A posteriori error estimates of hp spectral element method for parabolic optimal control problems

: In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive ﬁnite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a L 2 ( H 1 ) − L 2 ( L 2 ) posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a L 2 ( L 2 ) − L 2 ( L 2 ) posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.


Introduction
Optimal control problems are frequently used in practical problems of physical, social, economic processes, and other fields, and the numerical solution of optimal control problems is of great significance for better performance in these fields [30]. Consequently, it is particularly important to need some effective numerical methods to approximate the solution of the optimal control problem. As we all know, finite element method is one of the most commonly used numerical methods to solve optimal control problems. Applying finite element methods, the emergence of errors has captured the attention of scholars. One of the main sources of errors is the error caused by the discretisation of the model, so a large number of researchers have analyzed it in all aspects by using the finite element method. Bonifacius and Pieper, Lu and huang have studied the prior error estimates of the nonlinear optimal control problem [29,30]. Also, Boulaaras has analysed the posteriori error estimates of the finite element method for nonlinear optimal control problems [25,26]. Boulaaras, Touati Brahim, Bouzenada and et all used the Euler time scheme combined with Galerkin spatial method, a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved [27]. And Boulaaras and Haiour dealed with the semi-implicit scheme with respect to the t-variable combined with a finite element spatial approximation of evolutionary Hamilton-Jacobi-Bellman equations with nonlinear source terms [28]. Simultaneously, the spectral method, the finite volume method, the mixed finite element method and other numerical methods have also been applied to the approximate solution of the optimal control problem [1,5,6,8,10,13,18,19] and there are references.
It is common knowledge that the hp spectral element method, which combines the advantages of the spectral method and the hp finite element method, emphasizes the use of the hp-version adaptive by simply applying the spectral method for each element, because the spectral accuracy provides very accurate approximations when smoothing the solution, with relatively few unknowns. And the spectral element method can solve complex problems, for example, a posteriori error estimates for parameter identification problem, complex nonlinear optimal control problems and etc. A lot of literatures dealt with the optimal control problem and many solutions are proposed, such as the finite element method, mixed finite element method, spectral method and so on. For a brief introduction, there has been an amount of work on constrained optimal control problems for numerically solving via the finite element methods [14][15][16][17]. Also, the mixed finite element method for the optimal control problems [2-4, 7, 23, 24]. The hp spectral element method for optimal control problems seems to have not been much studied. Therefore, it is of great significance to solve the parabolic optimal control problem by using the hp spectral element method to solve the parabolic optimal control problem is of great significance.
Let us to introduce the hp spectral element method into the parabolic optimal control problem, which is due to the adaptation of hp-version, it can choose to segment an element (h-refinement) or increase its approximate order (p-refinement). For instance, some authors have studied the hp spectral element method for the optimal control problem controlled by elliptic equations. They have derived the a posteriori error estimation of the hp spectral element approximation of the optimal control problem, in which they used L 2 (Ω)-norm to estimate the control approximation error and H 1 (Ω)-norm of the state and common state approximation error [8]. In order to emphasize the hp spectral element method and its high precision, we study the hp spectral element method for optimal control problems governed by parabolic equations comparing with [8]. First, we propose a fully discrete scheme, which uses the backward Euler scheme in time, and then uses the hp spectral approximation in space. By using the Scott-Zhang type quasi-interpolation operator, we obtain a posteriori error estimate for the approximate solution of hp spectral elements of both the state and the co-state in L 2 (0, T ; H 1 (Ω))-norm or L 2 (0, T ; L 2 (Ω))-norm and the control in L 2 (0, T ; L 2 (Ω))-norm.
The remainder of this paper is organized as follows. In Section 2, We will use the spectral approximation in space and the inverse Euler scheme in time to construct the spectral approximation scheme for parabolic optimal control problems. In Section 3, a L 2 (H 1 )−L 2 (L 2 ) posteriori error estimate is derived for the parabolic optimal control problem. In Section 4, by using two auxiliary equations, we derive a L 2 (L 2 ) − L 2 (L 2 ) posteriori error estimates for parabolic optimal control problems. In the last section, the conclusions and some possible future work are briefly given.

