Virtual Element Discretization of Optimal Control Problem Governed by Brinkman Equations

Yanwei Li

School of Mathematics and Statistics, Shandong Normal University, Jinan, China.

DOI: 10.4236/eng.2023.152010                

Abstract

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 L² and H¹ norm are derived. The theoretical findings are illustrated by the numerical experiments.

Keywords

Virtual Element Method, Optimal Control Problem, Brinkman Equations, A Priori Error Estimate

1. Introduction

In this paper we consider virtual element discretization of the following optimal control problem: find $\left(y\mathrm{,}p\mathrm{,}u\right)\in V\times Q\times U_{ad}$ satisfying

$\underset{u\in U_{ad}}{\text{min}}J\left(y,u\right)=\frac{1}{2}{\Vert y-y_d\Vert }_{L^2\left(\Omega \right)}^2+\frac{\gamma }{2}{\Vert u\Vert }_{L^2\left(\Omega \right)}^2$ (1.1)

subject to

$(\begin{array}{ll}-\Delta y+\nabla p+{\mathbb{K}}^{-1}y=f+u\hfill & \text{in}\Omega \mathrm{,}\hfill \\ \nabla \cdot y=0\hfill & \text{in}\Omega \mathrm{,}\hfill \\ y=0\hfill & \text{on}\Gamma \mathrm{,}\hfill \end{array}$ (1.2)

where $J\left(y,u\right)$ is the objective functional, $y_d$ is the desired state, $\gamma >0$ is the regularization parameter, and $\Omega$ is a bounded domain in ${ℝ}^2$ with the boundary $\Gamma$ . We suppose that the $\mathbb{K}$ is a uniformly symmetric positive definite tensor, i.e. there exist two positive constants ${\lambda }_1,{\lambda }_2>0$ such that

${\lambda }_1{\eta }^{\text{T}}\eta \le {\eta }^{\text{T}}{\mathbb{K}}^{-1}\eta \le {\lambda }_2{\eta }^{\text{T}}\eta \mathrm{.}$

The admissible control set $U_{ad}$ is defined by

$U_{ad}=\left\{u\in L^2\left(\Omega \right)\mathrm{<mo>\; :\; </mo>}{u}_a\le u\left(x\right)\le u_b\text{a}\mathrm{.}\text{e}\text{.}\text{in}\Omega \right\}\mathrm{.}$

The quantities $u_a\mathrm{,}{u}_b\in {ℝ}^2$ are constant vectors and the inequality $u_a\le u\left(x\right)\le u_b$ 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 [1] . In the past decades, developing numerical methods for optimal control model governed by Brinkman equations has become a hot topic. For example, a discontinuous finite volume method for the approximation of distributed optimal control problems governed by the Brinkman equations was derived in [2] . In [3] the author investigated adaptive hybridizable discontinuous Galerkin methods for the gradient-velocity-pressure formulation of Brinkman equations and extended to solve the Brinkman optimal control problem. In [4] the author studied an optimal control problem constrained by the unsteady Stokes-Brinkman equation involving random data. For more models, we can refer to [5] [6] .

The virtual element method (VEM), first introduced in [7] , is regarded as an extension of finite element method. Unlike finite element method, the VEM has the advantages including: it can deal with highly general polygonal/polyhedral meshes; the basis function needn’t to be explicit expression, etc. VEM has been widely applied to approximate various PDEs [8] [9] [10] [11] [12] . There are many crucial literatures about the VEM framework for Brinkman problems. A mixed virtual element method for the Brinkamn equations was discussed in [13] . In [14] , the divergence free virtual element space in [11] was extended to solve the Brinkman equations. In [15] , the authors presented two stable virtual element methods for the Brinkman equations.

For the literature on the application of virtual element method to optimal control problem, we can refer to [16] and [17] . The authors study the virtual element discrete scheme of the elliptic optimal control problem and give a priori and a posteriori error analysis. There is still a gap in combination of the virtual element method and optimal control problem governed by Brinkman equations. Thus, in this paper, we aim to apply the VEM to approximate optimal control problem governed by Brinkman equations with pointwise control constraint. By making use of the virtual element projection operators the virtual element discrete scheme of the optimal control problem is developed, where the piecewise $L^2$ projection of the discrete state is used in the cost functional to guarantee the computability of the discrete adjoint state equation. Then, we derive a priori error estimates for state, adjoint state and control variables in $L^2$ and $H^1$ norm. Finally numerical experiments on three polygonal meshes are given to verify the theoretical findings.

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 $H^s\left(K\right)$ and $H^s\left(K\right)$ equipped with seminorm ${\left|\cdot \right|}_{s\mathrm{,}K}$ and norm ${\Vert \cdot \Vert }_{s\mathrm{,}K}$ , while ${\left(\cdot \mathrm{,}\cdot \right)}_{\mathrm{0,}K}$ will denote the $L^2\left(K\right)$ or $L^2\left(K\right)$ inner product for scale and vector.

2. Preliminaries

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:

$V\mathrm{<mo>\; :\; </mo>}=H_0^1\left(\Omega \right)\mathrm{,}Q\mathrm{<mo>\; :\; </mo>}=L_0^2\left(\Omega \right)=\left\{q\in L^2\left(\Omega \right)\text{s}\text{.t}\text{.}{\displaystyle {\int }_{\Omega }}q\text{d}\Omega =0\right\}\mathrm{.}$

We endow the space $V$ with the norm

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}v{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\mathrm{<mo>\; :\; </mo>}={\Vert \nabla v\Vert }_0^2+{\Vert {\mathbb{K}}^{-1/2}v\Vert }_0^2$

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 $\left(y\mathrm{,}p\mathrm{,}u\right)\in V\times Q\times U_{ad}$ satisfying

$\underset{u\in U_{ad}}{\text{min}}J\left(y\mathrm{,}u\right)$

s.t.

$(\begin{array}{l}A\left(y\mathrm{,}v\right)-b\left(v\mathrm{,}p\right)=\left(f+u\mathrm{,}v\right)\forall v\in V\mathrm{,}\\ b\left(y\mathrm{,}q\right)=0\forall q\in Q\mathrm{,}\end{array}$

where

$A\left(y\mathrm{,}v\right)=a\left(y\mathrm{,}v\right)+d\left(y\mathrm{,}v\right)\mathrm{,}a\left(y\mathrm{,}v\right)={\displaystyle {\int }_{\Omega }}\nabla y\mathrm{<mo>\; :\; </mo>}\nabla v\text{d}\Omega \mathrm{,}$

$d\left(y\mathrm{,}v\right)={\displaystyle {\int }_{\Omega }}{\mathbb{K}}^{-1}y\cdot v\text{d}\Omega \mathrm{,}b\left(v\mathrm{,}p\right)={\displaystyle {\int }_{\Omega }}p\text{div}v\text{d}\Omega \mathrm{.}$

Additionally, we introduce the kernel:

$Z\mathrm{<mo>\; :\; </mo>}=\left\{v\in V\text{s}\text{.t}\text{.}b\left(v\mathrm{,}q\right)=0\forall q\in Q\right\}\mathrm{.}$

Following [14] , we can obtain that:

· $A\left(\cdot \mathrm{,}\cdot \right)$ and $b\left(\cdot \mathrm{,}\cdot \right)$ are continuous, i.e.

$\left|A\left(u\mathrm{,}v\right)\right|\le C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}u{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}v{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\forall u\mathrm{,}v\in V\mathrm{,}$

$\left|b\left(v\mathrm{,}q\right)\right|\le C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}v{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V{\Vert q\Vert }_Q\forall v\in V\mathrm{,}q\in Q\mathrm{.}$

· $A\left(\cdot \mathrm{,}\cdot \right)$ is coercive on the kernel $Z$ , i.e.

$A\left(v\mathrm{,}v\right)\ge \mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}v{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V^2\forall v\in Z\mathrm{.}$

· $b\left(\cdot \mathrm{,}\cdot \right)$ satisfies the inf-sup condition, i.e.

