Space-time finite element discretization of parabolic optimal control problems with energy regularization

1 Introduction

In this paper, we consider and analyze continuous space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of the following parabolic optimal control problem: For a given target function $u_{d} \in L^{2} (Q)$ , we want to minimize the cost functional

J (u, z) := \frac{1}{2} \int_{Q} {| u - u_{d} |}_{d}^{2} d x d t + \frac{1}{2} ϱ ∥ z ∥_{L^{2} (0, T; H^{- 1} (Ω))}^{2}

(1)

subject to the linear parabolic state equation

\partial_{t} u - Δ_{x} u = z i n Q, u = 0 o n Σ, u = 0 o n Σ_{0},

(2)

where $Q := Ω \times (0, T)$ is the space-time domain with the lateral boundary $Σ := \partial Ω \times (0, T)$ , and $Σ_{0} := Ω \times {0}$ . Moreover, $Ω \subset R^{d}$ , $d = 2, 3$ , is a bounded Lipschitz domain, $T > 0$ is the final time, and $ϱ > 0$ is some regularization parameter.

In our recent paper [18], we have considered the related standard optimal control problem with regularization in $L^{2} (Q)$ , i.e.,

J (u, z) := \frac{1}{2} \int_{Q} {| u - u_{d} |}_{d}^{2} d x d t + \frac{1}{2} ϱ ∥ z ∥_{L^{2} (Q)}^{2},

(3)

subject to (2). In both cases, we can numerically solve the corresponding parabolic forward-backward optimality systems at once. This allows not only for a more efficient solution of the global system, but also for parallelization in space and time, and for adaptive discretizations simultaneously in space and time. In contrast to classical time-stepping methods or discontinuous Galerkin (dG) methods which are defined with respect to time slices or slabs, see, e.g., the monographs [16] and [31], and the review article [9] on parallel-in-time methods, we use fully unstructured simplicial space-time meshes for the numerical solution of the parabolic state equation (2), see the recent review article [29] and the related references therein.

The standard approach for distributed control problems is to consider the control $z$ in $L^{2} (Q)$ . There is a huge number of publications on the standard setting (3) with $L^{2} (Q)$ -regularization. We here only refer to the monographs [5, 12, 32], to the more recent papers [10, 21, 22] on discontinuous (dG) and continuous Galerkin time-slice finite element methods, [23] on full space-time dG finite element methods, [11] on space-time adaptive wavelet methods, [13, 17] on multiharmonic methods, [1] on proper orthogonal decomposistion, [7]

on low-rank tensor method, and to our very recent paper

[18] on completely unstructured space-time finite element methods for optimal control of parabolic equations based on

L^{2} (Q)

-regularization, and the references given therein. However, since the state

u \in L^{2} (0, T; H_{0}^{1} (Ω))

is well defined as the solution of the forward heat equation for

z \in L^{2} (0, T; H^{- 1} (Ω))

, we may also consider the tracking type functional as given in (1). Applying integration by parts also in time to derive a variational formulation for the adjoint equation, we end up, in contrast to the case of

L^{2}

regularization, with a positive definite but skew-symmetric bilinear form describing the optimality system. In this paper, we provide a complete numerical analysis for both the continuous and discrete system.

The rest of the paper is structured as follows. In Section 2, we introduce some notation and state some preliminary results on the solvability and numerical analysis of the parabolic initial-boundary value problem that serves as state equation in the optimal control problem. In Section 3, we analyze the unique solvability of the continuous optimality system, whereas Section 4 is devoted to the numerical analysis of the space-time finite element approximation. Numerical results are presented in Section 5 Finally, some conclusions are drawn in Section 6.

2 Preliminaries

In this section, we introduce basic notations, and summarize some recent results on space-time finite element methods for the numerical solution of the state equation (2). For the mathematical analysis of parabolic initial boundary value problems in space-time Sobolev spaces, see [14, 15], and [20, 33] for Bochner spaces of abstract functions, mapping the time interval $(0, T)$ to some Hilbert or Banach space.

