Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

1.   Introduction

There are numerous research on finite element methods (FEMs) solving elliptic optimal control problems (OCPs) with control constraint. A systematic introduction can be seen in ^{[1,2,3,4,5,6]}. Due to the low regularity of the control variable, it is usually approximated by piecewise constant functions. Then the optimal priori error estimate is $\mathcal{O}(h)$ ^[7,8,9]. In order to improve the efficiency and accuracy of FEMs for solving such problems, many experts have considered its superconvergence ^[10,11,12], a posteriori error estimation ^[13,14,15], adaptive algorithm ^[16,17] and variational discretization technique ^[18,19,20].

In the past two decades, many scholars have proposed different numerical methods for elliptic or parabolic OCPs, such as FEMs ^[21], space-time FEMs ^[22,23], characteristic FEMs ^[24,25], mixed finite element methods (MFEMs) ^[26,27], splitting positive definite mixed FEMs ^[28,29], finite volume methods ^[30,31], spectral method ^[32,33,34], virtual element methods (VEMs) ^[35,36,37]. Unfortunately, almost of these numerical methods for parabolic OCPs use the backward Euler scheme (BES) for time discretization, and the error estimates of time variable is $\mathcal{O}(k)$. In recent years, there are also a few works using the Crank-Nicolson scheme (CNS) for time discretization ^{[38,39,40,41]}, which can improve the error convergence order of the time variable to $\mathcal{O}(k^2)$.

To the best of our knowledge, all reported fully discrete MFEMs for parabolic OCPs use BES for time discretization and optimal error estimates of time variable is $\mathcal{O}(k)$. The purpose of this paper is to develop a MFEM combined with CNS approximation of parabolic OCPs and establish optimal a priori error estimates $\mathcal{O}(h^2+k^2)$.

We are concerned with the following parabolic OCPs: Find $(y, \boldsymbol{p}, u)$ such that

where $\Omega\subset{\bf{R}^2}$ is a rectangle, $J = (0, T]$. Let $U = L^2(J; L^2(\Omega))$, $f, y_{d}\in U$, $\boldsymbol{p}_{d}\in U^{2}$ and $y_{0}\in H^{1}(\Omega)$. $K$ is a closed convex subset of $U$ defined by

Throughout the paper, we adopt the standard notation $W^{m, p}(\Omega)$ for Sobolev spaces on $\Omega$ with a norm $\|\cdot\|_{m, p}$ given by $\|v\|_{m, p}^{p} = \sum\limits_{|\alpha|\leq m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p}$, a semi-norm $|\cdot|_{m, p}$ given by $| v|_{m, p}^{p} = \sum\limits_{|\alpha| = m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p}$. For $p = 2$, we denote $H^m(\Omega) = W^{m, 2}(\Omega), H_0^m(\Omega) = \{v\in W^{m, p}(\Omega): v|_{\partial \Omega} = 0\}$, and $\|\cdot\|_{m} = \|\cdot\|_{m, 2}, \, \|\cdot\| = \|\cdot\|_{0, 2}$. We denote by $L^s(J; W^{m, p}(\Omega))$ all $L^s$ integrable functions from $J$ into $W^{m, p}(\Omega)$ with norm $\|v\|_{L^s(J; W^{m, p}(\Omega))} = \Big(\int_0^T||v||_{W^{m, p}(\Omega)}^sdt\Big)^{1/s}\, \rm{for}\, s\in [1, \infty),$ and the standard modification for $s = \infty$. For ease of presentation, we denote $\|v\|_{L^s(J; W^{m, p}(\Omega))}$ by $\| v\|_{L^s(W^{m, p})}$. Similarly, one can define the spaces $H^l(W^{m, p})$. In addition $C$ denotes a general positive constant.

The layout of this paper is as follows. In Section 2, we construct a MFEM combined with CNS approximation of the parabolic OCPs (1.1)–(1.5). In Section 3, we introduce some useful intermediate variables and important error estimates. In Section 4, we derive a priori error estimates for the control, state and co-state. In Section 5, we provide some numerical examples to illustrate our theoretical results.

