Error estimates of finite volume method for Stokes optimal control problem

In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}-norm error estimates. The approximate orders for the state, costate, and control variables are O(h2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(h^{2})$\end{document} in the sense of L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}-norm. Furthermore, we derive H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}$\end{document}-norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.


Introduction
Flow control problems have been widely used in science and engineering. Finite element [2,5], finite difference [28], and spectral [7] methods have been employed to numerically solve them. Finite volume method is an effective discretization technique for partial differential equations. Due to its local conservative property and other attractive properties, the finite volume method is widely used in the numerical approximation fluid dynamics. Since the method was proposed, there have been a lot of studies of mathematical theory for the finite volume method in the literature [1,3,4,6,10,11,14,15]. Bank and Rose obtained some results for elliptic boundary value problems that the finite volume element approximation was comparable with the finite element approximation in H 1 -norm which can be found in [1]. In [15], the authors presented the optimal L 2 -error estimate for second-order elliptic boundary value problems under the assumption that f ∈ H 1 , they also obtained the H 1 -norm and maximum-norm error estimates for those problems. In [6], Chatzipantelidis proposed a nonconforming finite volume method and obtained the L 2 -norm and H 1 -norm error estimates for elliptic boundary value problems in two dimensions. The authors discussed a priori estimates for a linear elliptic optimal control problem in [26], they derived the optimal order error estimates in L 2 and L ∞ -norm for the state, costate, and control variables, and the optimal H 1 and W 1,∞ -norm error estimates for the state and costate variables. At the same time, there are some other literature works to study optimal control problems [8,9,16,[19][20][21][22][23][24][25].
In fact, finite volume methods lie somewhere between finite difference and finite element methods, they have a flexibility similar to that of finite element methods for handling complicated solution domain geometries and boundary conditions, and they have a simplicity for implementation comparable to finite difference methods with triangulations of a simple structure. The finite volume methods and finite element methods are commonly employed in computational fluid mechanics and computational solid dynamics, where the finite volume method is traditionally associated with computational fluid mechanics and the finite element method is associated with computational solid dynamics. In general, two different functional spaces (one for the trial space and one for the test space) are used in the finite volume method. Owing to the two different spaces, the numerical analysis of the finite volume method is more difficult than that of the finite element method and finite difference method. In [18], the authors developed a family of stabilized discontinuous finite volume element methods for the Stokes equations. A priori error estimates are derived for the velocity and pressure in the energy norm, and convergence rates are predicted for velocity in the L 2 -norm under the assumption that the source term was locally in H 1 .
In [13], the authors established a general framework for analyzing the class of finite volume methods for the Stokes equations. Under the framework, optimal L 2 error estimates for velocity were obtained for the first time for several different finite volume methods.
In recent years, the authors studied the Legendre-Galerkin in spectral approximation of distributed optimal control problems governed by Stokes equations. They derived a priori error estimates in both H 1 and L 2 norms for the Legendre-Galerkin approximation of the unconstrained control problems in [7]. However, a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations have few papers to study.
In this paper, we mainly establish finite volume schemes for Stokes optimal control problem and obtain some optimal order error estimates. Firstly, we use the finite volume method to discretize the state and adjoint equation of the optimal control problem. Then, applying the variational discretization concept [17], the control variable is not discretized directly, but discretized by a projection of the discrete costate variable. At last, we obtain some optimal order error estimates under some reasonable assumptions.
In this paper, we consider the following Stokes optimal control problem: where ⊂ R 2 is a bounded convex polygon domain with boundary ∂ , f ∈ L 2 ( ) 2 or H 1 0 ( ) 2 , υ > 0 is a given constant, and y, u are unknown functions, U ad is denoted by The bilinear form b(·, ·) relating the functional spaces for velocity and pressure satisfies the following Babuška-Brezzi condition (see [27] for example): there exists a constant ς > 0 such that The weak formulation associated with the state equations (1.1)-(1.4) is given as follows: The paper is organized as follows. In Sect. 2, we present some notations and describe the finite volume method briefly. In Sect. 3, we analyze the error estimates between the exact solution and the finite volume element approximation. Finally, we give a conclusion and some possible future work in Sect. 4.