Following the latter approach, we define

$X$	$:=$	$L^{2} (0, T; H_{0}^{1} (Ω)) \cap H_{0,}^{1} (0, T; H^{- 1} (Ω))$
	$=$	${v \in L^{2} (0, T; H_{0}^{1} (Ω)) : \partial_{t} v \in L^{2} (0, T; H^{- 1} (Ω)), v = 0 on Σ_{0}},$
$Y$	$:=$	$L^{2} (0, T; H_{0}^{1} (Ω)), Y^{*} := L^{2} (0, T; H^{- 1} (Ω)),$

using the standard Sobolev spaces $H_{0}^{1} (Ω)$ and its dual $H^{- 1} (Ω)$ . Note that we have $X = {v \in W (0, T) : v = 0 on Σ_{0}}$ as used in [20]. The related norms are given by

∥ u ∥_{X} := [∥ w_{u} ∥_{Y}^{2} + ∥ u ∥_{Y}^{2}]^{1 / 2}, ∥ v ∥_{Y} := ∥ \nabla_{x} v ∥_{L^{2} (Q)},

where $w_{u} \in Y$ is the unique solution of the variational formulation [27]

\int_{Q} \nabla_{x} w_{u} \cdot \nabla_{x} v d x d t = ⟨ \partial_{t} u, v ⟩_{Q}, \forall v \in Y .

(4)

The standard weak formulation of the initial boundary value problem (2) reads as follows: Given $z \in Y^{*}$ , find $u \in X$ such that

b (u, v) = ⟨ z, v ⟩_{Q}, \forall v \in Y,

(5)

with the bilinear form $b (\cdot, \cdot) : X \times Y \to R$ ,

b (u, v) := \int_{Q} [\partial_{t} u v + \nabla_{x} u \cdot \nabla_{x} v] d x d t, \forall (u, v) \in X \times Y,

(6)

and the linear form $⟨ z, \cdot ⟩_{Q} : Y \to R$ with the duality pairing $⟨ z, v ⟩_{Q}$ as extension of the inner product in $L^{2} (Q)$ . Similarly, the first integral in (6) has to be understood as duality pairing as well.

The bilinear form $b (\cdot, \cdot)$ is bounded,

| b (u, v) | \leq \sqrt{2} ∥ u ∥_{X} ∥ v ∥_{Y}, \forall (u, v) \in X \times Y,

(7)

and satisfies the inf-sup stability condition [27, Theorem 2.1]

inf 0 \neq u \in X sup 0 \neq v \in Y \frac{b (u, v)}{∥ u ∥_{X} ∥ v ∥_{Y}} \geq \frac{1}{2 \sqrt{2}} .

(8)

Moreover, for $v \in Y ∖ {0}$ , we define

˜ u (x, t) = \int_{0}^{t} v (x, s) d s, (x, t) \in Q

to obtain

b (˜ u, v) = ∥ v ∥_{L^{2} (Q)}^{2} + \frac{1}{2} ∥ \nabla_{x} ˜ u (T) ∥_{L^{2} (Ω)}^{2} > 0.

Hence, we can apply the Nec̆as-Babuška theorem [2, 24] to conclude that the variational problem (5) is well-posed, see also [3, 6, 8].

For the finite element discretization of the variational formulation (5), we introduce conforming space-time finite element spaces $X_{h} \subset X$ and $Y_{h} \subset Y$ , where we assume $X_{h} \subseteq Y_{h}$ . In particular, we may use $X_{h} = Y_{h} = S_{h}^{1} (Q_{h}) \cap X$ spanned by continuous and piecewise linear basis functions which are defined with respect to some admissible decomposition $T_{h} (Q)$ of the space-time domain $Q$ into shape regular simplicial finite elements $τ_{ℓ}$ , and which are zero at the initial time $t = 0$ and at the lateral boundary $Σ$ , where $h$ denotes a suitable mesh-size parameter, see, e.g., [6, 8, 27]. Then the finite element approximation of (5) is to find $u_{h} \in X_{h}$ such that

b (u_{h}, v_{h}) = ⟨ z, v_{h} ⟩_{Q}, \forall v_{h} \in Y_{h} .

