Lower error bounds

The following theorem provides upper bounds for errors in the state, co-state and control variables.

Theorem 3.2.1. Let(y, q)and(yh, qh)be the solutions of (3.5)-(3.6) and (3.11)-(3.12), respectively. Let z and zh be the solutions of the co-state equations (3.8) and (3.16), respectively. Then,

kyh−yk²L²(Ω)+kzh−zk²L²(Ω)+kqh−qk²L²(Ω)≤ 3 2(2c1−1)

nC₃²η₁²+C₄²(η²₂+η²₃) +C₅²η₄²o , where η1 is defined in Lemma 3.2.3 and ηi|i=2,...,4 are defined in Lemma 3.2.4.

Remark 3.2.1. Note that Theorem 3.2.1 is valid when ω =δxc, where δxc denotes the Dirac delta function concentrated at a point xc. If xc ∈ K then the bound for the state reduces to

kyh−yk²L²(Ω) ≤ 3 2(2c1−1)

nC₃²η₁²+C₄²(˜η₂²+η₃²) +C₅²η₄²o ,

where η˜2 :=hKkgk^L∞(K), η1 is defined in Lemma 3.2.3 andηi|^i=3,4 are defined in Lemma 3.2.4. If xc is the vertex of the element K then the bound for the state reduces to

kyh−yk²L²(Ω) ≤ 3 2(2c1−1)

C₃²η₁²+ ˆC₄²η₃²+C₅²η₄² , where Cˆ₄ =CR˜C₁max{C_I,0, C_I,e}.

wherew^◦_e is the interior ofwe andλ^K_Pi^j is the barycentric coordinate ofx associated with the triangle Kj and the point Pi, extended to the wholewe.

The following lemma provides some properties of the edge bubble function be(x) which will be used in deriving lower bound for the state variable.

Lemma 3.3.1. For e∈ Eh, let be(x) and we be defined as above. Then,

∂be(x)

∂ne

= 0 on we, C6he ≤

Z

e

be(x)de ≤C7he,

|b_e(x)|H^m(we) ≤ C₈h^1−m_e , m= 1,2, kbe(x)kL²(we) ≤ C9,

where Ci|i=6,...,9 are positive constants depend on the polynomial degree and shape pa- rameter of Th, and n_e is the unit outward normal to the edge e.

We now define the element bubble function as follows: Let K be a triangle of Th

containing xc (if xc lies on an inner edge, any of the two triangles sharing the edge can be chosen as K). Let

wK :=∪{K⁰ ∈ T^h : K⁰∩K 6=∅}, (3.34) and L:=dist(xc, ∂wK), where ∂wK denotes the boundary of wK. Notice that, because of the shape regularity of the mesh, hK ≤CL. Let bxc(x) be a smooth bubble function defined on Ω with support in wK and satisfying















0≤bxc(x)≤1, ∀x∈Ω,

bxc(x) = 1, |x−xc| ≤L/4, ∀x∈Ω, b_xc(x) = 0, |x−x_c| ≥3L/4, ∀x∈Ω.

(3.35)

Lemma 3.3.2. Let b_xc(x) and w_K be defined as above and x_c ∈K. Then, we have kbxc(x)kL^∞(wK) ≤ C10,

|bxc(x)|^Hm(wK) ≤ C11h^1−m_K , m = 1,2, kbxc(x)kL²(wK) ≤ C12,

where the positive constants Ci|^i=10,11,12 depend on the polynomial degree and shape reg- ularity of the triangulation T^h.

Bubble functions for the co-state: Let bK(x) be the standard third order polyno- mial on K scaled such that bK = λ1λ2λ3, where {λ1, λ2, λ3} denote the barycentric coordinates on K. Then, bK(x) satisfies the following properties:

supp b_K(x)⊂K, 0≤b_K(x)≤1, max

x∈K b_K(x) = 1, Let βK =ϕbK for all polynomials ϕ of degree 1, then βK ∈H²(K) and

C13kϕkL²(K) ≤ kb

1 2

KϕkL²(K) (3.36)

kβKkH²(K) ≤ C14h⁻²_K kβKkL²(K), ∀K ∈ T^h, (3.37) where C13 and C14 are positive constants depend on the polynomial degree and shape regularity of the triangulation T^h.