$\exists \beta >\mathrm{0,}\text{such}\text{that}\underset{v\in V\mathrm{,}v\ne 0}{\max }\frac{b\left(v\mathrm{,}p\right)}{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}v{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V}\ge \beta {\Vert p\Vert }_Q\forall p\in Q\mathrm{.}$

We introduce the following Lagrangian functional:

$\mathcal{L}\left(y\mathrm{,}p\mathrm{,}u\mathrm{,}z\mathrm{,}\varphi \right)\mathrm{<mo>\; :\; </mo>}=J\left(y\mathrm{,}u\right)-A\left(y\mathrm{,}z\right)+b\left(z\mathrm{,}p\right)-b\left(y\mathrm{,}\varphi \right)+\left(f+u\mathrm{,}z\right)\mathrm{.}$

Then the following continue first order optimality condition can be obtained by computing the derivatives of $\mathcal{L}\left(\cdot \mathrm{,}\cdot \mathrm{,}\cdot \mathrm{,}\cdot \mathrm{,}\cdot \right)$ with respect to $\left(y\mathrm{,}p\mathrm{,}u\mathrm{,}z\mathrm{,}\varphi \right)$ :

$(\begin{array}{ll}A\left(y\mathrm{,}w\right)-b\left(w\mathrm{,}p\right)=\left(f+u\mathrm{,}w\right)\hfill & \forall w\in V\mathrm{,}\hfill \\ b\left(y\mathrm{,}\psi \right)=0\hfill & \forall \psi \in Q\mathrm{,}\hfill \end{array}$ (2.1)

$(\begin{array}{ll}A\left(w\mathrm{,}z\right)+b\left(w\mathrm{,}\varphi \right)=\left(y-y_d\mathrm{,}w\right)\hfill & \forall w\in V\mathrm{,}\hfill \\ b\left(z\mathrm{,}\psi \right)=0\hfill & \forall \psi \in Q\mathrm{,}\hfill \end{array}$ (2.2)

$\left(\gamma u+z\mathrm{,}v-u\right)\ge 0\forall v\in U_{ad}\mathrm{,}$ (2.3)

where $\left(z\mathrm{,}\varphi \right)$ is the adjoint state variable. Following [18] , the variational inequality (2.3) is equivalent to

$u=P_{U_{ad}}\left(-\frac{1}{\gamma }z\right)\mathrm{,}$

where

$P_{U_{ad}}\left(u\right)=\text{max}\left\{u_a\mathrm{,}\text{min}\left\{u\mathrm{,}{u}_b\right\}\right\}$

denotes the projection onto the admissible set $U_{ad}$ .

Let ${\mathcal{T}}_h$ be a sequence of decompositions of $\Omega$ into general polygonal elements K with

$h_K\mathrm{<mo>\; :\; </mo>}=\text{diameter}\left(K\right)\mathrm{,}h\mathrm{<mo>\; :\; </mo>}=\underset{K\in {\mathcal{T}}_h}{\max }{h}_K\mathrm{.}$

Assumption 2.1. We assume that there exists two positive constants c and $\rho$ such that, every $K\in {\mathcal{T}}_h$ satisfies the following assumptions:

(A1) Each element K is star-shaped with respect to a ball of radius $\ge c{h}_K$ ,

(A2) The distance $D_h$ between any two points of each element K satisfies $D_h\ge \rho h_K$ .

The bilinear forms $A\left(\cdot \mathrm{,}\cdot \right)$ and $b\left(\cdot \mathrm{,}\cdot \right)$ , the norms $\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}\cdot {\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V$ and ${\Vert \cdot \Vert }_Q$ , can be decomposed into local contributions, i.e.:

$A\left(y\mathrm{,}v\right)=\mathrm{<mo>\; :\; </mo>}\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{A}^K\left(y\mathrm{,}v\right)=\mathrm{<mo>\; :\; </mo>}\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\left(a^K\left(y\mathrm{,}v\right)+d^K\left(y\mathrm{,}v\right)\right)\forall y\mathrm{,}v\in V\mathrm{,}$

$b\left(v\mathrm{,}p\right)=\mathrm{<mo>\; :\; </mo>}\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{b}^K\left(v\mathrm{,}p\right)\forall v\in V\mathrm{,}p\in Q\mathrm{,}$

and

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}v{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V=\mathrm{<mo>\; :\; </mo>}{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}v{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}\right)}^{\frac{1}{2}}\forall v\in V\mathrm{,}{\Vert q\Vert }_Q=\mathrm{<mo>\; :\; </mo>}{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert q\Vert }_{Q\mathrm{,}K}^2\right)}^{\frac{1}{2}}\forall q\in Q\mathrm{.}$

Definition 2.1. For all $K\in {\mathcal{T}}_h$ , we define the energy projection operator ${\Pi }_2^{\nabla }$ : $H^1\left(K\right)\to {\left[{\mathbb{P}}_2\left(K\right)\right]}^2$ as follows:

$(\begin{array}{l}{a}^K\left(v_h-{\Pi }_2^{\nabla }{v}_h\mathrm{,}p\right)=\mathrm{0,}\forall v_h\in H^1\left(K\right)\mathrm{,}p\in {\left[{\mathbb{P}}_2\left(K\right)\right]}^2\mathrm{,}\\ {\displaystyle {\int }_K}{v}_h\text{d}K={\displaystyle {\int }_K}{\Pi }_2^{\nabla }{v}_h\text{d}K\mathrm{.}\end{array}$

It obviously holds ${\Pi }_2^{\nabla }p=p$ for all $p\in {\left[{\mathbb{P}}_2\left(K\right)\right]}^2$ .

Definition 2.2. For all $K\in {\mathcal{T}}_h$ , we define the $L^2$ projection operator ${\Pi }_k^0$ : $L^2\left(K\right)\to {\left[{\mathbb{P}}_k\left(K\right)\right]}^2$ as follows:

$\left(P_k^0{v}_h-v_h\mathrm{,}p\right)=\mathrm{0,}\forall v_h\in L^2\left(K\right)\mathrm{,}p\in {\left[{\mathbb{P}}_k\left(K\right)\right]}^2\mathrm{.}$

For $k\in \mathbb{N}$ , we define the following spaces:

· ${\mathbb{P}}_k\left(K\right)$ : the set of polynomials on K of degree $\le k$ , usually, ${\mathbb{P}}_{-1}=\left\{0\right\}$ ,

· ${\mathbb{B}}_k\left(K\right)\mathrm{<mo>\; :\; </mo>}=\left\{v\in C^0\left(\partial K\right)\text{s}\text{.t}{v}_{\mathrm{<mo>\; |\; </mo>}e}\in {\mathbb{P}}_k\left(e\right)\mathrm{,}\forall e\subset \partial K\right\}$ ,

· ${\mathcal{G}}_k\left(K\right)\mathrm{<mo>\; :\; </mo>}=\nabla \left({\mathbb{P}}_{k+1}\left(K\right)\right)\subseteq {\left[{\mathbb{P}}_k\left(K\right)\right]}^2$ ,

· ${\mathcal{G}}_k{\left(K\right)}^{\perp }\subseteq {\left[{\mathbb{P}}_k\left(K\right)\right]}^2$ is the $L^2$ -orthogonal complement to ${\mathcal{G}}_k\left(K\right)$ .

In [14] the following local virtual element space was introduced

$V_h^k\mathrm{<mo>\; :\; </mo>}=\left\{v\in U_h^k\text{s}\text{.t}\text{.}\left(v-{\Pi }_2^{\nabla }v\mathrm{,}{g}_2^{\perp }\right)=\mathrm{0,}\forall g_2^{\perp }\in {\mathcal{G}}_2{\left(K\right)}^{\perp }/{\mathcal{G}}_0{\left(K\right)}^{\perp }\right\}\mathrm{,}$

where

$\begin{array}{l}{U}_h^K\mathrm{<mo>\; :\; </mo>}=\{v\in H^1\left(K\right)\text{s}\text{.t}{v}_{\mathrm{<mo>\; |\; </mo>}\partial K}\in {\left[{\mathbb{B}}_2\left(\partial K\right)\right]}^2\mathrm{,}-\Delta v+\nabla s\in {\mathcal{G}}_2{\left(K\right)}^{\perp }\\ \text{and}\text{div}v\in {\mathbb{P}}_1\left(K\right)\mathrm{,}\text{for}\text{some}s\in L^2\left(K\right)\}\mathrm{,}\end{array}$

and ${\mathcal{G}}_2{\left(K\right)}^{\perp }/{\mathcal{G}}_0{\left(K\right)}^{\perp }$ denotes the polynomials in ${\mathcal{G}}_2{\left(K\right)}^{\perp }$ that are $L^2$ -orthogonal to all polynomials in ${\mathcal{G}}_0{\left(K\right)}^{\perp }$ .