2.   MFEM combined with CNS approximation of parabolic OCPs

In this section, we shall consider a MFEM combined with CNS approximation of parabolic OCPs (1.1)–(1.5). For simplicity, we shall take the following state spaces $\boldsymbol{L} = H^{1}(J; \boldsymbol{V})$ and $Q = H^{1}(J; W)$, where $\boldsymbol{V}$ and $W$ are defined as follows:

Furthermore, we define the space

Then OCPs (1.1)–(1.5) can be recast as the following weak form: Find $(y, \boldsymbol{p}, u)\in Q\times\boldsymbol{L}\times K$ such that

It follows from ^[3] that OCPs (2.1)–(2.4) has a unique solution $(y, \boldsymbol{p}, u)$, and that a triplet $(y, \boldsymbol{p}, u)\in Q\times\boldsymbol{L}\times K$ is the solution of (2.1)–(2.4) if and only if there is a co-state $(z, \boldsymbol{q})\in Q\times \boldsymbol{L}$ such that $(y, \boldsymbol{p}, z, \boldsymbol{q}, u)$ satisfies the following optimality conditions:

We introduce a pointwise projection $P_{[a, b]}$, which satisfies: For any $\varphi\in W$,

Then the variational inequality (2.11) can be equivalently expressed as

We use the Raviart-Thomas mixed finite element of the order $m = 1$ for space discretization. Let $\mathcal{T}_{h}$ be a regular triangulations of the domain $\Omega$, $h_{e}$ denotes the diameter of $e$ and $h = \max\limits_{e\in \mathcal{T}_{h}} \{h_{e}\}$. Let $P_m(e)$ indicates the space of polynomials of total degree no more than $m$ on $e$ and $\boldsymbol{V}_h\times W_h\subset \boldsymbol{V}\times W$ denote Raviart-Thomas mixed finite element spaces ^[1,2] associated with the triangulations $\mathcal{T}_{h}$ of $\Omega$, namely,

We shall use the CNS for time discretization. Let $N$ be a positive integer, $k = T/N$ and $t_n = nk, n = 0, 1, \cdots, N$. Set $I_n = [t_n, t_{n+1}], n = 0, 1, \cdots, N-1$. For any function $\varphi$, we define $\varphi^n = \varphi(x, t_n)$,

and discrete time-dependent norms

Then a MFEM combined with CNS approximation of (2.1)–(2.4) is as follows: Find $(y_h, \boldsymbol{p}_h, u_h)\in W_h\times \boldsymbol{V}_h\times K^\prime$ such that

where $R_h$ is a $L^2$ projection operator, which will be specific later.

Like in ^[6], the OCPs (2.13)–(2.16) has a unique solution $\left(y_h^n, \boldsymbol{p}_h^n, u_h^n\right), n = 0, 1, \cdots, N$ and the triplet $\left(y_h^n, \boldsymbol{p}_h^n, u_h^n\right)\in W_h\times\boldsymbol{V}_h\times K^\prime, n = 0, 1, \cdots, N$ is the solution of (2.13)–(2.16) if and only if there is a co-state $\left(z_h^n, \boldsymbol{q}_h^n\right)\in W_h\times\boldsymbol{V}_h, (n = N, \cdots, 1, 0)$ such that $\left(y_h^n, \boldsymbol{p}_h^n, z_h, \boldsymbol{q}_h^n, u_h^n\right), (n = 0, 1, \cdots, N)$ satisfies the following optimality conditions:

Here, we use the variational discretization technique for the variational inequality. Similarly to (2.12), the variational inequality (2.23) can be equivalently rewritten as

This means that, we can obtain $u_h^{n+\frac{1}{2}}$ from $z_h^{n+\frac{1}{2}}$ by using the relation (2.24).

The following projection operators are commonly used in the following error estimates of MFEMs approximation of OCPs. First, we define the standard $L^{2}(\Omega)$-projection ^[2] $R_h:\ W\rightarrow W_h$, which satisfies: For any $\phi \in W$,

Second, we define the Fortin projection ^[2] $\Pi_h:\ \boldsymbol{V} \rightarrow \boldsymbol{V}_h$, which satisfies: For any $\boldsymbol{q}\in \boldsymbol{V}$,

3.   Error estimates of intermediate variables

In this section, we will introduce some important intermediate variables and error estimates. For any $\tilde{u}\in K$, we define variables $(y_h^n(\tilde{u}), \boldsymbol{p}_h^n(\tilde{u}), z_h^{n}(\tilde{u}), \boldsymbol{q}_h^{n}(\tilde{u})), n = 0, 1, \cdots, N$, associated with $\tilde{u}$, which satisfies

According to standard Raviart-Thomas mixed finite element approximation error analysis like in ^[20,26], we can derive the following error estimates.

