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A Priori Error Analysis of the Petrov Galerkin Crank Nicolson Scheme for Parabolic Optimal Control Problems Dominik Meidner and Boris Vexler Fakultt fr Mathematik Technische Universitt Mnchen October 10 - 14, 2011 Boris Vexler Crank

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A Priori Error Analysis of the Petrov Galerkin Crank Nicolson Scheme for Parabolic Optimal Control Problems

Dominik Meidner and Boris Vexler

Fakultät für Mathematik Technische Universität München

October 10 - 14, 2011

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 1

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Optimal control problem

Cost functional

Minimize J(q, u) = 1 2 T

  • Ω

(u(t, x) − ˆ u(t, x))2 dx dt + α 2 T |q(t)|2 dt

State equation

∂tu − ∆u = f + Gq in (0, T) × Ω, u = 0

  • n (0, T) × ∂Ω,

u = u0 in {0} × Ω with G : Q = L2(0, T; RD) → L2(0, T; H1(Ω)) and (Gq)(t, x) = D

i=1 qi(t)gi(x),

gi ∈ V = H1

0(Ω).

Control constraints

qa ≤ q(t) ≤ qb

  • a. e. in (0, T).

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 2

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Optimal control problem

Cost functional

Minimize J(q, u) = 1 2 T

  • Ω

(u(t, x) − ˆ u(t, x))2 dx dt + α 2 T |q(t)|2 dt

State equation

∂tu − ∆u = f + Gq in (0, T) × Ω, u = 0

  • n (0, T) × ∂Ω,

u = u0 in {0} × Ω with G : Q = L2(0, T; RD) → L2(0, T; H1(Ω)) and (Gq)(t, x) = D

i=1 qi(t)gi(x),

gi ∈ V = H1

0(Ω).

Control constraints

qa ≤ q(t) ≤ qb

  • a. e. in (0, T).

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 2

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Optimal control problem

Cost functional

Minimize J(q, u) = 1 2 T

  • Ω

(u(t, x) − ˆ u(t, x))2 dx dt + α 2 T |q(t)|2 dt

State equation

∂tu − ∆u = f + Gq in (0, T) × Ω, u = 0

  • n (0, T) × ∂Ω,

u = u0 in {0} × Ω with G : Q = L2(0, T; RD) → L2(0, T; H1(Ω)) and (Gq)(t, x) = D

i=1 qi(t)gi(x),

gi ∈ V = H1

0(Ω).

Control constraints

qa ≤ q(t) ≤ qb

  • a. e. in (0, T).

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 2

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A priori error analysis

Temporal discretization of the state

→ discretization parameter k

Spatial discretization of the state

→ discretization parameter h

Treatment of the control? → Goal:

Error estimate

¯ q − ˜ qkhQ = O(k2 + h2). → optimal control ¯ q is not smooth (due to control constraints) → Crank Nicolson scheme is of second order in k → Adjoint scheme?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 3

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A priori error analysis

Temporal discretization of the state

→ discretization parameter k

Spatial discretization of the state

→ discretization parameter h

Treatment of the control? → Goal:

Error estimate

¯ q − ˜ qkhQ = O(k2 + h2). → optimal control ¯ q is not smooth (due to control constraints) → Crank Nicolson scheme is of second order in k → Adjoint scheme?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 3

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A priori error analysis

Temporal discretization of the state

→ discretization parameter k

Spatial discretization of the state

→ discretization parameter h

Treatment of the control? → Goal:

Error estimate

¯ q − ˜ qkhQ = O(k2 + h2). → optimal control ¯ q is not smooth (due to control constraints) → Crank Nicolson scheme is of second order in k → Adjoint scheme?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 3

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A priori error analysis

Temporal discretization of the state

→ discretization parameter k

Spatial discretization of the state

→ discretization parameter h

Treatment of the control? → Goal:

Error estimate

¯ q − ˜ qkhQ = O(k2 + h2). → optimal control ¯ q is not smooth (due to control constraints) → Crank Nicolson scheme is of second order in k → Adjoint scheme?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 3