We need to introduce the edge bubble function for the co-state. Let ˜be(x) be the fourth order polynomial for the edgee, wheree =∂K1∩∂K2andK1 be the triangle with vertices (x0, y0), (x1, y1), (x2, y2),K2be the triangle with vertices (x0, y0), (x2, y2), (x3, y3) and ebe the common edge joining (x0, y0), (x2, y2). So, we define the edge bubble func- tion ˜be(x) as follows:

˜be(x) :=











λˆ0λˆ2ˆλ⁰₀λˆ⁰₂, K1∪K2, 0, Ω\K1∪K2,

where ˆλ₀,ˆλ₂ corresponds to the triangle K₁ and ˆλ⁰₀,λˆ⁰₂ corresponds to the triangle K₂.

Denote ∆1 :=

x0 y0 1 x₁ y₁ 1 x2 y2 1

, ∆2 :=

x0 y0 1 x₂ y₂ 1 x3 y3 1

, ∆01 :=

x0 y0 1 x₁ y₁ 1

¯

x y¯ 1

, ∆12 :=

¯

x y¯ 1 x₁ y₁ 1 x2 y2 1 ,

∆₂₃:=

¯

x y¯ 1 x2 y2 1 x3 y3 1

and ∆₀₃:=

x0 y0 1

¯

x y¯ 1 x3 y3 1

. Then, set

ˆλ0 := ∆12

∆1+ ∆2

, λˆ2 := ∆01

∆1+ ∆2

, λˆ⁰₀ := ∆23

∆1+ ∆2

and ˆλ⁰₂ := ∆03

∆1 + ∆2

. Let ˜K =K1 ∪K2. Then for βe = ˜ϕ˜be(x) satisfies ^∂β_∂n^e

e = 0 on∂K, where˜ ne is the unit normal to the edge e, for all polynomials ˜ϕ of degree 1. So, we have βe ∈H₀²( ˜K).

Then, from the standard scaling arguments, it can be shown that (cf. [3, 87])

kβekL²( ˜K) ≤ C15h^1/2_e kβekL²(e), (3.38) C₁₆kϕ˜kL²(e) ≤ k˜b

1

e2ϕ˜kL²(e), (3.39) kβekH²( ˜K) ≤ C17h⁻²_e kβekL²( ˜K), (3.40) where C15, C16 and C17 are positive constants depend on the polynomial degree and shape regularity of the triangulation T^h.

Now, we are in a position to derive local lower bounds for error in the state variable.

Theorem 3.3.1. Let x_c ∈K and w_K be defined in (3.34). Let Eh^K be the set of edges e of triangles K ⊂wK such that e*∂wK. Then

hKgK(xc) ≤ hK

n

C10kgK−gkL^∞(wK)+C12

kq−qhkL²(wK)+kqh− AyhkL²(wK)

o

+C11ky−yhkL²(wK)+ X

e∈E_h^K

h²_e

h∂yh

∂nA

i L²(e), and

h²_e

h∂yh

∂nA

i

L²(e) ≤ 1 C6

nC8ky−yhkL²(we)+C9he

kqh− AyhkL²(we)+kq−qhkL²(we)

o, where gK denotes the mean value ofg on the element K,i.e., gK(x) := _|K|¹ R

Kg(x)dx.

Proof. Letbxc be the bubble function defined by (3.35). Using (3.6) and integration by parts, we obtain

g_K(x_c) = ≺g_Kδ_xc, b_xc

= ≺(gK−g)δxc, bxc + Z

wK

yA^∗bxcdx− Z

wK

q bxcdx

= ≺(gK−g)δxc, bxc + Z

wK

(y−yh)A^∗bxcdx+ Z

wK

Ayhbxcdx

+ X

e∈E_h^K

Z

e

h∂y_h

∂nA

i

bxcde− X

e∈E_h^K

Z

e

h ∂b_xc

∂nA^∗

i

yhde− Z

wK

q bxcdx.

