Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems

We study a posteriori error analysis of linear-quadratic boundary control problems under bilateral box constraints on the control which acts through a Neumann type boundary condition. We adopt the hybridizable discontinuous Galerkin method as discretization technique, and the flux variables, the scalar variables and the boundary trace variables are all approximated by polynomials of degree k. As for the control variable, it is discretized by the variational discretization concept. Then an efficient and reliable a posteriori error estimator is introduced, and we prove that the error estimator provides an upper bound and a lower bound for the error. Finally, numerical results are presented to illustrate the performance of the obtained a posteriori error estimator.

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1. Introduction

Many optimization processes in science and engineering lead to optimal control problems where the sought state is a solution of a partial differential equation. The complexity of such problem needs special care in order to obtain efficient numerical approximations for the optimization problem. One particular method is adaptive finite element method, which can reduces the computational cost and boosts the accuracy of the numerical solutions by locally refining the meshes around the singularity.

Although the adaptive finite element method has become a popular approach for numerical solutions of partial differential equations since the work of Babuška and Rheinboldt [1], it has only quiet recently become popular for constrained optimal control problems. The pioneer work concerning a posteriori error analysis for distributed optimal control problems is published by Liu and Yan [25] for residual-type error estimators and Becker, Kapp, and Rannacher [2] for goal-oriented error estimators. Here, we further refer readers to [20, 27, 31, 32, 33] for residual-type estimators and [3, 21] for goal-oriented approach. Recently, in order to guarantee the performance of the a posteriori error estimator theoretically, many scholars have tried to prove the convergence of an adaptive finite element algorithm for distributed optimal control problems in [14, 15, 23, 28].

Compared to distributed optimal control problems, there exists limited work on a posteriori error analysis for boundary optimal control problems. In [26], the convex Neumann boundary control problem was considered on polygonal or Lipschitz piecewise domain. Then a residual-type a posteriori error estimator was introduced, and the authors proved that the estimator provided an upper bound for the errors in the state and the control. In [19], by introducing a Lagrange multiplier, the authors derived an efficient and reliable residual-type a posteriori error estimator for Neumann boundary control problems on polygonal domain. In [22], Kohls, Rösch and Siebert derived a unifying framework for the a posteriori error analysis of control constrained linear-quadratic optimal control problems for the full and variational discretizations. In [4], Benner and Yücel investigated symmetric interior penalty Galerkin methods for Neumann boundary control problems with an extra coefficient in cost functional. By invoking a Lagrange multiplier associated with the control constraints, an efficient and reliable residual-type a posteriori error estimator was obtained for the errors in the state, adjoint, control and co-control. As for Dirichlet boundary control problems, we just mention [8, 16] and references therein for more details on a posteriori error analysis.

Recently, the hybridizable discontinuous Galerkin (HDG) methods [9]

, which keep the advantages of discontinuous Galerkin (DG) methods and result in a system with significantly reduced degrees of freedom, have been proposed for convection diffusion problem

[13], interface problem [6], flow problem[29], optimal control problem [5, 17], and so on. In [10, 11, 12], Cockburn and Zhang studied HDG methods for second order elliptic problems, and an a posteriori error estimator with postprocessing solutions was obtained. To the best of our knowledge, there exists no work on residual-type a posteriori error analysis of HDG methods for boundary control problems.

In this paper, we investigate a posteriori error analysis of Neumann optimal control problems under bilateral box constraints on the control. The HDG method is used as discretization technique, and the flux variables, the scalar variables and the boundary trace variables are discretized by polynomials of degree . As for the control variable, we adopt the variational discretization concept proposed by Hinze in [18] for approximation. Then an efficient and reliable residual-type a posteriori error estimator without any postprocessing solutions is introduced, and we prove that the error estimator provides not only an upper bound but also a lower bound up to data oscillations for the errors. Finally, numerical experiments are presented to validate the performance of the obtained estimator.

The remainder of the paper is arranged as follows: In Section 2 we introduce the model problem and the associated optimality system. In Section 3 the discrete optimality system is given, and we prove that the discrete scheme has a unique solution. Then we prove the reliability and efficiency of the error estimator in Section 4 and Section 5 respectively. Numerical experiments are presented in Section 6 to validate the performance of the obtained estimator. Finally, some conclusions are provided in Section 7.