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A priori error analysis

Temporal discretization of the state

→ discretization parameter k

Spatial discretization of the state

→ discretization parameter h

Treatment of the control? → Goal:

Error estimate

¯ q − ˜ qkhQ = O(k2 + h2). → optimal control ¯ q is not smooth (due to control constraints) → Crank Nicolson scheme is of second order in k → Adjoint scheme?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 3

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Existing literature

Error estimates for parabolic problems without constraints:

Gunzburger, Hou, Hinze, Deckelnick, Chrysafinos, Winther, . . . Apel and Flaig 2011 (Crank-Nicolson)

Error estimates for parabolic problems with control constraints

Lasiecka and Malanowski 1977/78, Malanowski 1981 Rösch 2004 Meidner and Vexler 2008 Neitzel and Vexler 2011 (semilinear) Meidner and Vexler 2011 (Petrov-Galerkin-Crank-Nicolson) → O(k2 + h2)

Error estimates for parabolic problems with state constraints

Deckelnick and Hinze 2010 Meidner, Rannacher, and Vexler 2010

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 4

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Existing literature

Error estimates for parabolic problems without constraints:

Gunzburger, Hou, Hinze, Deckelnick, Chrysafinos, Winther, . . . Apel and Flaig 2011 (Crank-Nicolson)

Error estimates for parabolic problems with control constraints

Lasiecka and Malanowski 1977/78, Malanowski 1981 Rösch 2004 Meidner and Vexler 2008 Neitzel and Vexler 2011 (semilinear) Meidner and Vexler 2011 (Petrov-Galerkin-Crank-Nicolson) → O(k2 + h2)

Error estimates for parabolic problems with state constraints

Deckelnick and Hinze 2010 Meidner, Rannacher, and Vexler 2010

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 4

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Existing literature

Error estimates for parabolic problems without constraints:

Gunzburger, Hou, Hinze, Deckelnick, Chrysafinos, Winther, . . . Apel and Flaig 2011 (Crank-Nicolson)

Error estimates for parabolic problems with control constraints

Lasiecka and Malanowski 1977/78, Malanowski 1981 Rösch 2004 Meidner and Vexler 2008 Neitzel and Vexler 2011 (semilinear) Meidner and Vexler 2011 (Petrov-Galerkin-Crank-Nicolson) → O(k2 + h2)

Error estimates for parabolic problems with state constraints

Deckelnick and Hinze 2010 Meidner, Rannacher, and Vexler 2010

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 4

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Existing literature

Error estimates for parabolic problems without constraints:

Gunzburger, Hou, Hinze, Deckelnick, Chrysafinos, Winther, . . . Apel and Flaig 2011 (Crank-Nicolson)

Error estimates for parabolic problems with control constraints

Lasiecka and Malanowski 1977/78, Malanowski 1981 Rösch 2004 Meidner and Vexler 2008 Neitzel and Vexler 2011 (semilinear) Meidner and Vexler 2011 (Petrov-Galerkin-Crank-Nicolson) → O(k2 + h2)

Error estimates for parabolic problems with state constraints

Deckelnick and Hinze 2010 Meidner, Rannacher, and Vexler 2010

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 4

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Optimality system and regularity

Optimality system

∂t¯ u − ∆¯ u = f + G ¯ q, u(0) = u0 −∂t¯ z − ∆¯ z = ¯ u − ˆ u, z(T) = 0 (α¯ q + G ∗¯ z, δq − ¯ q) ≥ 0 ∀δq ∈ Qad → ¯ q = PQad

  • − 1

αG ∗¯

z

  • with the pointwise projection PQad : Q → Qad

Assumption 1

Ω is polygonal and convex f , ˆ u ∈ H1(0, T; L2(Ω)), f (0), ˆ u(T) ∈ H1

0(Ω)

u0, ∆u0 ∈ H1

0(Ω)