hp spectral element approximation
In this section, the hp spectral element method and the backward Euler discretisation approximation for distributed convex optimal control problems governed by parabolic equations is investigated as follows: where Ω is bounded open subset in R 2 with a Lipschitz boundary ∂Ω, and B is a linear continuous operator from X to L 2 (0, T ; Y ). Now K is a set defined by Obviously f, y d ∈ L 2 (0, T ; H), y 0 ∈ H 1 0 (Ω) and A(·) = (a i, j (·)) n×n ∈ (C ∞ (Ω)) n×n , such that there exists a constant c > 0 satisfying We shall take the state space W = L 2 (0, T ; Y) with Y = H 1 0 (Ω), the control space X = L 2 (0, T ; U) with U = L 2 (Ω) to fixed the idea. Then there holds a(y, ω) = Ω (A∇y) · ∇ωdx, ∀ y, ω ∈ Y, On the basis of the assumptions on A, there exist constants c > 0 and C > 0 such that Then a weak formula of the convex optimal control problem reads: where y ∈ W, u ∈ X, u(t) ∈ K subject to Apparently, the optimal control problem (2.5)-(2.7) has a unique solution (y, u), and a pair (y, u) is the solution of (2.5)-(2.7) if and only if there is a co-state p ∈ W such that the triplet (y, p, u) satisfies the following optimality conditions [12]: where B * is the adjoint operators of B. Now, let's consider the hp spectral element approximation of the parabolic optimal control problem (2.5)-(2.7). As we all know, the spectral element method proposed by Patera combines the advantages of Galerkin spectral method and finite element method by a simple application of the spectral method per element [21]. Also, it is similar to the finite element method that the domain is divided into N τ non-overlapping subdomains elements τ i , 1 ≤ i ≤ N τ : Considering the hp spectral element approximation of (2.5)-(2.7), we set T = {τ} be a local quasiuniform partitioning of Ω into non-overlapping regular element τ. We denote by theτ = (−1, 1) 2 the reference element, and let E(T ) denote all edges, and E 0 (T ) denote all edges which do not lie on the boundary ∂Ω. Each element τ can be the image of the reference elementτ under an affine map F τ :τ → τ. We write h τ := diam τ. If we assume that the triangulation is γ-shape regular, we have For γ-shape regular meshes T on the domain Ω, we associate with each element τ ∈ T a polynomial degree p τ ∈ N 0 . Moreover, these polynomial degrees {p τ } are collected into the polynomial degree vector p p p = {p τ }. Therefore, we can define the spaces of hp spectral element approximation U p p p (T , Ω), S p p p (T , Ω), S p p p 0 (T , Ω) as described below: where P p τ (τ) denotes the spaces of polynomials inτ of degree ≤ p τ in each variable, respectively. As to polynomial degree distribution p p p, similar to (2.11), we assume that the polynomial degrees of neighboring elements are comparable. As a result, there exists a constant γ > 0 such that Let K h,p p p (T , Ω) := K U p p p (T , Ω) be the space of hp spectral element approximation for the control, and S p p p 0 (T , Ω) be the space of hp spectral element approximation for the state and co-state. Then the semi-discrete hp spectral element approximation of (2.5)-(2.7) is as follows: where y hp ∈ H 1 (0, T ; S p p p 0 (T , Ω)) and y h,p p p 0 ∈ S p p p 0 (T , Ω) is a hp spectral element approximation of y 0 . It follows that the optimal control problem (2.13)-(2.15) has a unique solution (y hp , u hp ) and that a pair (y hp , u hp ) is the solution of (2.13)-(2.15) if and only if there is a co-state p hp such that the triplet (y hp , p hp , u hp ) satisfies the following optimality conditions: Now, we shall consider the fully discrete hp spectral element approximation for above semi-discrete problem by using the backward Euler scheme in time.
For i = 1, 2, · · · , M, we construct the hp spectral element approximation spaces S p p p i0 (T , Ω) ⊂ H 1 0 (Ω) (similar as S p p p 0 (T , Ω)) on the i-th time step. Similarly, we construct the hp spectral element approximation spaces K h,p p p i (T , Ω) ⊂ K (similar as K h,p p p (T , Ω)) on the i-th time step. Then the fully discrete hp spectral element approximation scheme (2.21)-(2.23) is to It follows that the optimal control problem (2.21)-(2.23) has a unique solution M, satisfies the following optimality conditions: For any function w ∈ C(0, T ; . Then the optimality conditions (2.24)-(2.28) can be restated as : In the follows, we introduce a lemma which generalize the well-known Clément-type interpolation operators of [22] to the hp context. Lemma 2.1. (Scott-Zhang type quasi-interpolation). Let T be a γ-shape regular triangulation (see (2.11)) of a domain Ω ∈ R 2 and p p p be a polynomial degree distribution which is comparable (see (2.12)). Then there exists a linear operatorΠ : H 1 0 (Ω) → S p p p 0 (T , Ω), and there exists a constant C > 0, which depends only on γ, such that for every u ∈ H 1 0 (Ω) and all elements τ ∈ T and all edges e ∈ E(τ),