Throughout this paper, let with or without subscript be a generic positive constant independent of the mesh size. For ease of exposition, we denote by .

2. The Neumann boundary control problem

Let be a polygonal or polyhedral domain with boundary . Before we introduce the model problem, let us summarize some notation. For bounded and open set or , we denote the usual Sobolev spaces by with norm and seminorm . The Hilbertian Sobolev spaces are abbreviated by with norm and seminorm . For , coincides with , and the inner product is denoted by for and for . Furthermore, we define .

Based on the domain , we consider the following Neumann boundary control problem

subject to the elliptic equations

 −Δy+y =fin Ω, (2a) ∇y⋅n =u+gon ∂Ω, (2b)

where the regularization parameter is a positive constant, , , , n

is the unit vector normal to the boundary

. The set of constraints is given by

where and are assumed to be constant, and that .

From [24], we know that the Neumann boundary control problem (1)-(2) admits a unique solution , and there exists an adjoint-state such that

 −Δy+y =fin Ω, (3a) ∇y⋅n =u+gon ∂Ω, (3b) −Δz+z =y−ydin Ω, (3c) ∇z⋅n =0on ∂Ω, (3d) ⟨αu+z,v−u⟩∂Ω ≥0∀v∈Uad. (3e)

Moreover, the variational inequality (3e) is equivalent to the projection formula

where is the -projection onto . Then let and , the optimality system (3) can be rewritten in a mixed form as follows:

 p+∇y =0in Ω, (5a) ∇⋅p+y =fin Ω, (5b) −p⋅n =u+gon ∂Ω, (5c) q+∇z =0in Ω, (5d) ∇⋅q+z =y−ydin Ω, (5e) −q⋅n =0on ∂Ω, (5f) ⟨αu+z,v−u⟩∂Ω ≥0∀v∈Uad. (5g)

3. The HDG discretization

Let be a conforming and shape regular partition of the domain . For each , we denote the set of its faces. Then we define . Denote the set of all interior faces of and the set of all boundary faces of . Then we define . For any and , and denote the diameters of the element and the face respectively. Furthermore, we define the mesh-dependent inner product by

 (w,v)Th=∑K∈Th(w,v)K,⟨w,v⟩∂Th=∑K∈Th⟨w,v⟩∂K.

For vector-valued functions, the notations are similarly defined by the dot product.

Based on the partition , we define the discontinuous finite element spaces for the flux variables, the scalar variables and the boundary trace variables as following

 Vkh={v∈(L2(Ω))d:v|K∈(Pk(K))d, ∀K∈Th}, Wkh={w∈L2(Ω):w|K∈Pk(K), ∀K∈Th}, Mkh={μ∈L2(Eh):μ|F∈Pk(F), ∀F∈Eh},

where is the set of polynomials of degree no larger than on the domain . In this paper, we adopt the variational concept proposed by Hinze [18] for the control variable, which suggests to approximate the state equation but not the control variable. Therefore the control variable will be implicitly discretized by formula (4). Then the HDG scheme of the system (5) reads as follows: Find , and such that

 (ph,r1)Th−(yh,∇⋅r1)Th+⟨ˆyh,r1⋅n⟩∂Th =0, (6a) −(ph,∇w1)Th+(yh,w1)Th+⟨ˆph⋅n,w1⟩∂Th =(f,w1)Th, (6b) ⟨ˆph⋅n,μ1⟩∂Th∖∂Ω =0, (6c) −⟨ˆph⋅n,μ1⟩∂Ω =⟨uh+g,μ1⟩∂Ω, (6d) (qh,r2)Th−(zh,∇⋅r2)Th+⟨ˆzh,r2⋅n⟩∂Th =0, (6e) −(qh,∇w2)Th+(zh,w2)Th+⟨ˆqh⋅n,w2⟩∂Th =(yh−yd,w2)Th, (6f) ⟨ˆqh⋅n,μ2⟩∂Th∖∂Ω =0, (6g) −⟨ˆqh⋅n,μ2⟩∂Ω =0, (6h) ⟨αuh+ˆzh,v−uh⟩∂Ω ≥0, (6i)

for any , and . Similarly, we know that the inequality (6i) is equivalent to the following projection formula

Here the normal component of numerical fluxes and is defined as

 ˆph⋅n=p⋅n+τ1(yh−ˆyh)on ∂Th, ˆqh⋅n=q⋅n+τ2(zh−ˆzh)on ∂Th,

for stabilization parameters and .