Regularity

¯ q ∈ W 1,∞(0, T; RD), ¯ u, ¯ z ∈ H1(0, T; H2(Ω)) ∩ H2(0, T; L2(Ω))

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 5

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Optimality system and regularity

Optimality system

∂t¯ u − ∆¯ u = f + G ¯ q, u(0) = u0 −∂t¯ z − ∆¯ z = ¯ u − ˆ u, z(T) = 0 (α¯ q + G ∗¯ z, δq − ¯ q) ≥ 0 ∀δq ∈ Qad → ¯ q = PQad

  • − 1

αG ∗¯

z

  • with the pointwise projection PQad : Q → Qad

Assumption 1

Ω is polygonal and convex f , ˆ u ∈ H1(0, T; L2(Ω)), f (0), ˆ u(T) ∈ H1

0(Ω)

u0, ∆u0 ∈ H1

0(Ω)

Regularity

¯ q ∈ W 1,∞(0, T; RD), ¯ u, ¯ z ∈ H1(0, T; H2(Ω)) ∩ H2(0, T; L2(Ω))

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 5

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Optimality system and regularity

Optimality system

∂t¯ u − ∆¯ u = f + G ¯ q, u(0) = u0 −∂t¯ z − ∆¯ z = ¯ u − ˆ u, z(T) = 0 (α¯ q + G ∗¯ z, δq − ¯ q) ≥ 0 ∀δq ∈ Qad → ¯ q = PQad

  • − 1

αG ∗¯

z

  • with the pointwise projection PQad : Q → Qad

Assumption 1

Ω is polygonal and convex f , ˆ u ∈ H1(0, T; L2(Ω)), f (0), ˆ u(T) ∈ H1

0(Ω)

u0, ∆u0 ∈ H1

0(Ω)

Regularity

¯ q ∈ W 1,∞(0, T; RD), ¯ u, ¯ z ∈ H1(0, T; H2(Ω)) ∩ H2(0, T; L2(Ω))

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 5

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Temporal discretization

→ cG(1) Petrov-Galerkin method

Partitioning of the time interval ¯ I = [0, T]: ¯ I = {0} ∪ I1 ∪ I2 ∪ · · · ∪ IM with subintervals Im = (tm−1, tm] of size km and time points 0 = t0 < t1 < · · · < tM−1 < tM = T Ansatz space (continuous) X 1

k =

  • vk ∈ C(¯

I, V )

  • vk
  • Im ∈ P1(Im, V ), m = 1, 2, . . . , M
  • ,

Test space (discontinuous) ˜ X 0

k =

  • vk ∈ L2(I, V )
  • vk
  • Im ∈ P0(Im, V ), m = 1, 2, . . . , M, vk(0) ∈ V
  • Boris Vexler

Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 6

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Temporal discretization

→ cG(1) Petrov-Galerkin method

Partitioning of the time interval ¯ I = [0, T]: ¯ I = {0} ∪ I1 ∪ I2 ∪ · · · ∪ IM with subintervals Im = (tm−1, tm] of size km and time points 0 = t0 < t1 < · · · < tM−1 < tM = T Ansatz space (continuous) X 1

k =

  • vk ∈ C(¯

I, V )

  • vk
  • Im ∈ P1(Im, V ), m = 1, 2, . . . , M
  • ,

Test space (discontinuous) ˜ X 0

k =

  • vk ∈ L2(I, V )
  • vk
  • Im ∈ P0(Im, V ), m = 1, 2, . . . , M, vk(0) ∈ V
  • Boris Vexler

Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 6

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Temporal discretization

→ cG(1) Petrov-Galerkin method

Partitioning of the time interval ¯ I = [0, T]: ¯ I = {0} ∪ I1 ∪ I2 ∪ · · · ∪ IM with subintervals Im = (tm−1, tm] of size km and time points 0 = t0 < t1 < · · · < tM−1 < tM = T Ansatz space (continuous) X 1

k =

  • vk ∈ C(¯

I, V )

  • vk
  • Im ∈ P1(Im, V ), m = 1, 2, . . . , M
  • ,

Test space (discontinuous) ˜ X 0

k =

  • vk ∈ L2(I, V )
  • vk
  • Im ∈ P0(Im, V ), m = 1, 2, . . . , M, vk(0) ∈ V
  • Boris Vexler

Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 6

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Temporal discretization

Bilinear form: B(uk, φ) := (∂tuk, φ)I×Ω + (∇uk, ∇φ)I×Ω + (uk,0, φ−

0 ).

