0 | Rate |
| 100 | 1.000E−01 | 2.65768E−03 | |
| 400 | 5.000E−02 | 3.18677E−04 | 3.06 |
| 900 | 3.333E−02 | 9.55500E−05 | 2.97 |
| 1600 | 2.500E−02 | 4.04384E−05 | 2.99 |
conclusion in Theorem 4.5.
The author would like to thank editor and referees for their valuable advices for the improvement of this article.
The author declares no conflicts of interest regarding the publication of this paper.
Li, Y.W. (2023) Virtual Element Discretization of Optimal Control Problem Governed by Brinkman Equations. Engineering, 15, 114-133. https://doi.org/10.4236/eng.2023.152010