For the pressure space we adopt the finite-dimensional space

$Q_h^K\mathrm{<mo>\; :\; </mo>}={\mathbb{P}}_1\left(K\right)\mathrm{.}$

Then we define the global virtual element spaces:

$V_h\mathrm{<mo>\; :\; </mo>}=\left\{v\in V\text{s}\text{.t}{v|}_K\in V_h^K\mathrm{,}\forall K\in {\mathcal{T}}_h\right\}$

and

$Q_h\mathrm{<mo>\; :\; </mo>}=\left\{q\in Q\text{s}\text{.t}{q|}_K\in Q_h^K\mathrm{,}\forall K\in {\mathcal{T}}_h\right\}\mathrm{.}$

We remark that the above spaces have the following relation

$\text{div}{V}_h\subseteq Q_h\mathrm{.}$

This implies an exactly divergence-free discrete velocity.

Lemma 2.3. (See [8] )There exists a positive constant C such that, for all $K\in {\mathcal{T}}_h$ and all smooth enough functions $\phi$ defined on K, it holds:

${\Vert \phi -P_k^0\phi \Vert }_{m\mathrm{,}K}\le C{h}_K^{s-m}{\left|\phi \right|}_{s\mathrm{,}K}\mathrm{,}m=\mathrm{0,1,}s\in \mathbb{N}\mathrm{,}m\le s\le k+1.$

3. Virtual Element Approximation

The virtual element discrete scheme of (1.2) can be defined as follows:

$(\begin{array}{ll}{A}_h\left(y_h\left(u\right)\mathrm{,}{v}_h\right)-b\left(v_h\mathrm{,}{p}_h\left(u\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(f+u\mathrm{,}{\Pi }_2^0{v}_h\right)}_{\mathrm{0,}K}\hfill & \forall v_h\in V_h\mathrm{,}\hfill \\ b\left(y_h\left(u\right)\mathrm{,}{q}_h\right)=0\hfill & \forall q_h\in Q_h\mathrm{,}\hfill \end{array}$

where

$\begin{array}{c}{A}_h\left(y_h\left(u\right)\mathrm{,}{v}_h\right)\mathrm{<mo>\; :\; </mo>}=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{A}_h^K\left(y_h\left(u\right)\mathrm{,}{v}_h\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\left(a_h^K\left(y_h\left(u\right)\mathrm{,}{v}_h\right)+d_h^K\left(y_h\left(u\right)\mathrm{,}{v}_h\right)\right)\mathrm{,}\end{array}$

$a_h^K\left(y_h\left(u\right)\mathrm{,}{v}_h\right)\mathrm{<mo>\; :\; </mo>}=a^K\left({\Pi }_2^{\nabla }{y}_h\left(u\right)\mathrm{,}{\Pi }_2^{\nabla }{v}_h\right)+S^K\left(y_h\left(u\right)-{\Pi }_2^{\nabla }{y}_h\left(u\right)\mathrm{,}{v}_h-{\Pi }_2^{\nabla }{v}_h\right)\mathrm{,}$

$d_h^K\left(y_h\left(u\right)\mathrm{,}{v}_h\right)\mathrm{<mo>\; :\; </mo>}=d^K\left({\Pi }_2^0{y}_h\left(u\right)\mathrm{,}{\Pi }_2^0{v}_h\right)+R^K\left(y_h\left(u\right)-{\Pi }_2^0{y}_h\left(u\right)\mathrm{,}{v}_h-{\Pi }_2^0{v}_h\right)\mathrm{.}$

Here, $R^K\left(\cdot \mathrm{,}\cdot \right)$ and $S^K\left(\cdot \mathrm{,}\cdot \right)$ are symmetric stabilizing bilinear forms satisfying

$c_1{a}^K\left(v_h\mathrm{,}{v}_h\right)\le S^K\left(v_h\mathrm{,}{v}_h\right)\le c_2{a}^K\left(v_h\mathrm{,}{v}_h\right)\mathrm{,}\forall v_h\in V_h\text{with}{\Pi }_2^{\nabla }{v}_h=\mathrm{0,}$

$c_3{d}^K\left(v_h\mathrm{,}{v}_h\right)\le R^K\left(v_h\mathrm{,}{v}_h\right)\le c_4{d}^K\left(v_h\mathrm{,}{v}_h\right)\mathrm{,}\forall v_h\in V_h\text{with}{\Pi }_2^0{v}_h=0.$

where $c_1\mathrm{,}{c}_2\mathrm{,}{c}_3$ and $c_4$ are positive constants independent of h. One can refer to [14] for the example of construction of $R^K\left(\cdot \mathrm{,}\cdot \right)$ and $S^K\left(\cdot \mathrm{,}\cdot \right)$ . Moreover, the bilinear form $A_h^K\left(\cdot \mathrm{,}\cdot \right)$ satisfies:

· Consistency:

$A_h^K\left(p\mathrm{,}{v}_h\right)=A^K\left(p\mathrm{,}{v}_h\right)\mathrm{,}\forall p\in {\left[{\mathbb{P}}_2\left(K\right)\right]}^2\mathrm{,}{v}_h\in V_h^K\mathrm{.}$

· Stability:

${\alpha }_{\mathrm{<mo>\; *\; </mo>}}{A}^K\left(v_h\mathrm{,}{v}_h\right)\le A_h^K\left(v_h\mathrm{,}{v}_h\right)\le {\alpha }^{\mathrm{<mo>\; *\; </mo>}}{A}^K\left(v_h\mathrm{,}{v}_h\right)\mathrm{,}\forall v_h\in V_h^K\mathrm{.}$

Here, ${\alpha }_{\mathrm{<mo>\; *\; </mo>}}$ and ${\alpha }^{\mathrm{<mo>\; *\; </mo>}}$ are two positive constants independent of h. By the stability of $A_h^K\left(\cdot \mathrm{,}\cdot \right)$ and the coercive of $A\left(\cdot \mathrm{,}\cdot \right)$ , we obtain that the bilinear form $A_h\left(\cdot \mathrm{,}\cdot \right)$ is coercive, i.e.:

$A_h\left(v_h\mathrm{,}{v}_h\right)\ge C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{v}_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V^2\forall v_h\in V_h\mathrm{.}$ (3.1)

Next, the bilinear $b\left(\cdot \mathrm{,}\cdot \right)$ satisfies the inf-sup condition [14] .

Lemma 3.1. Given the discrete spaces $V_h$ and $Q_h$ , there exists a positive constant $\tilde{\beta }$ independent of h with

$\underset{v_h\in V_h\mathrm{,}{v}_h\ne 0}{\text{sup}}\frac{b\left(v_h\mathrm{,}{q}_h\right)}{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{v}_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V}\ge \tilde{\beta }{\Vert q_h\Vert }_Q\forall q_h\in Q_h\mathrm{.}$

Then the virtual element approximation of optimal control problem (1.1)-(1.2) is to find $\left(y_h\mathrm{,}{p}_h\mathrm{,}{u}_h\right)\in V_h\times Q_h\times U_{ad}$ such that

$\underset{u_h\in U_{ad}}{\text{min}}J\left(y_h\mathrm{,}{u}_h\right)\mathrm{<mo>\; :\; </mo>}=\frac{1}{2}{\displaystyle \underset{K\in {\mathcal{T}}_h}{\sum }}{\displaystyle {\int }_K}{\left({\Pi }_2^0{y}_h-y_d\right)}^2\text{d}K+\frac{\gamma }{2}{\displaystyle {\int }_{\Omega }}{u}_h^2\text{d}\Omega$

subject to

$(\begin{array}{ll}{A}_h\left(y_h\mathrm{,}{v}_h\right)-b\left(v_h\mathrm{,}{p}_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(f+u_h\mathrm{,}{\Pi }_2^0{v}_h\right)}_{\mathrm{0,}K}\hfill & \forall v_h\in V_h\mathrm{,}\hfill \\ b\left(y_h\mathrm{,}{q}_h\right)=0\hfill & \forall q_h\in Q_h\mathrm{.}\hfill \end{array}$ (3.2)

Here the control variable is implicitly discretized (see [19] ), and the minimization problem is defined on infinite dimensional set $U_{ad}$ , instead of virtual element space. In order to balance the convergence rates of state and control variables, in the discrete state equation we adopt the $L^2$ projection ${\Pi }_2^0$ .