Temporal discretization of the state: uk ∈ X 1

k : B(uk, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

k .

Temporal discretization of the adjoint state: zk ∈ ˜ X 0

k : B(φ, zk) = (uk − ˆ

u, φ)I×Ω ∀φ ∈ X 1

k .

→ Discrete adjoint state is piecewise constant.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 7

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Temporal discretization

Bilinear form: B(uk, φ) := (∂tuk, φ)I×Ω + (∇uk, ∇φ)I×Ω + (uk,0, φ−

0 ).

Temporal discretization of the state: uk ∈ X 1

k : B(uk, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

k .

Temporal discretization of the adjoint state: zk ∈ ˜ X 0

k : B(φ, zk) = (uk − ˆ

u, φ)I×Ω ∀φ ∈ X 1

k .

→ Discrete adjoint state is piecewise constant.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 7

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Temporal discretization

Bilinear form: B(uk, φ) := (∂tuk, φ)I×Ω + (∇uk, ∇φ)I×Ω + (uk,0, φ−

0 ).

Temporal discretization of the state: uk ∈ X 1

k : B(uk, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

k .

Temporal discretization of the adjoint state: zk ∈ ˜ X 0

k : B(φ, zk) = (uk − ˆ

u, φ)I×Ω ∀φ ∈ X 1

k .

→ Discrete adjoint state is piecewise constant.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 7

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Temporal discretization

Bilinear form: B(uk, φ) := (∂tuk, φ)I×Ω + (∇uk, ∇φ)I×Ω + (uk,0, φ−

0 ).

Temporal discretization of the state: uk ∈ X 1

k : B(uk, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

k .

Temporal discretization of the adjoint state: zk ∈ ˜ X 0

k : B(φ, zk) = (uk − ˆ

u, φ)I×Ω ∀φ ∈ X 1

k .

→ Discrete adjoint state is piecewise constant.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 7

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Temporal discretization

cG(1) discretization of the state is a variant of the Crank Nicolson scheme

→ second order convergence

Consistent discretization of the adjoint state → piecewise constants

→ only first order convergence expected

→ Variational discretization (no control discretization)

→ only first order convergence ¯ q − ¯ qkQ = O(k).

→ How to achieve O(k2) ?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 8

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Temporal discretization

cG(1) discretization of the state is a variant of the Crank Nicolson scheme

→ second order convergence

Consistent discretization of the adjoint state → piecewise constants

→ only first order convergence expected

→ Variational discretization (no control discretization)

→ only first order convergence ¯ q − ¯ qkQ = O(k).

→ How to achieve O(k2) ?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 8

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Temporal discretization

cG(1) discretization of the state is a variant of the Crank Nicolson scheme

→ second order convergence

Consistent discretization of the adjoint state → piecewise constants

→ only first order convergence expected

→ Variational discretization (no control discretization)

→ only first order convergence ¯ q − ¯ qkQ = O(k).

→ How to achieve O(k2) ?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 8

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Temporal discretization

cG(1) discretization of the state is a variant of the Crank Nicolson scheme

→ second order convergence

Consistent discretization of the adjoint state → piecewise constants

→ only first order convergence expected

→ Variational discretization (no control discretization)

→ only first order convergence ¯ q − ¯ qkQ = O(k).

→ How to achieve O(k2) ?