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where h e is the length of the edge e and p e = max(p τ , p τ ), where τ, τ are elements sharing the edge e, ω τ , ω e are patches covering τ and e with a few layers, respectively. See [20] for more details on ω τ and ω e .
3. A L 2 (H 1 ) − L 2 (L 2 ) posteriori error estimates In this section, we shall derive a L 2 (H 1 ) − L 2 (L 2 ) posteriori error estimates for the hp spectral approximation of the optimal control problem governed by parabolic equations. Set According to [11], it can be shown that where p(U hp ) is the solution of the auxiliary equations: where p(U hp ) is defined by (3.4)-(3.7) and Proof. According to the definition of norm · L 2 (0,T ;L 2 (Ω)) , there are (3.10) Moreover, note that U hp ∈ K h,p p p (T , Ω) ⊂ K. It follows from (2.10) that  We first estimate I 1 here. It is clear that for any sufficiently small positive number δ. Then for I 2 form (3.12), we obtain for any sufficiently small positive number δ. Thus, applying Eqs (3.12) and (3.14) gives the estimate This proves (3.8). where Proof. Let e p = p(U hp )−P hp and e p I =Πe p , whereΠ be the Scott-Zhang type quasi-interpolator defined as in Lemma 2.1. Note that (p(U hp ) − P hp )(x, T ) = 0, hence Then there holds the estimate:  Employing Lemma 2.1, the first estimate J 1 becomes as where δ is an arbitrary positive number, C(δ) is a constant dependent on δ. Similarly,  Similarly, let e y = y(U hp ) − Y hp , e y I =Πe y , whereΠ be the Scott-Zhang type quasi-interpolator defined as in Lemma 2.1. Note that .

Conclusions
In this paper, a completely discrete scheme is proposed, which uses the inverse Euler scheme in time and the hp spectral element approximation in space to solve the parabolic optimal control problem (2.5)-(2.7). By using the Scott-Zhang type quasi-interpolation operator, we obtain a L 2 (H 1 ) − L 2 (L 2 ) posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable. And two auxiliary equations are introduced, we derive a L 2 (L 2 ) − L 2 (L 2 ) posteriori error estimates for parabolic optimal control problems.
A fully discrete scheme is proposed for improve the accuracy and construct an adaptive finite element algorithm in this paper, which uses the inverse Euler scheme in time and the hp spectral element approximation in space to solve the parabolic optimal control problem (2.5)-(2.7). Our main results as follows: (1) We extend the elliptic optimal control problem to the parabolic optimal control problem, by using the Scott-Zhang type quasi-interpolation operator and get two kind of posteriori error estimates for parabolic optimal control problems. (2) For the general elliptic problem, only a L 2 (H 1 ) − L 2 (L 2 ) posteriori error estimate of the elliptic optimal control problem is deduced, however, we derive a L 2 (H 1 ) − L 2 (L 2 ) and L 2 (L 2 ) − L 2 (L 2 ) posteriori error estimates for parabolic optimal control problem. (3) The two kinds of error estimates we obtained are very useful for us to construct adaptive finite element approximation.
These results and the techniques used can be generalized to optimal control problems with more general objective functions. Furthermore, we well consider the hp spectral element approximation for a posteriori error estimates of nonlinear optimal control problems, nonlinear parabolic optimal control problems and hyperbolic optimal control problems and etc.