Since h

∂b_xc

∂nA∗

i

e= 0, it now follows that gK(xc) = ≺(gK−g)δxc, bxc +

Z

wK

(y−yh)A^∗bxcdx+ Z

wK

(Ayh−qh)bxcdx

+ X

e∈E_h^K

Z

e

h∂yh

∂nA

ibxcde+ Z

wK

(qh−q)bxcdx.

An application of the Cauchy Schwarz inequality and the shape regularity of the mesh yields

gK(xc) ≤ kgK−gkL^∞(wK)kbxckL^∞(wK)+ky−yhkL²(wK)kA^∗bxckL²(wK)

+kqh− AyhkL²(wK)kbxckL²(wK)+ X

e∈E_h^K

he

h∂yh

∂n_A i

L²(e)

+kq−qhkL²(wK)kbxckL²(wK). An use of Lemma 3.3.2 implies

gK(xc) ≤ C10kgK−gk^L∞(wK)+C11h⁻¹_K ky−yhk^L2(wK)+C12kqh− Ayhk^L2(wK)

+ X

e∈E_h^K

h_e

h∂y_h

∂nA

i

L²(e)+C₁₂kq−q_hkL²(wK),

which completes the proof of the first inequality. To prove the second inequality, we use (3.6) and integration by parts to obtain

Z

we

(y−yh)A^∗bedx+ Z

we

(qh−q)bedx = ≺gδxc, be − Z

we

yhA^∗bedx+ Z

we

qhbedx

= ≺gδxc, be + Z

we

(qh− Ayh)bedx

− Z

e

h∂yh

∂nA

i

bede+ Z

e

h ∂be

∂nA^∗

i yhde.

Using the fact be(xc) = 0 and h

∂be

∂nA∗

i

e= 0, we have Z

we

(y−yh)A^∗bedx+ Z

we

(qh −q)bedx= Z

we

(qh− Ayh)bedx− Z

e

h∂y_h

∂nA

i

bede.(3.41) Since

Z

e

h∂yh

∂nA

ibede

≥ C6he

h∂yh

∂nA

i L²(e), the equation (3.41) becomes

C6he

h∂yh

∂nA

i

L²(e) ≤

Z

we

(qh− Ayh)bedx +

Z

we

(y−yh)A^∗bedx +

Z

we

(q−qh)bedx

≤ kq_h− Ay_hkL²(we)kb_ekL²(we)+ky−y_hkL²(we)kA^∗b_ekL²(we)

+kq−qhkL²(we)kbekL²(we). Finally, using Lemma 3.3.1 we obtain

C6he

h∂yh

∂nA

i

L²(e) ≤C8h⁻¹_e ky−yhkL²(we)+C9

nkqh− AyhkL²(we)+kq−qhkL²(we)

o , and this completes the proof of the theorem.

In the following theorem, we derive local lower bounds for the co-state variable.

Theorem 3.3.2. Let(y, q)and(y_h, q_h)be the solutions of (3.5)-(3.6) and (3.11)-(3.12), respectively. Let z and zh be the solutions of the co-state equations (3.8) and (3.16), respectively. Then,

h²_Kk(¯y_h−y¯_d)− A^∗z_hkL²(K) ≤ C₁₃⁻²n

C₁₄kz−z_hkL²(K)+h²_K

ky_h−y_d−(¯y_h−y¯_d)kL²(K)

+ky−yhkL²(K)

o , and

h

3

e2

h ∂zh

∂n_A^∗ i

L²(e) ≤ C₁₆⁻²C15

nh

1

e2

k(¯yh−y¯d)− A^∗zhkL²( ˜K)+k(yh−yd)−(¯yh−y¯d)kL²( ˜K)

+ky−yhkL²( ˜K)

+C17kz−zhkL²( ˜K)

o,

where ¯v is the mean value of v on the element K such thatv¯|^K = _|K|¹ R

Kv(x)dx.