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 8

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Temporal discretization

Adjoint discretization

zk

Piecewise linear interpolation at the midpoints

→ Superconvergence result (Meidner & Vexler 2011) z − πkzkL2(I×Ω) = O(k2)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 9

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Temporal discretization

Adjoint discretization

zk

Piecewise linear interpolation at the midpoints

zk πkzk → Superconvergence result (Meidner & Vexler 2011) z − πkzkL2(I×Ω) = O(k2)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 9

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Temporal discretization

Adjoint discretization

zk

Piecewise linear interpolation at the midpoints

zk πkzk → Superconvergence result (Meidner & Vexler 2011) z − πkzkL2(I×Ω) = O(k2)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 9

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Superconvergence result

Theorem (Meidner & Vexler 2011)

Let z be solution of −∂tz − ∆z = g, z(T) = 0 with g ∈ H1(0, T; L2(Ω)), g(T) ∈ H1

0(Ω) and zk ∈ ˜

X 0

k be defined by

B(φ, zk) = (g, φ)I×Ω ∀φ ∈ X 1

k .

Then there holds z − πkzkL2(I×Ω) ≤ c k2 ∂2

t zL2(I×Ω) + ∂t∆zL2(I×Ω)

  • Boris Vexler

Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 10

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Superconvergence result: Proof

Step 1: Semidiscrete stability estimates

→ diagonal testing is not possible! ∂tvkI×Ω + Pk∆vkI×Ω + ∇vkI×Ω ≤ cf I×Ω with L2-Projection Pk : L2(0, T; L2(Ω)) → ˜ X 0

k .

Step 2: Supercloseness for the midpoint interpolation

→ duality arguments & stability estimates (step 1) Πkz − zkL2(I×Ω) ≤ c k2 ∂2

t zL2(I×Ω) + ∂t∆zL2(I×Ω)

  • Step 3: Stability of πk in L2(I × Ω) for semidiscrete functions

→ πk is in general not stable w.r.t. L2(I × Ω), but πkykI×Ω ≤ cykI×Ω ∀yk ∈ ˜ X 0

k

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 11

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Superconvergence result: Proof

Step 1: Semidiscrete stability estimates

→ diagonal testing is not possible! ∂tvkI×Ω + Pk∆vkI×Ω + ∇vkI×Ω ≤ cf I×Ω with L2-Projection Pk : L2(0, T; L2(Ω)) → ˜ X 0

k .

Step 2: Supercloseness for the midpoint interpolation

→ duality arguments & stability estimates (step 1) Πkz − zkL2(I×Ω) ≤ c k2 ∂2

t zL2(I×Ω) + ∂t∆zL2(I×Ω)

  • Step 3: Stability of πk in L2(I × Ω) for semidiscrete functions

→ πk is in general not stable w.r.t. L2(I × Ω), but πkykI×Ω ≤ cykI×Ω ∀yk ∈ ˜ X 0

k

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 11

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Superconvergence result: Proof

Step 1: Semidiscrete stability estimates

→ diagonal testing is not possible! ∂tvkI×Ω + Pk∆vkI×Ω + ∇vkI×Ω ≤ cf I×Ω with L2-Projection Pk : L2(0, T; L2(Ω)) → ˜ X 0

k .

Step 2: Supercloseness for the midpoint interpolation

→ duality arguments & stability estimates (step 1) Πkz − zkL2(I×Ω) ≤ c k2 ∂2

t zL2(I×Ω) + ∂t∆zL2(I×Ω)

  • Step 3: Stability of πk in L2(I × Ω) for semidiscrete functions

→ πk is in general not stable w.r.t. L2(I × Ω), but πkykI×Ω ≤ cykI×Ω ∀yk ∈ ˜ X 0

k

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 11

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Postprocessing strategy

State discretization

uk

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 12

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Postprocessing strategy

State discretization

uk

Adjoint and control discretization

zk

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 12

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Postprocessing strategy

State discretization

uk

Adjoint discretization and interpolation

zk πkzk

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 12

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Postprocessing strategy

Intermediate step: − 1

απkG ∗zk

−α−1πkG ∗zk

Postprocessed ˜ qk = PQad(− 1

απkG ∗zk)