We introduce the following Lagrangian functional:

$\begin{array}{c}\mathcal{L}\left(y_h\mathrm{,}{p}_h\mathrm{,}{u}_h\mathrm{,}{z}_h\mathrm{,}{\varphi }_h\right)\mathrm{<mo>\; :\; </mo>}=J\left(y_h\mathrm{,}{u}_h\right)-A_h\left(y_h\mathrm{,}{z}_h\right)+b\left(z_h\mathrm{,}{p}_h\right)-b\left(y_h\mathrm{,}{\varphi }_h\right)\\ +{\displaystyle \underset{K\in {\mathcal{T}}_h}{\sum }}{\left(f+u_h\mathrm{,}{\Pi }_2^0{z}_h\right)}_{\mathrm{0,}K}\mathrm{.}\end{array}$

Then the following discrete first order optimality condition can be obtained by computing the derivatives of $\mathcal{L}\left(\cdot \mathrm{,}\cdot \mathrm{,}\cdot \mathrm{,}\cdot \mathrm{,}\cdot \right)$ with respect to $\left(y_h\mathrm{,}{p}_h\mathrm{,}{u}_h\mathrm{,}{z}_h\mathrm{,}{\varphi }_h\right)$ :

$(\begin{array}{ll}{A}_h\left(y_h\mathrm{,}{w}_h\right)-b\left(w_h\mathrm{,}{p}_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(f+u_h\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\hfill & \forall w_h\in V_h\mathrm{,}\hfill \\ b\left(y_h\mathrm{,}{\psi }_h\right)=0\hfill & \forall {\psi }_h\in Q_h\mathrm{,}\hfill \end{array}$ (3.3)

$(\begin{array}{ll}{A}_h\left(w_h\mathrm{,}{z}_h\right)+b\left(w_h\mathrm{,}{\varphi }_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h-y_d\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\hfill & \forall w_h\in V_h\mathrm{,}\hfill \\ b\left(z_h\mathrm{,}{\psi }_h\right)=0\hfill & \forall {\psi }_h\in Q_h\mathrm{,}\hfill \end{array}$ (3.4)

${\displaystyle \underset{K\in {\mathcal{T}}_h}{\sum }}{\left(\gamma u_h+{\Pi }_2^0{z}_h\mathrm{,}{v}_h-u_h\right)}_{\mathrm{0,}K}\ge 0\forall v_h\in U_{ad}\mathrm{.}$ (3.5)

4. A Priori Error Estimates

Lemma 4.1. (See [20] )For the state equation, there exists a positive constant C, the state variables admit the following estimates

${\Vert y\Vert }_2+{\Vert p\Vert }_1\le C{\Vert f\Vert }_0\mathrm{.}$

To achieve a priori error estimates, we introduce some auxiliary problems: $\forall \left(w_h\mathrm{,}{\psi }_h\right)\in V_h\times Q_h$ ,

$(\begin{array}{l}{A}_h\left(y_h\left(u\right)\mathrm{,}{w}_h\right)-b\left(w_h\mathrm{,}{p}_h\left(u\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(f+u\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(y_h\left(u\right)\mathrm{,}{\psi }_h\right)=\mathrm{0,}\end{array}$ (4.1)

$(\begin{array}{l}{A}_h\left(w_h\mathrm{,}{z}_h\left(y\right)\right)+b\left(w_h\mathrm{,}{\varphi }_h\left(y\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y-y_d\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(z_h\left(y\right)\mathrm{,}{\psi }_h\right)=\mathrm{0,}\end{array}$ (4.2)

$(\begin{array}{l}{A}_h\left(w_h\mathrm{,}{z}_h\left(u\right)\right)+b\left(w_h\mathrm{,}{\varphi }_h\left(u\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h\left(u\right)-y_d\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(z_h\left(u\right)\mathrm{,}{\psi }_h\right)=0.\end{array}$ (4.3)

Additionally, we introduce the discrete kernel:

$Z_h\mathrm{<mo>\; :\; </mo>}=\left\{v_h\in V_h\text{s}\text{.t}\text{.}b\left(v_h\mathrm{,}{q}_h\right)=0\forall q_h\in Q_h\right\}\mathrm{.}$

Lemma 4.2. (See [14] ) Let $y\mathrm{,}z\in V\cap H^{s+1}\left(\Omega \right)$ with $0\le s\le 2$ . Under the Assumption (2.1) on the decomposition ${\mathcal{T}}_h$ , there exist $y_I\mathrm{,}{z}_I\in V_h$ such that

${\Vert y-y_I\Vert }_{\mathrm{0,}K}+h_K\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-y_I{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}\le C{h}_K^{s+1}{\left|y\right|}_{s+\mathrm{1,}K}\mathrm{,}$

${\Vert z-z_I\Vert }_{\mathrm{0,}K}+h_K\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}z-z_I{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}\le C{h}_K^{s+1}{\left|z\right|}_{s+\mathrm{1,}K}\mathrm{,}$

where C is a positive constant independent of h.

Lemma 4.3. Let $\left(y\mathrm{,}p\right)$ and $\left(y_h\left(u\right)\mathrm{,}{p}_h\left(u\right)\right)$ be the solutions (2.1) and (4.1), respectively. Under the Assumption 2.1, we have the following estimates

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-y_h\left(u\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{h}^2\left({\left|f+u\right|}_1+{\left|y\right|}_3\right)\mathrm{,}$

${\Vert p-p_h\left(u\right)\Vert }_0\le C{h}^2\left({\left|f+u\right|}_1+{\left|y\right|}_3+{\left|p\right|}_2\right)\mathrm{,}$

${\Vert y-y_h\left(u\right)\Vert }_0\le C{h}^3\left({\left|f+u\right|}_1+{\left|y\right|}_3+{\left|p\right|}_2\right)\mathrm{.}$

Proof. Note that $\left(y_h\left(u\right)\mathrm{,}{p}_h\left(u\right)\right)$ is the virtual element approximation of $\left(y\mathrm{,}p\right)$ . We observe that, if $y\in V$ is the velocity solution to Equation (2.1), then it is also the solution to the following problem: find $y\in Z$ , such that

$A\left(y\mathrm{,}w\right)=\left(f+u\mathrm{,}w\right)\forall w\in Z\mathrm{.}$

Analogously, if $y_h\left(u\right)\in V_h$ is the velocity solution to Equation (4.1), then it is also the solution to problem: find $y_h\left(u\right)\in Z_h$ , such that

$A_h\left(y_h\left(u\right)\mathrm{,}{w}_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(f+u\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\forall w_h\in Z_h\mathrm{.}$

Therefore, by using the same techniques of Theorem 4.6 and Theorem 4.7 in [11] , we can derive the first and second estimate in this lemma. Now we just give the proof of the last one.