Proof. Multiplying (3.8) by v ∈ H₀¹(Ω). We form L²-inner product over Ω. Then, an application of the Green’s theorem leads to

a(z−zh, v) = (y−yd, v)−a(zh, v)

= (y−yd, v)− X

K∈Th

Z

KA^∗zhv dx− X

e∈Eh

Z

e

h ∂zh

∂nA^∗

iv de

= (y−yh, v) + X

K∈Th

Z

K

((¯yh−y¯d)− A^∗zh)v dx−X

e∈Eh

Z

e

h ∂zh

∂nA^∗

iv de

+ X

K∈Th

Z

K

(yh−yd−(¯yh−y¯d))v dx. (3.42)

Note that

a(z−z_h, v) = X

K∈Th

Z

KAv(z−z_h)dx+X

e∈Eh

Z

e

h ∂v

∂nA

i(z−z_h)de. (3.43)

Set βK =bK((¯yh −y¯d)− A^∗zh) and choose v =βK in (3.42) and (3.43), and compare both the equations to obtain

Z

KAβK(z−zh)dx = Z

K

((¯yh−y¯d)− A^∗zh)²bKdx+ Z

K

(yh−yd−(¯yh−y¯d))βKdx +

Z

K

(y−yh)βKdx,

where we have used supp βK ⊂K. An application of the triangle inequality yields

Z

K

((¯y_h−y¯_d)− A^∗z_h)²b_Kdx =

Z

KAβ_K(z−z_h)dx− Z

K

(y−y_h)β_Kdx

− Z

K

(yh−yd−(¯yh−y¯d))βKdx

≤ kβKkH²(K)kz−zhkL²(K)+ky−yhkL²(K)kβKkL²(K)

+kyh−yd−(¯yh−y¯d)kL²(K)kβKkL²(K). The property of βK, kβKkH²(K) ≤C14h⁻²_K kβKkL²(K) gives

Z

K

((¯y_h−y¯_d)− A^∗z_h)²b_Kdx

≤ C₁₄h⁻²_K kβ_KkL²(K)kz−z_hkL²(K)

+ky−yhkL²(K)kβKkL²(K)

+kyh−yd−(¯yh−y¯d)kL²(K)kβKkL²(K),(3.44) and using (3.36), we find that

Z

K

((¯yh−y¯d)− A^∗zh)²bKdx≥C₁₃² k(¯yh−y¯d)− A^∗zhk²L²(K). (3.45) We now combine (3.44) and (3.45) to have

C₁₃² k(¯yh−y¯d)− A^∗zhk²L²(K) ≤

C14h⁻²_Kkz−zhk^L2(K)+ky−yhk^L2(K) +kyh−yd−(¯yh−y¯d)kL²(K)

k(¯yh−y¯d)− A^∗zhkL²(K), where we have used the fact kβ_KkL²(K) ≤ k(¯y_h−y¯_d)− A^∗z_hkL²(K). This proves the first inequality. Next, to prove the second inequality, let βe =h

∂zh

∂nA∗

i˜be and choose v =βe in (3.42) and (3.43). We compare both the equations to have

Z

K˜

Aβe(z−zh)dx = Z

K˜

(y−yh)βedx+ Z

K˜

((¯yh−y¯d)− A^∗zh)βedx

− Z

e

h ∂z_h

∂nA^∗

i2

˜bede+ Z

K˜

(yh−yd−(¯yh−y¯d))βedx

− Z

e

h∂βe

∂nA

izhde.

Since h

∂βe

∂nA

i

e = 0, it now leads to

Z

e

h ∂zh

∂nA^∗

i2

˜bede =

Z

K˜

((¯yh−y¯d)− A^∗zh)βedx+ Z

K˜

(yh−yd−(¯yh−y¯d))βedx +

Z

K˜

(y−yh)βedx− Z

K˜ Aβe(z−zh)dx

≤

k(¯yh−y¯d)− A^∗zhkL²( ˜K)+kyh−yd−(¯yh−y¯d)kL²( ˜K)

+ky−yhkL²( ˜K)

kβekL²( ˜K)+kβekH²( ˜K)kz−zhkL²( ˜K).