Finite volume element approximation
As is shown in [15], the partition T h is quasi-uniform, i.e., there exists a positive constant C such that For the convex polygon , we consider a quasi-uniform triangulation T h consisting of closed triangle elements K such that¯ = K∈T h K . We use N h to denote the set of all nodes or vertices of T h , N t denote the number of triangles in the primal partition. To define the dual partition T * h of T h , we divide each K ∈ T h into three quadrilaterals by connecting the barycenter C K of K with line segments to the midpoints of edges of K as is shown in Fig. 1. The control volume V i consists of the quadrilaterals sharing the same vertex z i as is shown in Fig. 2.
The dual partition T * h consists of the union of the control volume V i . Let h = max{h K }, where h K is the diameter of the triangle K . The dual partition T * h is also quasi-uniform. We define the finite dimensional space V h associated with T h for the trial functions by and define the finite dimensional space Q h associated with the dual partition T * h for the test functions by where V z is a dual element and P l (K) or P l (V ) consists of all the polynomials with degree less than or equal to l defined on K or V . Let R h be the following finite dimensional space for pressure: To connect the trial space and the test space, we define a transfer operator I h : V h → Q h as follows: where χ i is the characteristic function of V i . It is well known (see [12] for example) that there exists a positive constant C such that, The finite volume scheme of (1.6)-(1.8) is defined as the solution of the problem: find where the bilinear forms a(y h , I h w h ) and b(I h w h , r h ) are defined by where n is the unit outward normal vector to ∂V i . The bilinear form a(·, ·) is not symmetric though the problem is self-adjoint. Then, for all w h , v h ∈ V h , there exist positive constants C and h 0 ≥ 0 such that [11], for all 0 < h < h 0 , It is well known (see, e.g., [20]) that the optimal control problem (1.1)-(1.4) has a unique solution (y, r, u), and that if a triplet (y, r, u) is the solution of (1.1)-(1.4), then there is a co-state (p, s) ∈ H 1 0 ( ) 2 × L 2 0 ( ) such that (y, r, p, s, u) satisfies the following optimality conditions:

L 2 error estimates
In this section, we consider the error analysis of the finite volume element approximation. Let (y h (u), r h (u), p h (y), s h (y)) be the solution of Proof Subtracting (2.11)-(2.12) from (3.1)-(3.2), we have Then we can obtain By using (2.2), we have It is clear that we obtain Then, we deal with this term r h (u)r h by using (1.5) Similarly, we can obtain This completes the proof.
Proof Similar to the proof of Theorem 4.1 in [29], directly apply Lemma 3.1 to readily derive the following estimates: Then we will estimate the derivation of L 2 -estimates for yy h (u) and pp h (y). Let us consider the dual problem: find (η, ρ) such that

16)
∇ · η = 0, in , (3.17) which is uniquely solvable; moreover, the following H 2 ( ) × H 1 ( )-regularity is satisfied: Let η I ∈ V h be the usual continuous piecewise linear interplant; it is not hard to see that there exists a constant c independent of h such that Let denote the L 2 -projection from L 2 0 ( ) to Q h , we can get Multiplying (3.16) by yy h (u), integrating by parts, we can get For the first term of (3.28), we can obtain Then we estimate E 2 as follows: By using (1.2), we can obtain According to the quality of , we have Now, we estimate all terms on the right-hand side of (3.38). From Lemma 3.2 and δ-Cauchy inequality, we have By applying (y h (u)y h , I h (y hy h (u))) ≤ 0 and Lemma 3.2, it is clear that According to (2.1) and Lemma 3.2, we obtain Proof Using the triangle inequality, we have Next, we will discuss the error estimates of the numerical solutions of the state and costate in H 1 -norm. Theorem 3.3 Assume that A ∈ W 2,∞ ( ) and f , y d ∈ L 2 ( ) 2 . Let (y, r, p, s, u) and (y h , r h , p h , s h , u h ) ∈ V h × R h × V h × R h × U ad be the solutions of (2.6)-(2.10) and (2.11)-(2.15), respectively. Then there exists h 0 > 0 such that, for all 0 < h ≤ h 0 , yy h 1 + pp h 1 ≤ Ch.

Conclusion and future works
In this paper, we considered a priori error estimates for the finite volume element approximation of Stokes optimal control problem. Then we used the finite volume method to discretize the state and adjoint equation of the system. Under some reasonable assumptions, we obtained some optimal order error estimates. The approximate orders for the state, costate, and control variables were O(h 2 ), and the approximate orders for the state and costate variables was O(h) in the sense of L 2 -norm and H 1 -norm. To our best knowledge, in the context of optimal control problems, these a priori error estimates of the finite volume method for Stokes optimal control problem are new.
In future, we shall consider a posteriori error estimates and superconvergence of the finite volume element solutions for Stokes optimal control problem.