Let $\left(r\mathrm{,}t\right)\in V\times Q$ be the solution to the dual problem

$(\begin{array}{ll}-\Delta r-\nabla t+{\mathbb{K}}^{-1}r=y-y_h\left(u\right)\hfill & \text{in}\Omega \mathrm{,}\hfill \\ \nabla \cdot r=0\hfill & \text{in}\Omega \mathrm{,}\hfill \\ r=0\hfill & \text{on}\Gamma \mathrm{.}\hfill \end{array}$ (4.4)

From Lemma 4.1 we know that $r$ satisfies the regularity bound

${\Vert r\Vert }_2\le C{\Vert y-y_h\left(u\right)\Vert }_0$

and consequently for any interpolation $r_I$ as in Lemma 4.2, we have

$\begin{array}{l}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}r-r_I{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V={\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}r-r_I{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{h}_K^2{\left|r\right|}_{\mathrm{2,}K}^2\right)}^{\frac{1}{2}}\\ \le Ch{\left|r\right|}_2\\ \le Ch{\Vert y-y_h\left(u\right)\Vert }_0\mathrm{.}\end{array}$

Because of $\text{div}{V}_h\subseteq Q_h$ , we obtain that the discrete velocity solution of state equation is divergence-free. Thus, we get $\nabla \cdot \left(y-y_h\left(u\right)\right)=0$ . Further, multiplying (4.4) by $y-y_h\left(u\right)$ and integrating leads to

$\begin{array}{c}{\Vert y-y_h\left(u\right)\Vert }_0^2=\left(y-y_h\left(u\right)\mathrm{,}-\Delta r-\nabla t+{\mathbb{K}}^{-1}r\right)\\ =A\left(y-y_h\left(u\right)\mathrm{,}r\right)\\ =A\left(y-y_h\left(u\right)\mathrm{,}r-r_I\right)+A\left(y-y_h\left(u\right)\mathrm{,}{r}_I\right)\\ =A\left(y-y_h\left(u\right)\mathrm{,}r-r_I\right)+A\left(y\mathrm{,}{r}_I\right)-A\left(y_h\left(u\right)\mathrm{,}{r}_I\right)\end{array}$

$\begin{array}{c}=A\left(y-y_h\left(u\right)\mathrm{,}r-r_I\right)+b\left(r_I\mathrm{,}p\right)-b\left(r_I\mathrm{,}{p}_h\left(u\right)\right)\\ +A_h\left(y_h\left(u\right)\mathrm{,}{r}_I\right)-A\left(y_h\left(u\right)\mathrm{,}{r}_I\right)\\ +\left(f+u\mathrm{,}{r}_I\right)-\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left({\Pi }_2^0\left(f+u\right)\mathrm{,}{r}_I\right)}_{\mathrm{0,}K}\mathrm{.}\end{array}$ (4.5)

We label these as $T_1\mathrm{,}{T}_2\mathrm{,}{T}_3$ and $T_f$ , respectively, and bound them separately.

Firstly, we can bound $T_1$ as follows

$\begin{array}{c}{T}_1\mathrm{<mo>\; :\; </mo>}=A\left(y-y_h\left(u\right)\mathrm{,}r-r_I\right)\\ \le C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-y_h\left(u\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}r-r_I{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\\ \le C{h}^2\left({\left|f+u\right|}_1+{\left|y\right|}_3\right)\cdot Ch{\Vert y-y_h\left(u\right)\Vert }_0\\ \le C{h}^3\left({\left|f+u\right|}_1+{\left|y\right|}_3\right){\Vert y-y_h\left(u\right)\Vert }_0\mathrm{.}\end{array}$

Due to $\nabla \cdot r=0$ the estimate of term $T_2$ follows

$\begin{array}{c}{T}_2\mathrm{<mo>\; :\; </mo>}=b\left(r_I\mathrm{,}p\right)-b\left(r_I\mathrm{,}{p}_h\left(u\right)\right)\\ =b\left(r_I\mathrm{,}p-p_h\left(u\right)\right)\\ =b\left(r_I-r\mathrm{,}p-p_h\left(u\right)\right)\\ \le \mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{r}_I-r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V{\Vert p-p_h\left(u\right)\Vert }_0\\ \le Ch{\Vert y-y_h\left(u\right)\Vert }_0\cdot C{h}^2\left({\left|f+u\right|}_1+{\left|y\right|}_3+{\left|p\right|}_2\right)\\ \le C{h}^3\left({\left|f+u\right|}_1+{\left|y\right|}_3+{\left|p\right|}_2\right){\Vert y-y_h\left(u\right)\Vert }_0\mathrm{.}\end{array}$

For the inconsistency term $T_3$ we have

$\begin{array}{c}{T}_3\mathrm{<mo>\; :\; </mo>}=A_h\left(y_h\left(u\right)\mathrm{,}{r}_I\right)-A\left(y_h\left(u\right)\mathrm{,}{r}_I\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\left(A_h^K\left(y_h\left(u\right)\mathrm{,}{r}_I\right)-A^K\left(y_h\left(u\right)\mathrm{,}{r}_I\right)\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\left(A_h^K\left(y_h\left(u\right)-{\Pi }_2^{0}y\mathrm{,}{r}_I-{\Pi }_1^{0}r\right)-A^K\left(y_h\left(u\right)-{\Pi }_2^{0}y\mathrm{,}{r}_I-{\Pi }_1^{0}r\right)\right)\mathrm{.}\end{array}$

Note that

$\begin{array}{l}{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-{\Pi }_2^{0}y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ ={\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y+y-{\Pi }_2^{0}y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-{\Pi }_2^{0}y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}+C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-{\Pi }_2^{0}y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\end{array}$

$\begin{array}{l}\le C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V+C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{h}_K^4{\left|y\right|}_{\mathrm{3,}K}^2\right)}^{\frac{1}{2}}\\ \le C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V+C{h}^2{\left|y\right|}_3\\ \le C{h}^2\left({\left|f+u\right|}_1+{\left|y\right|}_3\right)\end{array}$

and

$\begin{array}{l}{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{r}_I-{\Pi }_1^{0}r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ ={\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{r}_I-r+r-{\Pi }_1^{0}r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}r-r_I{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}r-{\Pi }_1^{0}r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}r-r_I{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}+C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}r-{\Pi }_1^{0}r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\end{array}$

$\begin{array}{l}\le C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{r}_I-r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V+C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{h}_K^2{\left|r\right|}_{\mathrm{2,}K}^2\right)}^{\frac{1}{2}}\\ \le C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{r}_I-r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V+Ch{\left|r\right|}_2\\ \le Ch{\Vert y-y_h\left(u\right)\Vert }_0\mathrm{.}\end{array}$

Then applying the Cauchy-Schwarz inequality [21] , the following conclusion can be drawn

$\begin{array}{l}\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\left(A_h^K\left(y_h\left(u\right)-{\Pi }_2^{0}y\mathrm{,}{r}_I-{\Pi }_1^{0}r\right)-A^K\left(y_h\left(u\right)-{\Pi }_2^{0}y\mathrm{,}{r}_I-{\Pi }_1^{0}r\right)\right)\\ \le \underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\left(\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-{\Pi }_2^{0}y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{r}_I-{\Pi }_1^{0}r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}\right)\\ \le {\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-{\Pi }_2^{0}y{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{r}_I-{\Pi }_1^{0}r{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_{V\mathrm{,}K}^2\right)}^{\frac{1}{2}}\\ \le C{h}^3\left({\left|f+u\right|}_1+{\left|y\right|}_3\right){\Vert y-y_h\left(u\right)\Vert }_0\mathrm{.}\end{array}$

Finally, the definition and estimate of the $L^2$ projection operator leads to the estimate of $T_f$

$\begin{array}{c}{T}_f\mathrm{<mo>\; :\; </mo>}=\left(f+u\mathrm{,}{r}_I\right)-\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left({\Pi }_2^0\left(f+u\right)\mathrm{,}{r}_I\right)}_{\mathrm{0,}K}\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(f+u-{\Pi }_2^0\left(f+u\right)\mathrm{,}{r}_I\right)}_{\mathrm{0,}K}\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(f+u-{\Pi }_2^0\left(f+u\right)\mathrm{,}{r}_I-{\Pi }_1^0{r}_I\right)}_{\mathrm{0,}K}\\ \le \underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert f+u-{\Pi }_2^0\left(f+u\right)\Vert }_{\mathrm{0,}K}{\Vert r_I-{\Pi }_1^0{r}_I\Vert }_{\mathrm{0,}K}\\ \le {\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert f+u-{\Pi }_2^0\left(f+u\right)\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r_I-{\Pi }_1^0{r}_I\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}\end{array}$

$\begin{array}{c}\le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{h}_K^2{\left|f+u\right|}_{\mathrm{1,}K}^2\right)}^{\frac{1}{2}}\cdot C{h}^2{\left|r\right|}_2\\ \le Ch{\left|f+u\right|}_1\cdot C{h}^2{\left|r\right|}_2\\ \le C{h}^3{\left|f+u\right|}_1{\Vert y-y_h\left(u\right)\Vert }_0\mathrm{.}\end{array}$