An application of (3.38) and (3.40) yields

Z

e

h ∂zh

∂nA^∗

i2

˜bede

≤

k(¯yh−y¯d)− A^∗zhkL²( ˜K)+kyh −yd−(¯yh−y¯d)kL²( ˜K)

+ky−yhkL²( ˜K)+C17h⁻²_e kz−zhkL²( ˜K)

kβekL²( ˜K)

≤ C₁₅h

1

e2

k(¯y_h−y¯_d)− A^∗z_hkL²( ˜K)+ky_h−y_d−(¯y_h−y¯_d)kL²( ˜K)

+ky−yhkL²( ˜K)+C17h⁻²_e kz−zhkL²( ˜K)

kβekL²(e)

≤ C15h

1

e2

k(¯yh−y¯d)− A^∗zhkL²( ˜K)+kyh−yd−(¯yh−y¯d)kL²( ˜K)

+ky−yhkL²( ˜K)+C17h⁻²_e kz−zhkL²( ˜K)

h ∂zh

∂nA^∗

i

L²(e). (3.46) In view of (3.39), we have

Z

e

h ∂zh

∂nA^∗

i2

˜bede≥C₁₆²

h ∂zh

∂nA^∗

i

2 L²(e), which combine with (3.46) completes the rest of the proof.

Now, we derive global lower bound for error in the control variable.

Theorem 3.3.3. Let(y, q)and(yh, qh)be the solutions of (3.5)-(3.6) and (3.11)-(3.12), respectively. Let z and zh be the solutions of the co-state equations (3.8) and (3.16), respectively. Then,

η₁² ≤C^∗

α²kq−q_hk²L²(Ω)+kz−z_hk²L²(Ω)

, where C^∗ is a positive constant and η1 is defined in Lemma 3.2.3.

Proof. From the optimality condition (3.9), we deduce that αq+z = 0 on Q^E_ad. An application of inverse property [24] yields

X

K∈Th

h²_K|αqh+zh|²H¹(K) ≤ X

K∈Th

h²_Kkαqh+zhk²H¹(K)

≤ C^∗ X

K∈Th

kαq_h+z_hk²L²(K)

=C^∗kαq_h+z_hk²L²(Ω)

=C^∗kαqh+zh−αq−zk²L²(Ω)

≤ C^∗

α²kq−qhk²L²(Ω)+kz−zhk²L²(Ω)

, and this completes the proof.

Concluding remarks. In this chapter, a posteriori error estimators for the state, co-state and control variables are obtained for EOCP with measure data. The error es- timators obtained in Theorem 3.2.1 are contributed from the approximation error of the state, co-state and control variables. Among them η₁ mainly indicates the approxima- tion error for the control, η2 and η3 are contributed by the state equation whereas η4 is contributed by the co-state equation. The local lower bound for the state and co-state variables are derived in Theorems 3.3.1-3.3.2. Further, a global lower bound for the control variable is derived in Theorem 3.3.3. The performance of our error estimators is reported in Chapter 7. These error estimators work satisfactorily in guiding the mesh refinement and save substantial computational work (see Table 7.2, Chapter 7).

4

POCP with Measure Data in Space: A Priori Error Analysis

In this chapter, we analyze finite element approximation for POCP (1.6)-(1.8) with measure data in space in a bounded convex domain in R^d(d = 2 or 3). The main mathematical difficulty of this problem is that the solution of the state equation exhibits low regularity due to the presence of measure data in the source term, which makes the the convergence analysis somewhat cumbersome. We prove the existence, uniqueness and regularity of the solution of the control problem. A priori error estimates for the state, co-state and control variables are derived for both spatially discrete and fully discrete approximations of optimal control problems. Moreover, L²(0, T;L²(Ω)) convergence properties for the state, co-state and control variables are established. We use piecewise linear and continuous finite elements for approximations of the state and co-state variables whereas the control variable is approximated by piecewise constant functions.