(9)

When replacing (4) by its finite element approximation to find $w_{u, h} \in Y_{h}$ such that

\int_{Q} \nabla_{x} w_{u, h} \cdot \nabla_{x} v_{h} d x d t = \int_{Q} \partial_{t} u v_{h} d x d t, \forall v_{h} \in Y_{h},

(10)

we can define a discrete norm

∥ u ∥_{X_{h}} := [∥ w_{u, h} ∥_{Y}^{2} + ∥ u ∥_{Y}^{2}]^{1 / 2} .

As in the continuous case, see (8), we can prove a discrete inf-sup condition, see [27, Theorem 3.1],

\frac{1}{2 \sqrt{2}} ∥ u_{h} ∥_{X_{h}} \leq sup 0 \neq v_{h} \in Y_{h} \frac{b (u_{h}, v_{h})}{∥ v_{h} ∥_{Y}}, \forall u_{h} \in X_{h} .

(11)

Hence, we conclude unique solvability of the Galerkin scheme (9), and we obtain the following quasi-optimal error estimate, see [27, Theorem 3.2]:

∥ u - u_{h} ∥_{X_{0, h}} \leq 5 inf z_{h} \in X_{0, h} ∥ u - z_{h} ∥_{X_{0}} .

(12)

In particular, when assuming $u \in H^{2} (Q)$ , this finally results in the energy error estimate, see [27, Theorem 3.3],

∥ u - u_{h} ∥_{L^{2} (0, T; H_{0}^{1} (Ω))} \leq c h | u |_{H^{2} (Q)} .

(13)

3 The first-order optimality system

We now consider the optimal control problem to minimize (1) subject to the heat equation (2). As in (4), we define $w_{z} \in Y$ as the unique solution of the variational problem

\int_{Q} \nabla_{x} w_{z} \cdot \nabla_{x} v d x d t = ⟨ z, v ⟩_{Q} \forall v \in Y,

(14)

to conclude

∥ z ∥_{L^{2} (0, T; H^{- 1} (Ω))}^{2} = ∥ \nabla_{x} w_{z} ∥_{L^{2} (Q)}^{2} = ⟨ z, w_{z} ⟩_{Q} .

Now, using standard arguments, we can write the first-order optimality system as the primal problem

\partial_{t} u - Δ_{x} u = z i n Q, u = 0 o n Σ, u = 0 o n Σ_{0},

the adjoint problem

- \partial_{t} p - Δ_{x} p = u - u_{d} i n Q, p = 0 o n Σ, p = 0 o n Σ_{T},

(15)

and the gradient equation

p + ϱ w_{z} = 0 i n Q .

(16)

Using the variational formulation (5) of the primal problem, inserting the definition (14), and the gradient equation (16), we get a first variational equation to find $(u, p) \in X \times Y$ such that

\frac{1}{ϱ} \int_{Q} \nabla_{x} p \cdot \nabla_{x} v d x d t + \int_{Q} [\partial_{t} u v + \nabla_{x} u \cdot \nabla_{x} v] d x d t = 0, \forall v \in Y .

On the other hand, when considering the variational formulation of the adjoint problem and integrating by parts also in time, we arrive at the second variational equation

- \int_{Q} [p \partial_{t} q + \nabla_{x} p \cdot \nabla_{x} q] d x d t + \int_{Q} u q d x d t = \int_{Q} u_{d} q d x d t, \forall q \in X .

Hence, we end up with a variational problem to find $(u, p) \in X \times Y$ such that

B (u, p; v, q) = ⟨ u_{d}, q ⟩_{L^{2} (Q)}, \forall (v, q) \in Y \times X,

(17)

with

B (u, p; v, q) := \frac{1}{ϱ} a (p, v) + b (u, v) - b (q, p) + c (u, q) .

(18)

Here, the bilinear form $b (\cdot, \cdot)$ is the same as used in Section 2,

a (p, v) := \int_{Q} \nabla_{x} p \cdot \nabla_{x} v d x d t, and c (u, q) := \int_{Q} u q d x d t .

Note that $a (p, p) = ∥ p ∥_{Y}^{2}$ and $c (u, u) = ∥ u ∥_{L^{2} (Q)}^{2}$ .

Theorem 1.

For $u_{d} \in X^{*}$ , the variational problem (17) admits a unique solution $(u, p) \in X \times Y$ satisfying the a priori estimates

∥ u ∥_{X} \leq \frac{8}{ϱ} ∥ u_{d} ∥_{X^{*}}, ∥ p ∥_{Y} \leq \sqrt{2} 8 ∥ u_{d} ∥_{X^{*}} .