Here, the estimate of ${\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r_I-{\Pi }_1^0{r}_I\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}$ is derived as follows:

$\begin{array}{l}{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r_I-{\Pi }_1^0{r}_I\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}\\ ={\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r_I-r+r-{\Pi }_1^{0}r+{\Pi }_1^{0}r-{\Pi }_1^0{r}_I\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r_I-r\Vert }_{\mathrm{0,}K}^2+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r-{\Pi }_1^{0}r\Vert }_{\mathrm{0,}K}^2+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert {\Pi }_1^{0}r-{\Pi }_1^0{r}_I\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(2\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r_I-r\Vert }_{\mathrm{0,}K}^2+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r-{\Pi }_1^{0}r\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}\end{array}$

$\begin{array}{l}\le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r_I-r\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}+C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert r-{\Pi }_1^{0}r\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}\\ \le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{h}_K^4{\left|r\right|}_{\mathrm{2,}D\left(K\right)}^2\right)}^{\frac{1}{2}}+C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{h}_K^4{\left|r\right|}_{\mathrm{2,}K}^2\right)}^{\frac{1}{2}}\\ \le C{h}^2{\left|r\right|}_2\mathrm{.}\end{array}$

Inserting above bounds into (4.5) yields the third estimate.

For the adjoint state variables, we have the following results.

Lemma 4.4. Let $\left(z\mathrm{,}\varphi \right)$ and $\left(z_h\left(y\right)\mathrm{,}{\varphi }_h\left(y\right)\right)$ be the solutions (2.2) and (4.2), respectively. Under the Assumption 2.1, we have the following estimates

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}z-z_h\left(y\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{h}^2\left({\left|y-y_d\right|}_1+{\left|z\right|}_3\right)\mathrm{,}$

${\Vert \varphi -{\varphi }_h\left(y\right)\Vert }_0\le C{h}^2\left({\left|y-y_d\right|}_1+{\left|z\right|}_3+{\left|\varphi \right|}_2\right)\mathrm{,}$

${\Vert z-z_h\left(y\right)\Vert }_0\le C{h}^3\left({\left|y-y_d\right|}_1+{\left|z\right|}_3+{\left|\varphi \right|}_2\right)\mathrm{.}$

Proof. Note that $\left(z_h\left(y\right)\mathrm{,}{\varphi }_h\left(y\right)\right)$ is the virtual element approximation of $\left(z\mathrm{,}\varphi \right)$ . In a similar way to state variables, the Equation (2.2) can be rewritten as: finding $z\in Z$ , such that

$A\left(w\mathrm{,}z\right)=\left(y-y_d\mathrm{,}w\right)\forall w\in Z\mathrm{,}$

while, the Equation (4.2) can be rewritten as: finding $z_h\left(y\right)\in Z_h$ , such that

$A_h\left(w_h\mathrm{,}{z}_h\left(y\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y-y_d\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\forall w_h\in Z_h\mathrm{.}$

Then by using the same techniques of Theorem 4.6 and Theorem 4.7 in [11] , we can derive the first and second estimate in this lemma. The last one can be derived by the similar argument to Lemma 4.3.

Theorem 4.5. (A priori error estimate)Suppose that $\left(y\mathrm{,}p\mathrm{,}u\mathrm{,}z\mathrm{,}\varphi \right)$ is the solution of (2.1)-(2.3), and $\left(y_h\mathrm{,}{p}_h\mathrm{,}{u}_h\mathrm{,}{z}_h\mathrm{,}{\varphi }_h\right)$ is the solution of (3.3)-(3.5). Under the Assumption 2.1, we derive

${\Vert y-y_h\Vert }_0+h\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-y_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{h}^3\mathrm{,}{\Vert p-p_h\Vert }_0\le C{h}^2\mathrm{,}$

${\Vert z-z_h\Vert }_0+h\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}z-z_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{h}^3\mathrm{,}{\Vert \varphi -{\varphi }_h\Vert }_0\le C{h}^2\mathrm{,}$

${\Vert u-u_h\Vert }_0\le C{h}^3\mathrm{,}$

where C is a positive constant independent of h.

Proof. We decompose the errors $y-y_h\mathrm{,}p-p_h\mathrm{,}z-z_h$ and $\varphi -{\varphi }_h$ into

$y-y_h=y-y_h\left(u\right)+y_h\left(u\right)-y_h\mathrm{,}p-p_h=p-p_h\left(u\right)+p_h\left(u\right)-p_h\mathrm{,}$

$z-z_h=z-z_h\left(y\right)+z_h\left(y\right)-z_h\mathrm{,}\varphi -{\varphi }_h=\varphi -{\varphi }_h\left(y\right)+{\varphi }_h\left(y\right)-{\varphi }_h\mathrm{.}$

Recalling Lemma 4.3, we know

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-y_h\left(u\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_0\le C{h}^3\mathrm{,\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-y_h\left(u\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{h}^2\mathrm{,}{\Vert p-p_h\left(u\right)\Vert }_0\le C{h}^2\mathrm{.}$

Moreover, by the governing equations of $y_h\left(u\right)$ and $y_h$ we have: $\forall \left(w_h\mathrm{,}{\psi }_h\right)\in V_h\times Q_h$ ,

$(\begin{array}{l}{A}_h\left(y_h\left(u\right)-y_h\mathrm{,}{w}_h\right)-b\left(w_h\mathrm{,}{p}_h\left(u\right)-p_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(u-u_h\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(y_h\left(u\right)-y_h\mathrm{,}{\psi }_h\right)=0.\end{array}$

Setting $w_h=y_h\left(u\right)-y_h$ and ${\psi }_h=p_h\left(u\right)-p_h$ gives

$A_h\left(y_h\left(u\right)-y_h\mathrm{,}{y}_h\left(u\right)-y_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(u-u_h\mathrm{,}{\Pi }_2^0\left(y_h\left(u\right)-y_h\right)\right)}_{\mathrm{0,}K}\mathrm{.}$

It follows from (3.1) that

$\begin{array}{l}C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V^2\le A_h\left(y_h\left(u\right)-y_h\mathrm{,}{y}_h\left(u\right)-y_h\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(u-u_h\mathrm{,}{\Pi }_2^0\left(y_h\left(u\right)-y_h\right)\right)}_{\mathrm{0,}K}\\ \le \underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert u-u_h\Vert }_{\mathrm{0,}K}{\Vert y_h\left(u\right)-y_h\Vert }_{\mathrm{0,}K}\\ \le {\Vert u-u_h\Vert }_0\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\mathrm{.}\end{array}$

We can deduce

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{\Vert u-u_h\Vert }_0\mathrm{.}$

Combining above inequalities leads to

${\Vert y-y_h\Vert }_0\le C\left(h^3+{\Vert u-u_h\Vert }_0\right)$ (4.6)

and

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}y-y_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C\left(h^2+{\Vert u-u_h\Vert }_0\right)\mathrm{.}$ (4.7)

Next we estimate ${\Vert p_h\left(u\right)-p_h\Vert }_0$ . Based on Lemma 3.1, we get

$\begin{array}{c}\tilde{\beta }{\Vert p_h\left(u\right)-p_h\Vert }_0\le \underset{w_h\in V_h\mathrm{,}{w}_h\ne 0}{\text{sup}}\frac{b\left(w_h\mathrm{,}{p}_h\left(u\right)-p_h\right)}{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{w}_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V}\\ =\underset{w_h\in V_h\mathrm{,}{w}_h\ne 0}{\text{sup}}\frac{A_h\left(y_h\left(u\right)-y_h\mathrm{,}{w}_h\right)+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(u_h-u\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}}{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{w}_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V}\\ \le C\left(\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{y}_h\left(u\right)-y_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V+{\Vert u_h-u\Vert }_0\right)\\ \le C{\Vert u-u_h\Vert }_0\mathrm{.}\end{array}$

By the triangle inequality, it holds

${\Vert p-p_h\Vert }_0\le C\left(h^2+{\Vert u-u_h\Vert }_0\right)\mathrm{.}$ (4.8)