→ cf. C. Meyer & A. Rösch 2004.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 13

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Postprocessing strategy

Intermediate step: − 1

απkG ∗zk

−α−1πkG ∗zk

Postprocessed ˜ qk = PQad(− 1

απkG ∗zk)

˜ qk −α−1πkG ∗zk → cf. C. Meyer & A. Rösch 2004.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 13

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Postprocessing strategy

Assumption 2

The boundaries of the active sets Ai = { t ∈ [0, T] | qi(t) = qa,i or qi(t) = qb,i } , i = 1, 2, . . . , D consist of a finite number of points.

Theorem

Let Assumptions 1 and 2 be fulfilled. Let ˜ qk be defined as ˜ qk = PQad

  • − 1

απkG ∗¯ zk

  • .

Then there holds: ¯ q − ˜ qkQ = O(k2).

◮ D. Meidner and B. Vexler. A priori error analysis of the Petrov Galerkin Crank Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim., accepted (2011)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 14

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Postprocessing strategy

Assumption 2

The boundaries of the active sets Ai = { t ∈ [0, T] | qi(t) = qa,i or qi(t) = qb,i } , i = 1, 2, . . . , D consist of a finite number of points.

Theorem

Let Assumptions 1 and 2 be fulfilled. Let ˜ qk be defined as ˜ qk = PQad

  • − 1

απkG ∗¯ zk

  • .

Then there holds: ¯ q − ˜ qkQ = O(k2).

◮ D. Meidner and B. Vexler. A priori error analysis of the Petrov Galerkin Crank Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim., accepted (2011)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 14

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Spatial discretization

→ Linear or bilinear finite elements for both state and the adjoint state Ansatz space (continuous in time) X 1

kh =

  • vkh ∈ C(¯

I, Vh)

  • vk
  • Im ∈ P1(Im, Vh), m = 1, 2, . . . , M
  • ,

Test space (discontinuous in time) ˜ X 0

kh =

  • vkh ∈ L2(I, Vh)
  • vkh
  • Im ∈ P0(Im, Vh), m = 1, 2, . . . , M, vkh(0) ∈ Vh
  • Space-time discretization of the state:

ukh ∈ X 1

kh : B(ukh, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

kh.

Space-time discretization of the adjoint state: zkh ∈ ˜ X 0

kh : B(φ, zkh) = (ukh − ˆ

u, φ)I×Ω ∀φ ∈ X 1

kh.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 15

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SLIDE 43

Spatial discretization

→ Linear or bilinear finite elements for both state and the adjoint state Ansatz space (continuous in time) X 1

kh =

  • vkh ∈ C(¯

I, Vh)

  • vk
  • Im ∈ P1(Im, Vh), m = 1, 2, . . . , M
  • ,

Test space (discontinuous in time) ˜ X 0

kh =

  • vkh ∈ L2(I, Vh)
  • vkh
  • Im ∈ P0(Im, Vh), m = 1, 2, . . . , M, vkh(0) ∈ Vh
  • Space-time discretization of the state:

ukh ∈ X 1

kh : B(ukh, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

kh.

Space-time discretization of the adjoint state: zkh ∈ ˜ X 0

kh : B(φ, zkh) = (ukh − ˆ

u, φ)I×Ω ∀φ ∈ X 1

kh.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 15

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SLIDE 44

Spatial discretization

→ Linear or bilinear finite elements for both state and the adjoint state Ansatz space (continuous in time) X 1

kh =

  • vkh ∈ C(¯

I, Vh)

  • vk
  • Im ∈ P1(Im, Vh), m = 1, 2, . . . , M
  • ,

Test space (discontinuous in time) ˜ X 0

kh =

  • vkh ∈ L2(I, Vh)
  • vkh
  • Im ∈ P0(Im, Vh), m = 1, 2, . . . , M, vkh(0) ∈ Vh
  • Space-time discretization of the state:

ukh ∈ X 1

kh : B(ukh, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

kh.