Proof.

Using the Riesz representation theorem, we introduce operators $A : Y \to Y^{*}$ , $B : X \to Y^{*}$ , and $C : X \to X^{*}$ , satisfying, for $u, q \in X$ and $p, v \in Y$ ,

⟨ A p, v ⟩_{Q} = a (p, v), ⟨ B u, v ⟩_{Q} = b (u, v), ⟨ C u, q ⟩_{Q} = c (u, q) .

Hence, we can write the variational problem (17) as operator equation

(\begin{matrix} \frac{1}{ϱ} A & B - B^{*} & C \end{matrix}) (\begin{matrix} p u \end{matrix}) = (\begin{matrix} 0 u_{d} \end{matrix}) .

Since the operator $A : Y \to Y^{*}$ is bounded and elliptic, we can determine $p = - ϱ A^{- 1} B u$ to obtain the Schur complement system

[C + ϱ B^{*} A^{- 1} B] u = u_{d} in X^{*} .

(19)

For $¯ ¯ ¯ p = A^{- 1} B u$ , we first have

∥ ¯ ¯ ¯ p ∥_{Y}^{2} = \int_{Q} | \nabla_{x} ¯ ¯ ¯ p |^{2} d x d t = a (¯ ¯ ¯ p, ¯ ¯ ¯ p) = ⟨ A ¯ ¯ ¯ p, ¯ ¯ ¯ p ⟩_{Q} = ⟨ B^{*} A^{- 1} B u, u ⟩_{Q} .

From the stability condition (8) for the state equation, see Section 2, we immediately get

\frac{1}{2 \sqrt{2}} ∥ u ∥_{X} \leq sup 0 \neq v \in Y \frac{⟨ B u, v ⟩_{Q}}{∥ v ∥_{Y}} = sup 0 \neq v \in Y \frac{⟨ A ¯ ¯ ¯ p, v ⟩_{Q}}{∥ v ∥_{Y}} \leq ∥ ¯ ¯ ¯ p ∥_{Y} .

Hence, we have

⟨ (C + ϱ B^{*} A^{- 1} B) u, u ⟩_{Q} \geq \frac{1}{8} ϱ ∥ u ∥_{X}^{2} for all u \in X .

Thus, we conclude unique solvability of the Schur complement system (19), and from

\frac{1}{8} ϱ ∥ u ∥_{X}^{2} \leq ⟨ (C + ϱ B^{*} A^{- 1} B) u, u ⟩_{Q} = ⟨ u_{d}, u ⟩_{Q} \leq ∥ u_{d} ∥_{X^{*}} ∥ u ∥_{X}

we obtain the first estimate. Now, the boundedness of $B$ , i.e., the boundedness (7) of the bilinear form $b (\cdot, \cdot)$ yields

∥ p ∥_{Y}^{2} = a (p, p) = - ϱ b (u, p) \leq \sqrt{2} ϱ ∥ u ∥_{X} ∥ p ∥_{Y},

i.e.,

∥ p ∥_{Y} \leq \sqrt{2} ϱ ∥ u ∥_{X} \leq \sqrt{2} 8 ∥ u_{d} ∥_{X^{*}} .

∎

Although unique solvability of the variational problem (17) already implies a related stability condition for the bilinear form $B (u, p; v, q)$ , we will present an alternative proof for this stability condition in order to be able to derive related results for the Galerkin discretization of (17).

Lemma 1.

The bilinear form (18) satisfies the stability condition

\frac{1}{16} ϱ [∥ u ∥_{X}^{2} + ∥ p ∥_{Y}^{2}]^{1 / 2} \leq sup 0 \neq (v, q) \in Y \times X \frac{B (u, p; v, q)}{[∥ v ∥_{Y}^{2} + ∥ q ∥_{X}^{2}]^{1 / 2}}

(20)

for all $(u, p) \in X \times Y$ , when assuming $ϱ \leq 1$ .

Proof.