In a similar way, from Lemma 4.4, we have

${\Vert z-z_h\left(y\right)\Vert }_0\le C{h}^3\mathrm{,\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}z-z_h\left(y\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{h}^2\mathrm{,}{\Vert \varphi -{\varphi }_h\left(y\right)\Vert }_0\le C{h}^2\mathrm{.}$

By the governing equations of $z_h\left(y\right)$ and $z_h$ we have: $\forall \left(w_h\mathrm{,}{\psi }_h\right)\in V_h\times Q_h$ ,

$(\begin{array}{l}{A}_h\left(w_h\mathrm{,}{z}_h\left(y\right)-z_h\right)+b\left(w_h\mathrm{,}{\varphi }_h\left(y\right)-{\varphi }_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y-y_h\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(z_h\left(y\right)-z_h\mathrm{,}{\psi }_h\right)=0.\end{array}$

Choosing $w_h=z_h\left(y\right)-z_h$ and ${\psi }_h={\varphi }_h\left(y\right)-{\varphi }_h$ , we obtain

$A_h\left(z_h\left(y\right)-z_h\mathrm{,}{z}_h\left(y\right)-z_h\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y-y_h\mathrm{,}{\Pi }_2^0\left(z_h\left(y\right)-z_h\right)\right)}_{\mathrm{0,}K}\mathrm{.}$

Further, it follows that

$\begin{array}{l}C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{z}_h\left(y\right)-z_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V^2\le A_h\left(z_h\left(y\right)-z_h\mathrm{,}{z}_h\left(y\right)-z_h\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y-y_h\mathrm{,}{\Pi }_2^0\left(z_h\left(y\right)-z_h\right)\right)}_{\mathrm{0,}K}\\ \le \underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert y-y_h\Vert }_{\mathrm{0,}K}{\Vert z_h\left(y\right)-z_h\Vert }_{\mathrm{0,}K}\\ \le {\Vert y-y_h\Vert }_0\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{z}_h\left(y\right)-z_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\mathrm{.}\end{array}$

This implies

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{z}_h\left(y\right)-z_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{\Vert y-y_h\Vert }_0\le C\left(h^3+{\Vert u-u_h\Vert }_0\right)\mathrm{.}$

Combining above inequalities gives

${\Vert z-z_h\Vert }_0\le C\left(h^3+{\Vert u-u_h\Vert }_0\right)$ (4.9)

and

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}z-z_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C\left(h^2+{\Vert u-u_h\Vert }_0\right)\mathrm{.}$ (4.10)

Using Lemma 3.1, we can derive

$\begin{array}{c}\tilde{\beta }{\Vert {\varphi }_h\left(y\right)-{\varphi }_h\Vert }_0\le \underset{w_h\in V_h\mathrm{,}{w}_h\ne 0}{\text{sup}}\frac{b\left(w_h\mathrm{,}{\varphi }_h\left(y\right)-{\varphi }_h\right)}{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{w}_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V}\\ =\underset{w_h\in V_h\mathrm{,}{w}_h\ne 0}{\text{sup}}\frac{A_h\left(w_h\mathrm{,}{z}_h-z_h\left(y\right)\right)+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y-y_h\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}}{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{w}_h{\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V}\\ \le C\left(\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{z}_h-z_h\left(y\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V+{\Vert y-y_h\Vert }_0\right)\\ \le C\left(h^3+{\Vert u-u_h\Vert }_0\right)\mathrm{.}\end{array}$

By the triangle inequality, we have

${\Vert \varphi -{\varphi }_h\Vert }_0\le C\left(h^2+{\Vert u-u_h\Vert }_0\right)\mathrm{.}$ (4.11)

Since the estimates of state and adjoint state variables both depend on the estimate of control variable, now it remains to estimate ${\Vert u-u_h\Vert }_0$ . Define

${{\hat{J}}^{\prime }}_h\left(u\right)\left(v-u\right)\mathrm{<mo>\; :\; </mo>}=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\left(\gamma u+{\Pi }_2^0{z}_h\left(u\right)\right)\left(v-u\right)\text{d}K\mathrm{.}$

We can prove that

${{\hat{J}}^{\prime }}_h\left(v\right)\left(v-u\right)-{{\hat{J}}^{\prime }}_h\left(u\right)\left(v-u\right)\ge \gamma {\Vert v-u\Vert }_0^2\mathrm{.}$ (4.12)

Note that

$\begin{array}{l}{{\hat{J}}^{\prime }}_h\left(v\right)\left(v-u\right)-{{\hat{J}}^{\prime }}_h\left(u\right)\left(v-u\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\left(\gamma v+{\Pi }_2^0{z}_h\left(v\right)-\gamma u-{\Pi }_2^0{z}_h\left(u\right)\right)\left(v-u\right)\text{d}K\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\gamma {\left(v-u\right)}^2\text{d}K+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\left({\Pi }_2^0\left(z_h\left(v\right)-z_h\left(u\right)\right)\right)\left(v-u\right)\text{d}K\\ =\gamma {\displaystyle {\int }_{\Omega }}{\left(v-u\right)}^2\text{d}\Omega +\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\left({\Pi }_2^0\left(z_h\left(v\right)-z_h\left(u\right)\right)\right)\left(v-u\right)\text{d}K\mathrm{.}\end{array}$

Using (4.1), we can derive: $\forall \left(w_h\mathrm{,}{\psi }_h\right)\in V_h\times Q_h$ ,

$(\begin{array}{l}{A}_h\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{w}_h\right)-b\left(w_h\mathrm{,}{p}_h\left(v\right)-p_h\left(u\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(v-u\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{\psi }_h\right)=0.\end{array}$

Let $w_h=z_h\left(v\right)-z_h\left(u\right)$ and ${\psi }_h={\varphi }_h\left(v\right)-{\varphi }_h\left(u\right)$ , then we obtain

$(\begin{array}{l}{A}_h\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{z}_h\left(v\right)-z_h\left(u\right)\right)-b\left(z_h\left(v\right)-z_h\left(u\right)\mathrm{,}{p}_h\left(v\right)-p_h\left(u\right)\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(v-u\mathrm{,}{\Pi }_2^0\left(z_h\left(v\right)-z_h\left(u\right)\right)\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{\varphi }_h\left(v\right)-{\varphi }_h\left(u\right)\right)=0.\end{array}$ (4.13)

Using (4.3), we can obtain: $\forall \left(w_h\mathrm{,}{\psi }_h\right)\in V_h\times Q_h$ ,

$(\begin{array}{l}{A}_h\left(w_h\mathrm{,}{z}_h\left(v\right)-z_h\left(u\right)\right)+b\left(w_h\mathrm{,}{\varphi }_h\left(v\right)-{\varphi }_h\left(u\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(z_h\left(v\right)-z_h\left(u\right)\mathrm{,}{\psi }_h\right)=0.\end{array}$

Taking $w_h=y_h\left(v\right)-y_h\left(u\right)$ and ${\psi }_h=p_h\left(v\right)-p_h\left(u\right)$ yields

$(\begin{array}{l}{A}_h\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{z}_h\left(v\right)-z_h\left(u\right)\right)+b\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{\varphi }_h\left(v\right)-{\varphi }_h\left(u\right)\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{\Pi }_2^0\left(y_h\left(v\right)-y_h\left(u\right)\right)\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(z_h\left(v\right)-z_h\left(u\right)\mathrm{,}{p}_h\left(v\right)-p_h\left(u\right)\right)=0.\end{array}$ (4.14)

According to (4.13)-(4.14) and the property of the $L^2$ projection, we deduce

$\begin{array}{l}\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(v-u\mathrm{,}{\Pi }_2^0\left(z_h\left(v\right)-z_h\left(u\right)\right)\right)}_{\mathrm{0,}K}\\ =A_h(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{z}_h\left(v\right)-z_h(\; u\; ))\end{array}$

$\begin{array}{l}=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h\left(v\right)-y_h\left(u\right)\mathrm{,}{\Pi }_2^0\left(y_h\left(v\right)-y_h\left(u\right)\right)\right)}_{\mathrm{0,}K}\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left({\Pi }_2^0\left(y_h\left(v\right)-y_h\left(u\right)\right)\mathrm{,}{\Pi }_2^0\left(y_h\left(v\right)-y_h\left(u\right)\right)\right)}_{\mathrm{0,}K}\ge 0.\end{array}$