Space-time discretization of the adjoint state: zkh ∈ ˜ X 0

kh : B(φ, zkh) = (ukh − ˆ

u, φ)I×Ω ∀φ ∈ X 1

kh.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 15

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SLIDE 45

Spatial discretization

→ Linear or bilinear finite elements for both state and the adjoint state Ansatz space (continuous in time) X 1

kh =

  • vkh ∈ C(¯

I, Vh)

  • vk
  • Im ∈ P1(Im, Vh), m = 1, 2, . . . , M
  • ,

Test space (discontinuous in time) ˜ X 0

kh =

  • vkh ∈ L2(I, Vh)
  • vkh
  • Im ∈ P0(Im, Vh), m = 1, 2, . . . , M, vkh(0) ∈ Vh
  • Space-time discretization of the state:

ukh ∈ X 1

kh : B(ukh, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

kh.

Space-time discretization of the adjoint state: zkh ∈ ˜ X 0

kh : B(φ, zkh) = (ukh − ˆ

u, φ)I×Ω ∀φ ∈ X 1

kh.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 15

slide-46
SLIDE 46

Spatial discretization

→ Linear or bilinear finite elements for both state and the adjoint state Ansatz space (continuous in time) X 1

kh =

  • vkh ∈ C(¯

I, Vh)

  • vk
  • Im ∈ P1(Im, Vh), m = 1, 2, . . . , M
  • ,

Test space (discontinuous in time) ˜ X 0

kh =

  • vkh ∈ L2(I, Vh)
  • vkh
  • Im ∈ P0(Im, Vh), m = 1, 2, . . . , M, vkh(0) ∈ Vh
  • Space-time discretization of the state:

ukh ∈ X 1

kh : B(ukh, φ) = (f + Gq, φ)I×Ω + (u0, φ− 0 )

∀φ ∈ ˜ X 0

kh.

Space-time discretization of the adjoint state: zkh ∈ ˜ X 0

kh : B(φ, zkh) = (ukh − ˆ

u, φ)I×Ω ∀φ ∈ X 1

kh.

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 15

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SLIDE 47

Postprocessing strategy

Theorem

Let Assumptions 1 and 2 be fulfilled. Let ˜ qkh be defined as ˜ qkh = PQad

  • − 1

απkG ∗¯ zkh

  • .

Then there holds: ¯ q − ˜ qkhQ = O(k2 + h2).

◮ D. Meidner and B. Vexler. A priori error analysis of the Petrov Galerkin Crank Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim., accepted (2011)

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SLIDE 48

Numerical example: temporal error

Problem with known exact solution

¯ q(t) := PQad

  • −π4

4

  • exp(aπ2t) − exp(aπ2T)
  • ,

¯ u(t, x1, x2) := −1 2 + aπ2wa(t, x1, x2), ¯ z(t, x1, x2) := wa(t, x1, x2) − wa(T, x1, x2). with wa(t, x1, x2) := exp(aπ2t) sin(πx1) sin(πx2), qa = −70, qb = −1, a = − √ 5, T = 0.1

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 17

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SLIDE 49

Numerical example: temporal error

10−2 10−1 100 100 101 102 103 M |¯ q − ¯ qkh|I |¯ q − ˜ qkh|I O(k) O(k2)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 18

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SLIDE 50

Numerical example: spatial error

101 102 103 104 10−4 10−3 10−2 10−1 100 N |¯ q − ¯ qkh|I |¯ q − ˜ qkh|I O(h2)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 19

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SLIDE 51

Conclusions

→ cG(1) Petrov-Galerkin discretization → Crank Nicolson scheme → consistent discretization of the adjoint state → second order convergence for the postprocessed control

◮ D. Meidner and B. Vexler. A priori error analysis of the Petrov Galerkin Crank Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim., accepted (2011)

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 20