For $u \in X \subset Y$ , let $w_{u} \in Y$ be the unique solution of the variational problem (4). For $p \in Y$ and arbitrary $α \in R_{+}$ , we have $v := u + w_{u} + α p \in Y$ . Choosing $q := α u \in X$ , we obtain

	$B (u, p; v, q)$	$=$	$\frac{1}{ϱ} a (p, u + w_{u} + α p) + b (u, u + w_{u} + α p) - b (α u, p) + c (u, α u)$
		$\geq$	$\frac{α}{ϱ} ∥ p ∥_{Y}^{2} + \frac{1}{ϱ} a (p, u + w_{u}) + b (u, u + w_{u}) .$

Following the proof of [27, Theorem 2.1], we use

$b (u, u + w_{u})$	$=$	$\int_{Q} [\partial_{t} u (u + w_{u}) + \nabla_{x} u \cdot \nabla_{x} (u + w_{u})] d x d t$
		$= \frac{1}{2} ∥ u (T) ∥_{L^{2} (Ω)}^{2} + \int_{Q} \nabla_{x} w_{u} \cdot \nabla_{x} w_{u} d x d t + ∥ u ∥_{Y}^{2} + \int_{Q} \nabla_{x} u \cdot \nabla_{x} w_{u} d x d t$
		$\geq ∥ w_{u} ∥_{Y}^{2} + ∥ u ∥_{Y}^{2} - ∥ u ∥_{Y}^{2} ∥ w_{u} ∥_{Y}^{2}$
		$\geq \frac{1}{2} [∥ w_{u} ∥_{Y}^{2} + ∥ u ∥_{Y}^{2}] = \frac{1}{2} ∥ u ∥_{X}^{2} .$

Moreover, for $γ \in R_{+}$ , we have

$a (p, u + w_{u})$	$=$	$\int_{Q} \nabla_{x} p \cdot \nabla_{x} (u + w_{u}) d x d t \geq - ∥ p ∥_{Y} ∥ u + w_{u} ∥_{Y}$
	$\geq$	$- \frac{1}{2 γ} ∥ p ∥_{Y}^{2} - \frac{1}{2} γ ∥ u + w_{u} ∥_{Y}^{2}$
	$\geq$	$- \frac{1}{2 γ} ∥ p ∥_{Y}^{2} - γ (∥ u ∥_{Y}^{2} + ∥ w_{u} ∥_{Y}^{2}) = - \frac{1}{2 γ} ∥ p ∥_{Y}^{2} - γ ∥ u ∥_{X}^{2} .$

Choosing $γ = \frac{1}{4} ϱ$ and $α = \frac{1}{4} ϱ + \frac{2}{ϱ}$ , we get

	$B (u, p; v, q)$	$\geq$	$\frac{1}{ϱ} (α - \frac{1}{2 γ}) ∥ p ∥_{Y}^{2} + (\frac{1}{2} - \frac{γ}{ϱ}) ∥ u ∥_{X}^{2}$
		$=$	$\frac{1}{4} [∥ u ∥_{X}^{2} + ∥ p ∥_{Y}^{2}] .$

On the other hand, we have

$∥ q ∥_{X}^{2} + ∥ v ∥_{Y}^{2}$	$=$	$α^{2} ∥ u ∥_{X}^{2} + ∥ u + w_{u} + α p ∥_{Y}^{2}$
	$\leq$	$α^{2} ∥ u ∥_{X}^{2} + (∥ u ∥_{Y} + ∥ w_{u} ∥_{Y} + α ∥ p ∥_{Y})^{2}$
	$\leq$	$α^{2} ∥ u ∥_{X}^{2} + (2 + α^{2}) (∥ u ∥_{Y}^{2} + ∥ w_{u} ∥_{Y}^{2} + ∥ p ∥_{Y}^{2})$
	$\leq$	$2 (1 + α^{2}) [∥ u ∥_{X}^{2} + ∥ p ∥_{Y}^{2}]$
	$=$	$2 (2 + \frac{1}{16} ϱ^{2} + \frac{4}{ϱ^{2}}) [∥ u ∥_{X}^{2} + ∥ p ∥_{Y}^{2}]$
	$\leq$	$(4 + \frac{1}{8} + 8) \frac{1}{ϱ^{2}} [∥ u ∥_{X}^{2} + ∥ p ∥_{Y}^{2}] \leq \frac{16}{ϱ^{2}} [∥ u ∥_{X}^{2} + ∥ p ∥_{Y}^{2}],$