For ease of exposition, we define operators by

 B(r1,w1,μ1;r2,w2,μ2;τ) = (r1,r2)Th−(w1,∇⋅r2)Th+⟨μ1,r2⋅n⟩∂Th +(∇⋅r1,w2)Th+(w1,w2)Th +⟨τ(w1−μ1),w2⟩∂Th−⟨r1⋅n+τ(w1−μ1),μ2⟩∂Th,

Then the HDG scheme (6) can be rewritten according to the operator : Find , and such that

 B(ph,yh,ˆyh;r1,w1,μ1;τ1) =(f,w1)Th+⟨uh+g,μ1⟩∂Ω, (7a) B(qh,zh,ˆzh;r2,w2,μ2;τ2) =(yh−yd,w2)Th, (7b) ⟨αuh+ˆzh,v−uh⟩∂Ω ≥0, (7c)

for any , and .

Theorem 3.1.

We assume that on and . Then the system (7) has a unique solution.

Proof.

Since the system (7) is finite dimensional, we only need to prove that the system (7) just has the zero solution for the case of . Let in (7a) and in (7b), we have

 0= B(ph,yh,ˆyh;qh,−zh,−ˆzh;τ1)+B(qh,zh,ˆzh;−ph,yh,ˆyh;τ2) = (yh,yh)Th−⟨uh,ˆzh⟩∂Ω≥(yh,yh)Th+α⟨uh,uh⟩∂Ω,

from (7c) and the assumption . Hence and . Furthermore, let in (7a) and in (7b), we have

 0= (ph,ph)Th+(yh,yh)Th+⟨τ1(yh−ˆyh),yh−ˆyh⟩∂Th, 0= (qh,qh)Th+(zh,zh)Th+⟨τ2(zh−ˆzh),zh−ˆzh⟩∂Th

Therefore , , , and . Then we conclude the proof. ∎

4. The residual-type a posteriori error estimator

4.1. Auxiliary results

Before we start to prove a posteriori error estimator for the model problem, we first provide some auxiliary results that will play an important role in the proof.

For each element and face , we denote and the -projections onto and for the nonnegative integer . Then, from [6] we have the following error estimates

Lemma 4.1.

For any and , we have

 ∥Πojv∥0,K≤ ∥v∥0,K∀v∈L2(K), ∥Π∂jv∥0,F≤ ∥v∥0,F∀v∈L2(F), ∥v−Πo0v∥0,K≲ hK∥∇v∥0,K∀v∈H1(K), ∥v−Πo0v∥0,∂K≲ h1/2K∥∇v∥0,K∀v∈H1(K).

We conclude this subsection by introducing a lemma that has been proved in [7].

Lemma 4.2.

Let be a face of the element , the unit vector normal to , and . Assume that is a given function in and . For any , we have

 ⟨∇v⋅nF,wh⟩F≲h−1/2F∥wh∥0,F(∥∇v∥0,K+hK∥Δv∥0,K).

4.2. Reliability of the error estimator

We begin this section by defining error estimators for each in the following

 ηs,K,1= ∥ph+∇yh∥0,K,ηas,K,1=∥qh+∇zh∥0,K, ηs,K,2= hK∥f−∇⋅ph−yh∥0,K,ηas,K,2=hK∥yh−yd−∇⋅qh−zh∥0,K, ηs,∂K= h−1/2K∥yh−ˆyh∥0,∂K,ηas,∂K=h−1/2K∥zh−ˆzh∥0,∂K.

Furthermore, we define

 η2s=∑K∈Th{η2s,K,1+η2s,K,2+η2s,∂K}, η2as=∑K∈Th{η2as,K,1+η2as,K,2+η2as,∂K}.