Lemma 3.1. Let $(\boldsymbol{p}_{h}, y_{h}, \boldsymbol{q}_{h}, z_{h}, u_h)$ and $(\boldsymbol{p}_{h}(u), y_{h}(u), \boldsymbol{q}_{h}(u), z_{h}(u))$ be the discrete solutions of (2.17)–(2.23) and (3.1)–(3.6) with $\tilde{u} = u$, respectively. Then we have

Proof. Let $\alpha = y_{h}-y_{h}(u)$ and $\boldsymbol{\beta} = \boldsymbol{p}_{h}-\boldsymbol{p}_{h}(u)$. From (2.17), (2.18), (3.1) and (3.2) with $\tilde{u} = u$, we have the following error equations

Selecting $w_h = \alpha^{n+\frac{1}{2}}$ and $\boldsymbol{v}_h = \boldsymbol{\beta}^{n+\frac{1}{2}}$ in (3.9) and (3.10), respectively. Then add those equations, we get

Note that $\left(d_t \alpha^{n}, \alpha^{n+\frac{1}{2}}\right) = \frac{\|\alpha^{n+1} \ \ \ \|^2-\|\alpha^n \ \|^2}{2k}$. From (3.11) and Young inequality, we obtain

Multiplying both sides of (3.12) by $2k$ and summing $n$ from $0$ to $M\; (0\leq M\leq N-1)$, we have

According to $\alpha^0 = 0$, $|||\alpha|||_{l^2(L^2)}\leq C|||\alpha|||_{l^{\infty}(L^2)}$ and (3.13), we get (3.7).

Set $\eta = z_h-z_h(u)$ and $\boldsymbol{\theta} = \boldsymbol{q}_h-\boldsymbol{q}_h(u)$. Subtract (3.4) and (3.5) from (2.20) and (2.21) to get the following error equations

Choosing $w_h = \eta^{n+\frac{1}{2}}$ and $\boldsymbol{v}_h = \boldsymbol{\theta}^{n+\frac{1}{2}}$ in (3.14) and (3.15), respectively. We derive

Note that $\eta^N = 0$ and $|||\eta|||_{l^2(L^2)}\leq C|||\eta|||_{l^{\infty}(L^2)}$, similarly to (3.11)–(3.13), we can arrive at

From (3.7) and (3.17), it is easy to get (3.8).

For convenience, we use the following notations

Lemma 3.2. Let $(\boldsymbol{p}, y, \boldsymbol{q}, z, u)$ and $(\boldsymbol{p}_h(u), y_h(u), \boldsymbol{q}_h(u), z_h(u))$ be the solutions of (2.5)–(2.11) and (3.1)–(3.6) with $\tilde{u} = u$ respectively. Assume that $y, z\in L^{2}(H^{2})$, $\boldsymbol{p}, \boldsymbol{q}\in L^{2}((H^{2})^2)$ and $y_{ttt}, z_{ttt} \in L^{2}(L^2)$, then we have

Proof. Set $t = \frac{t_{n+1}+t_{n}}{2}$ in (2.5) and (2.6) then subtract (3.1) and (3.2), we have

Taking $w_h = \upsilon_y^{n+\frac{1}{2}}$ and $\boldsymbol{v}_h = \boldsymbol{\vartheta}_{\boldsymbol{p}}^{n+\frac{1}{2}}$ in (3.20) and (3.21), respectively. According to the definition of $R_h$ and $\Pi_h$, we derive

From (3.22)–(3.23), we get

By using (2.28), Cauchy-Schwarz inequality, Young inequality and interpolation theory, we have

and

Combining (3.24)–(3.26), we obtain

Multiplying both sides of (3.27) by $2k$ and summing $n$ from $0$ to $M\; (0\leq M\leq N-1)$, we have

Noting that $\rho_y^0 = 0$. From (3.28), we arrive at

Then (3.18) follows from (2.26), (2.28), (3.29) and triangle inequality.

Analogously, we can define $\rho_z, \zeta_z, \upsilon_z$ and $\boldsymbol{\varrho}_{\boldsymbol{q}}, \boldsymbol{\xi}_{\boldsymbol{q}}, \boldsymbol{\vartheta}_{\boldsymbol{q}}$. Let $t = \frac{t_{n+1}+t_n}{2}$ in (2.8) and (2.9) then subtract (3.4) and (3.5), we get

Choosing $w_h = \upsilon_z^{n+\frac{1}{2}}$ and $\boldsymbol{v}_h = \boldsymbol{\vartheta}_{\boldsymbol{q}}^{n+\frac{1}{2}}$ in (3.30) and (3.31), respectively. From the definition of $R_h$ and $\Pi_h$, we arrive at

Similarly to (3.22)–(3.28), we can derive

Since $\rho_z^N = 0$, (3.19) follows from (2.26), (2.28), (3.18), (3.34) and triangle inequality.

4.   A priori error estimates

In this section, we shall derive a priori error estimates of the MFEM combined with CNS approximation scheme (2.17)–(2.23).