provided that $ϱ \leq 1$ . Therefore,

follows. This concludes the proof. ∎

4 Discretization

As before, let $X_{0, h} \subset X_{0}$ and $Y_{h} \subset Y$ be some conforming space-time finite element spaces satisfying $X_{0, h} \subseteq Y_{h}$ . Again, we choose $X_{0, h} = S_{h}^{1} (Q_{h}) \cap X_{0}$ , but now we use $Y_{h} = S_{h}^{1} (Q) \cap Y$ . By construction, we have $X_{0, h} \subset Y_{h}$ .

Instead of (10), we now consider the variational formulation to find $w_{u, h} \in Y_{h}$ such that

\int_{Q} \nabla_{x} w_{h} \cdot \nabla_{x} v_{h} d x d t = \int_{Q} \partial_{t} u v_{h} d x d t, \forall v_{h} \in Y_{h},

(21)

to define the discrete norm

∥ u ∥_{X_{0, h}} := [∥ w_{u, h} ∥_{Y}^{2} + ∥ u ∥_{Y}^{2}]^{1 / 2} .

The space-time finite element discretization of the variational formulation (17) is to find $(u_{h}, p_{h}) \in X_{0, h} \times Y_{h}$ such that

B (u_{h}, p_{h}; v_{h}, q_{h}) = ⟨ u_{d}, q_{h} ⟩_{L^{2} (Q)}, \forall (v_{h}, q_{h}) \in Y_{h} \times X_{0, h} .

(22)

As in the continuous case, see Theorem 1, and following [27, Section 3], we can confirm a discrete inf-sup condition for the bilinear form $B (\cdot, \cdot; \cdot, \cdot)$ .

Lemma 2.

The bilinear form (18) satisfies the discrete stability condition

\frac{1}{16} ϱ [∥ u_{h} ∥_{X_{0, h}}^{2} + ∥ p_{h} ∥_{Y}^{2}]^{1 / 2} \leq sup 0 \neq (v_{h}, q_{h}) \in Y_{h} \times X_{0, h} \frac{B (u_{h}, p_{h}; v_{h}, q_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}}

(23)

for all $(u_{h}, p_{h}) \in X_{0, h} \times Y_{h}$ , when assuming $X_{0, h} \subseteq Y_{h}$ and $ϱ \leq 1$ .

Proof.

Since the proof follows the lines of the proof of Theorem 1, we only sketch the most important steps.

For $u_{h} \in X_{0, h}$ , let $w_{u_{h}, h} \in Y_{h}$ be the unique finite element solution of the variational problem (21). For $p_{h} \in Y_{h}$ , and due to $X_{0, h} \subset Y_{h}$ , we then have $v_{h} := u_{h} + w_{u_{h}, h} + α p_{h} \in Y_{h}$ , $α \in R_{+}$ . Moreover, set $q_{h} = α u_{h} \in X_{0, h}$ .

As in the proof of Theorem 1, we now conclude

B (u_{h}, p_{h}; v_{h}, q_{h}) \geq \frac{1}{4} [∥ p_{h} ∥_{Y}^{2} + ∥ w_{u_{h}, h} ∥_{Y}^{2} + ∥ u_{h} ∥_{Y}^{2}] = \frac{1}{4} [∥ u_{h} ∥_{X_{0, h}}^{2} + ∥ p_{h} ∥_{Y}^{2}] .

On the other hand, and as in the proof of Theorem 1, we have

	$∥ q_{h} ∥_{X_{0, h}}^{2} + ∥ v_{h} ∥_{Y}^{2}$	$=$	$α^{2} ∥ u_{h} ∥_{X_{0, h}}^{2} + ∥ u_{h} + w_{u_{h}, h} + α p_{h} ∥_{Y}^{2}$
		$\leq$	$\frac{16}{ϱ^{2}} [∥ u_{h} ∥_{X_{0, h}}^{2} + ∥ p_{h} ∥_{Y}^{2}] .$

Now the assertion follows as in the continuous case. ∎

The discrete inf-sup condition (23) implies unique solvability of the space-time finite element scheme (22). By combining (22) with (17) and by using the inclusions $X_{0, h} \subset X_{0}$ and $Y_{h} \subset Y$ , we also conclude the Galerkin orthogonality

B (u - u_{h}, p - p_{h}; v_{h}, q_{h}) = 0, \forall (v_{h}, q_{h}) \in Y_{h} \times X_{0, h} .