Therefore, from (4.12), it follows

$\begin{array}{c}\gamma {\Vert u-u_h\Vert }_0^2\le {{\hat{J}}^{\prime }}_h\left(u\right)\left(u-u_h\right)-{{\hat{J}}^{\prime }}_h\left(u_h\right)\left(u-u_h\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\left(\gamma u+{\Pi }_2^0{z}_h\left(u\right)-\gamma u_h-{\Pi }_2^0{z}_h\left(u_h\right)\right)\left(u-u_h\right)\text{d}K\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\left(\gamma u+z+{\Pi }_2^0{z}_h\left(u\right)-z\right)\left(u-u_h\right)\text{d}K\\ +\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\displaystyle {\int }_K}\left(\gamma u_h+{\Pi }_2^0{z}_h\left(u_h\right)\right)\left(u_h-u\right)\text{d}K\end{array}$

$\begin{array}{l}=\left(\gamma u+z\mathrm{,}u-u_h\right)+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left({\Pi }_2^0{z}_h\left(u\right)-z\mathrm{,}u-u_h\right)}_{\mathrm{0,}K}\\ +\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(\gamma u_h+{\Pi }_2^0{z}_h\left(u_h\right)\mathrm{,}{u}_h-u\right)}_{\mathrm{0,}K}\\ \le 0+\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left({\Pi }_2^0{z}_h\left(u\right)-z\mathrm{,}u-u_h\right)}_{\mathrm{0,}K}+0.\end{array}$

This shows

${\Vert u-u_h\Vert }_0\le C{\left(\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert {\Pi }_2^0{z}_h\left(u\right)-z\Vert }_{\mathrm{0,}K}^2\right)}^{\frac{1}{2}}\mathrm{.}$ (4.15)

Note that

$\begin{array}{c}{\Vert {\Pi }_2^0{z}_h\left(u\right)-z\Vert }_{\mathrm{0,}K}\le {\Vert {\Pi }_2^0{z}_h\left(u\right)-{\Pi }_2^{0}z\Vert }_{\mathrm{0,}K}+{\Vert {\Pi }_2^{0}z-z\Vert }_{\mathrm{0,}K}\\ \le {\Vert z_h\left(u\right)-z\Vert }_{\mathrm{0,}K}+{\Vert {\Pi }_2^{0}z-z\Vert }_{\mathrm{0,}K}\mathrm{.}\end{array}$ (4.16)

By Lemma 2.3, we have ${\Vert {\Pi }_2^{0}z-z\Vert }_{\mathrm{0,}K}\le C{h}^3$ . We decompose the error $z_h\left(u\right)-z$ into

$z_h\left(u\right)-z=z_h\left(u\right)-z_h\left(y\right)+z_h\left(y\right)-z\mathrm{.}$

Applying Lemma 4.4 yields

${\Vert z-z_h\left(y\right)\Vert }_0\le C{h}^3\mathrm{.}$

By governing equations of $z_h\left(u\right)$ and $z_h\left(y\right)$ we have: $\forall \left(w_h\mathrm{,}{\psi }_h\right)\in V_h\times Q_h$

$(\begin{array}{l}{A}_h\left(w_h\mathrm{,}{z}_h\left(u\right)-z_h\left(y\right)\right)+b\left(w_h\mathrm{,}{\varphi }_h\left(u\right)-{\varphi }_h\left(y\right)\right)=\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h\left(u\right)-y\mathrm{,}{\Pi }_2^0{w}_h\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(z_h\left(u\right)-z_h\left(y\right)\mathrm{,}{\psi }_h\right)=0.\end{array}$

Setting $w_h=z_h\left(u\right)-z_h\left(y\right)$ and ${\psi }_h={\varphi }_h\left(u\right)-{\varphi }_h\left(y\right)$ , we obtain

$(\begin{array}{l}{A}_h\left(z_h\left(u\right)-z_h\left(y\right)\mathrm{,}{z}_h\left(u\right)-z_h\left(y\right)\right)+b\left(z_h\left(u\right)-z_h\left(y\right)\mathrm{,}{\varphi }_h\left(u\right)-{\varphi }_h\left(y\right)\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h\left(u\right)-y\mathrm{,}{\Pi }_2^0\left(z_h\left(u\right)-z_h\left(y\right)\right)\right)}_{\mathrm{0,}K}\mathrm{,}\\ b\left(z_h\left(u\right)-z_h\left(y\right)\mathrm{,}{\varphi }_h\left(u\right)-{\varphi }_h\left(y\right)\right)=0.\end{array}$

It can be deduced immediately by (3.1)

$\begin{array}{l}C\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{z}_h\left(u\right)-z_h\left(y\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V^2\le A_h\left(z_h\left(u\right)-z_h\left(y\right)\mathrm{,}{z}_h\left(u\right)-z_h\left(y\right)\right)\\ =\underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\left(y_h\left(u\right)-y\mathrm{,}{\Pi }_2^0\left(z_h\left(u\right)-z_h\left(y\right)\right)\right)}_{\mathrm{0,}K}\\ \le \underset{K\in {\mathcal{T}}_h}{{\displaystyle \sum }}{\Vert y_h\left(u\right)-y\Vert }_{\mathrm{0,}K}{\Vert z_h\left(u\right)-z_h\left(y\right)\Vert }_{\mathrm{0,}K}\\ \le {\Vert y_h\left(u\right)-y\Vert }_0{\Vert z_h\left(u\right)-z_h\left(y\right)\Vert }_1\mathrm{.}\end{array}$

Then we can derive

${\Vert z_h\left(u\right)-z_h\left(y\right)\Vert }_0\le \mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}{z}_h\left(u\right)-z_h\left(y\right){\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}}_V\le C{\Vert y_h\left(u\right)-y\Vert }_0\le C{h}^3\mathrm{.}$

Using the triangle inequality leads to

${\Vert z-z_h\left(u\right)\Vert }_0\le C{h}^3\mathrm{.}$ (4.17)

Combining (4.15), (4.16) and (4.17) results in

${\Vert u-u_h\Vert }_0\le C{h}^3\mathrm{.}$

Inserting above estimate into the estimates of state and adjoint state yields the final results.

5. Numerical Results

In this section, we present an example on domain $\Omega :=\left[0,1\right]\times \left[0,1\right]$ 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 Figure 1. The first sequence of meshes (labeled Distorted square) is the distorted square mesh. The second sequence of meshes (labeled Lloyd) is obtained by the Voronoi mesh generator (see [22] ).

Example 5.1. Consider the optimal control problem (1.1)-(1.2) on the square domain $\Omega$ . Let $u_a={\left(-\mathrm{0.015,}-0.015\right)}^{\text{T}}$ , $u_b={\left(\mathrm{0.015,0.015}\right)}^{\text{T}}$ , $\mathbb{K}=I$ , $\gamma =1$ . The exact solutions are chosen to be

$y=\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)=\left(\begin{array}{c}10x_1^2{\left(1-x_1\right)}^2{x}_2\left(1-x_2\right)\left(1-2x_2\right)\\ -10x_1\left(1-x_1\right)\left(1-2x_1\right)x_2^2{\left(1-x_2\right)}^2\end{array}\right)\mathrm{,}$

$p=10\left(2x_1-1\right)\left(2x_2-1\right)\mathrm{,}$

Figure 1. Two meshes.

$z=0.5y\mathrm{,}\varphi =-0.5p\mathrm{.}$

The control variable is given $u=P_{U_{ad}}\left(-\frac{1}{\gamma }z\right)$ . $f$ and $y_d$ can be determined from the exact solutions $y\mathrm{,}p\mathrm{,}z\mathrm{,}\varphi \mathrm{,}u$ .

In Tables 1-3, we show the numerical results about the state variables $y\mathrm{,}p$ , the adjoint state variables $z\mathrm{,}\varphi$ 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

Table 1. Errors and convergence rates of state variables on two meshes.

Table 2. Errors and convergence rates of adjoint state variables on two meshes.

Table 3. Errors and convergence rates of control variable on two meshes.

conclusion in Theorem 4.5.

Acknowledgements

The author would like to thank editor and referees for their valuable advices for the improvement of this article.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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