Theorem 4.1. Let $(\boldsymbol{p}, y, \boldsymbol{q}, z, u)$ be the solution of (2.5)–(2.11) and $(\boldsymbol{p}_h, y_h, \boldsymbol{q}_h, z_h, u_{h})$ be the solution of (2.17)–(2.23). Assume that all the conditions in Lemmas 3.1 and 3.2 are valid. Then, we have

Proof. From (2.11) and (2.23), we have

and

It follows from (4.2) and (4.3) that

It follows from (2.17)–(2.22) and (3.1)–(3.6) that

By using Hölder's inequality, Young's inequality and Lemma 3.2, we have

Combining (4.4)–(4.6), we obtain (4.1).

Theorem 4.2. Let $(\boldsymbol{p}, y, \boldsymbol{q}, z, u)$ and $(\boldsymbol{p}_{h}, y_{h}, \boldsymbol{q}_{h}, z_{h}, u_{h})$ be the solution of (2.5)–(2.11) and the solution of (2.17)–(2.23), respectively. Assume that all the conditions in Theorem 4.1 are valid. Then we have

Proof. By using the triangle inequality, Lemmas 3.1 and 3.2, Theorem 4.1, it is easy to get (4.7) and (4.8).

5.   Numerical experiments

In this section, we present two numerical examples to validate our theoretical results. The following parabolic OCPs were dealt numerically with codes developed based on AFEPack, which is a freely available software package and the details can be found in ^[42]. Their discretization schemes are described as (2.17)–(2.23) in Section 2. Let $\Omega = (0, 1)\times(0, 1)$ and $T = 1$.

Example 1. The data under testing are as follows:

In Table 1, we list errors of $|||u-u_h|||_{l^2(L^2)}$, $|||y-y_h|||_{l^{\infty}(L^2)}$, $|||\boldsymbol{p}-\boldsymbol{p}_h|||_{l^{2}(L^2)}$, $|||z-z_h|||_{l^{\infty}(L^2)}$ and $|||\boldsymbol{q}-\boldsymbol{q}_h|||_{l^{2}(L^2)}$ based on a sequence of uniformly refined meshes. In Figure 1, we show the relationship between $\log_{10}(error)$ and $\log_{10}(node)$. It is easy to see that $|||u-u_h|||_{l^2(L^2)} = \mathcal{O}(h^2+k^2)$, $|||y-y_h|||_{l^{\infty}(L^2)}+|||\boldsymbol{p}-\boldsymbol{p}_h|||_{l^{2}(L^2)} = \mathcal{O}(h^2+k^2)$ and $|||z-z_h|||_{l^{\infty}(L^2)}+|||\boldsymbol{q}-\boldsymbol{q}_h|||_{l^{2}(L^2)} = \mathcal{O}(h^2+k^2)$.

Example 2. The data under testing are as follows:

The numerical results based on a sequence of uniformly refined meshes are reported in Table 2. We show the relationship between $\log_{10}(error)$ and $\log_{10}(node)$ in Figure 2. It is easy to see that errors of $|||u-u_h|||_{l^2(L^2)}$, $|||y-y_h|||_{l^{\infty}(L^2)}$, $|||\boldsymbol{p}-\boldsymbol{p}_h|||_{l^{2}(L^2)}$, $|||z-z_h|||_{l^{\infty}(L^2)}$ and $|||\boldsymbol{q}-\boldsymbol{q}_h|||_{l^{2}(L^2)}$ estimates are $\mathcal{O}(h^2+k^2)$. It is consistent with the theoretical results in Section 4.

6.   Conclusions

In this paper, we investigate a MFEM combined with CNS approximation of constrained parabolic OCPs and obtain optimal priori error estimates, namely $|||u-u_h|||_{l^2(L^2)} = \mathcal{O}(h^2+k^2)$, $|||y-y_h|||_{l^{\infty}(L^2)}+|||\boldsymbol{p}-\boldsymbol{p}_h|||_{l^{2}(L^2)} = \mathcal{O}(h^2+k^2)$ and $|||z-z_h|||_{l^{\infty}(L^2)}+|||\boldsymbol{q}-\boldsymbol{q}_h|||_{l^{2}(L^2)} = \mathcal{O}(h^2+k^2)$. Our results are new.

Acknowledgments

This work is supported by the Natural Science Foundation of Hunan Province (2020JJ4323), the Scientific Research Foundation of Hunan Provincial Education Department (20A211), the construct program of applied characteristic discipline in Hunan University of Science and Engineering.

Conflict of interest

The author declare no conflict of interest in this paper.