(24)

Theorem 2.

Let $(u, p) \in X_{0} \times Y$ and $(u_{h}, p_{h}) \in X_{0, h} \times Y_{h}$ be the unique solutions of the variational problems (17) and (22), respectively, where $X_{0, h} \subset X_{0}$ and $Y_{h} \subset Y$ . Furthermore, assume that $(u, p) \in H^{2} (Q) \times H^{2} (Q)$ . Then there holds the discretization error estimate

ϱ [∥ u - u_{h} ∥_{X_{0, h}}^{2} + ∥ p - p_{h} ∥_{Y}^{2}]^{1 / 2} \leq c h [\frac{1}{ϱ} ∥ p ∥_{H^{2} (Q)} + ∥ u ∥_{H^{2} (Q)}] .

Proof.

For arbitrary $(z_{h}, r_{h}) \in X_{0, h} \times Y_{h}$ , the discrete inf-sup condition (23) and the Galerkin orthogonality (24) immediately yield the estimates

	$\frac{1}{16} ϱ [∥ u_{h} - z_{h} ∥_{X_{0, h}}^{2} + ∥ p_{h} - r_{h} ∥_{Y}^{2}]^{1 / 2}$
			$\leq sup 0 \neq (v_{h}, q_{h}) \in Y_{h} \times X_{0, h} \frac{B (u_{h} - z_{h}, p_{h} - r_{h}; v_{h}, q_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}}$
			$= sup 0 \neq (v_{h}, q_{h}) \in Y_{h} \times X_{0, h} \frac{B (u - z_{h}, p - r_{h}; v_{h}, q_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}} .$

Now we consider

	$\frac{B (u - z_{h}, p - r_{h}; v_{h}, q_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}}$
			$= \frac{\frac{1}{ϱ} a (p - r_{h}, v_{h}) + b (u - z_{h}, v_{h}) - b (q_{h}, p - r_{h}) + c (u - z_{h}, q_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}}$
			$\leq \frac{\frac{1}{ϱ} a (p - r_{h}, v_{h}) + b (u - z_{h}, v_{h})}{∥ v_{h} ∥_{Y}} - \frac{b (q_{h}, p - r_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}} + \frac{c (u - z_{h}, q_{h})}{∥ q_{h} ∥_{X_{0, h}}}$
			$\leq \frac{1}{ϱ} ∥ p - r_{h} ∥_{Y} + \sqrt{2} ∥ u - z_{h} ∥_{X_{0}} + c ∥ u - z_{h} ∥_{L^{2} (Q)} - \frac{b (q_{h}, p - r_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}} .$

Integrating by parts in time and using $q_{h} = 0$ in $Σ_{0}$ , we obtain

$b (q_{h}, p - r_{h})$	$=$	$\int_{Q} [\partial_{t} q_{h} (p - r_{h}) + \nabla_{x} q_{h} \cdot \nabla_{x} (p - r_{h})] d x d t$
		$= \int_{Ω} q_{h} (T) (p (T) - r_{h} (T)) d x + \int_{Q} [- q_{h} \partial_{t} (p - r_{h}) + \nabla_{x} q_{h} \cdot \nabla_{x} (p - r_{h})] d x d t$
		$\leq ∥ q_{h} (T) ∥_{L^{2} (Ω)} ∥ p (T) - r_{h} (T) ∥_{L^{2} (Ω)} + \sqrt{2} ∥ q_{h} ∥_{Y} [∥ \partial_{t} (p - r_{h}) ∥_{Y^{*}} + ∥ p - r_{h} ∥_{Y}] .$