Next, we consider the following auxiliary problem: Find and such that

 p(uh)+∇y(uh) =0in Ω, (8a) ∇⋅p(uh)+y(uh) =fin Ω, (8b) −p(uh)⋅n =uh+gon ∂Ω, (8c) q(uh)+∇z(uh) =0in Ω, (8d) ∇⋅q(uh)+z(uh) =y(uh)−ydin Ω, (8e) −q(uh)⋅n =0on Ω. (8f)

Now the error can be bounded by and .

Lemma 4.3.

Let and be the solutions of problems (5) and (7) respectively. Moreover let as defined above. Then the following error estimate holds

 ∥u−uh∥0,∂Ω+∥p−p(uh)∥0,Ω+∥y−y(uh)∥0,Ω+∥q−q(uh)∥0,Ω +∥z−z(uh)∥0,Ω≲{∑K∈Thη2as,∂K}1/2+∥zh−z(uh)∥1,Ω
Proof.

From (5), (8) and integration by parts to yield

 ⟨z−z(uh),u−uh⟩∂Ω=∥y−y(uh)∥20,Ω. (9)

Obviously, is the solution of system (5a)- (5c) with and , and is the solution of system (5d)-(5f) with . Therefore we have

 ∥p−p(uh)∥0,Ω+∥y−y(uh)∥0,Ω ≲∥u−uh∥0,∂Ω, (10) ∥q−q(uh)∥0,Ω+∥z−z(uh)∥0,Ω ≲∥y−y(uh)∥0,Ω, (11)

by the trace theorem. From (5g), (7c) and (9), we obtain

 α∥u−uh∥20,∂Ω≤ ⟨ˆzh−z(uh),u−uh⟩∂Ω ≤ ∥zh−z(uh)∥1,Ω∥u−uh∥0,∂Ω +⟨zh−ˆzh,(p−p(uh))⋅n⟩∂Ω ≲ ∥zh−z(uh)∥1,Ω∥u−uh∥0,∂Ω (12) +∥y−y(uh)∥0,Ω),

by the trace theorem and Lemma 4.2. Then we can conclude the proof by combining (10)-(12). ∎

Lemma 4.4.

Let and be the solutions of problems (8) and (7), then the following error estimates hold

 ∥∇(yh−y(uh))∥0,K≤ηs,K,1+∥ph−p(uh)∥0,K, (13) ∥∇(zh−z(uh))∥0,K≤ηas,K,1+∥qh−q(uh)∥0,K, (14)

for each .

Proof.

Since and in each , we can obtain the error estimates (13) and (14) directly by the triangle inequality. ∎

Now we are ready to prove a posteriori error estimators for and .

Lemma 4.5.

Let and be the solutions of problems (8) and (7), then we have

 ∥ph−p(uh)∥0,Ω+∥yh−y(uh)∥0,Ω+∥τ1/21(yh−ˆyh)∥0,∂Th≲ηs, (15) ∥qh−q(uh)∥0,Ω+∥zh−z(uh)∥0,Ω+∥τ1/22(zh−ˆzh)∥0,∂Th≲ηs+ηas, (16)

for on each for all , where

 ∥⋅∥20,∂Th=∑K∈Th∥⋅∥20,∂K.
Proof.

According to the definition of the operator to infer that

 B(p(uh)−ph,y(uh)−yh,y(uh)−ˆyh;r,w,μ;τ1)=0,

for any . Then from the above equality and the definition of the operator , we have

 ∥ph−p(uh)∥20,Ω+∥yh−y(uh)∥20,Ω+∥τ1/21(yh−ˆyh)∥20,∂Th = B(p(uh)−ph,y(uh)−yh,y(uh)−ˆyh;p(uh)−ph,y(uh)−yh,y(uh)−ˆyh;τ1) = B(p(uh)−ph,y(uh)−yh,y(uh)−ˆyh;δp,δy,δˆy;τ1),

where , , and for any . By integration by parts we yield

 B(p(uh)−ph,y(uh)−yh,y(uh)−ˆyh;δp,δy,δˆy;τ1) = −(ph+∇yh,δp)Th+(f−∇⋅ph−yh,δy)Th +⟨yh−ˆyh,δp⋅n⟩∂Th+⟨τ1(ˆyh−yh),δy⟩∂Th −⟨(p(uh)−ph)⋅n+τ1(ˆyh−yh),δˆy⟩∂T<