Therefore, the inequalities

	$\frac{b (q_{h}, p - r_{h})}{[∥ v_{h} ∥_{Y}^{2} + ∥ q_{h} ∥_{X_{0, h}}^{2}]^{1 / 2}}$	$\leq$	$\frac{b (q_{h}, p - r_{h})}{∥ q_{h} ∥_{Y}}$
			$\leq \frac{∥ q_{h} (T) ∥_{L^{2} (Ω)}}{∥ q_{h} ∥_{Y}} ∥ p (T) - r_{h} (T) ∥_{L^{2} (Ω)} + \sqrt{2} [∥ \partial_{t} (p - r_{h}) ∥_{Y^{*}} + ∥ p - r_{h} ∥_{Y}]$

follow. Let $τ_{ℓ}^{μ} \subset R^{μ}$ with $μ = d$ or $μ = d + 1$ be a shape regular simplicial finite element with mesh size $h_{ℓ}$ . For a piecewise linear finite element function, we then have the equivalence

\int_{τ_{ℓ}^{μ}} [v_{h} (x)]^{2} d x ≃ h_{ℓ}^{μ} μ + 1 \sum k = 1 v_{ℓ_{k}}^{2},

where the $v_{ℓ_{k}}$ are the local nodal values of $v_{h}$ . Hence, we can write

∥ q_{h} (T) ∥_{L^{2} (Ω)}^{2} = \sum τ_{ℓ}^{n} \in Σ_{T} ∥ q_{h} (T) ∥_{L^{2} (τ_{ℓ}^{n})}^{2} ≃ \sum τ_{ℓ}^{n} \in Σ_{T} h_{ℓ}^{n} n + 1 \sum k = 1 v_{ℓ_{k}}^{2}

as well as

∥ q_{h} ∥_{L^{2} (Q)}^{2} = \sum τ_{ℓ}^{n + 1} \in Q ∥ v_{h} ∥_{L^{2} (τ_{ℓ}^{n + 1})}^{2} ≃ \sum τ_{ℓ}^{n + 1} \in Q h_{ℓ}^{n + 1} n + 2 \sum k = 1 v_{ℓ_{k}}^{2} .

Thus, we conclude

∥ q_{h} (T) ∥_{L^{2} (Ω)}^{2} \leq c h_{T}^{- 1} ∥ q_{h} ∥_{L^{2} (Q)}^{2},

where $h_{T} ≃ h_{ℓ}$ is the globally quasi-uniform mesh size of all space-time finite elements sharing $Σ_{T}$ . Since we have $q_{h} = 0$ on $Σ$ , we finally obtain

∥ q_{h} (T) ∥_{L^{2} (Ω)}

Space-time finite element discretization of parabolic optimal control problems with energy regularization

Authors

Simultaneous space-time finite element methods for parabolic optimal control problems

Robust discretization and solvers for elliptic optimal control problems with energy regularization

Robust space-time finite element error estimates for parabolic distributed optimal control problems with energy regularization

Space-time formulation and time discretization of phase-field fracture optimal control problems

Adaptive finite element approximations for elliptic problems using regularized forcing data

Symplectic Runge-Kutta discretization of a regularized forward-backward sweep iteration for optimal control problems

Convergence analysis and a posteriori error estimates of reduced order solutions for optimal control problem of parameterized Maxwell system

1 Introduction

2 Preliminaries

3 The first-order optimality system

Theorem 1.

Proof.

Lemma 1.

Proof.

4 Discretization

Lemma 2.

Proof.

Theorem 2.

Proof.

Space-time finite element discretization of parabolic optimal control problems with energy regularization

Authors

Related Research

Simultaneous space-time finite element methods for parabolic optimal control problems

Robust discretization and solvers for elliptic optimal control problems with energy regularization

Robust space-time finite element error estimates for parabolic distributed optimal control problems with energy regularization

Space-time formulation and time discretization of phase-field fracture optimal control problems

Adaptive finite element approximations for elliptic problems using regularized forcing data

Symplectic Runge-Kutta discretization of a regularized forward-backward sweep iteration for optimal control problems

Convergence analysis and a posteriori error estimates of reduced order solutions for optimal control problem of parameterized Maxwell system

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1 Introduction

2 Preliminaries

3 The first-order optimality system

Theorem 1.

Proof.

Lemma 1.

Proof.

4 Discretization

Lemma 2.

Proof.

Theorem 2.

Proof.