An HDG Method for Tangential Boundary Control of Stokes Equations I: High Regularity
We propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a tangential Dirichlet boundary control problem for the Stokes equations with an penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, under certain assumptions on the domain and the target state, we prove a priori error estimates for the control for the HDG method. In the 2D case, our theoretical convergence rate for the control is superlinear and optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.
Let () be a Lipschitz polyhedral domain with boundary . For a given target state , we consider the following unconstrained Dirichlet boundary control problem for the Stokes equations:
where is the control space and is a fixed constant.
Control of fluid flows modeled by the Stokes or Navier-Stokes equations is an important and active area of interest. After the pioneering works by Glowinski and Lions  and Gunzburger [33, 36, 25, 38, 37, 34, 35], many important developments have been made both theoretically and computationally in the past decades. For an extensive body of literature devoted to this subject we refer to, e.g., [72, 39, 55, 3, 40, 9, 4, 7, 24, 23, 70, 62, 65, 71, 50, 11] and the references therein. Despite the large amount of existing work on numerical methods for fluid flow control problems, we are not aware of any contributions to the analysis and approximation of the tangential Stokes Dirichlet boundary control problem. Work on this problem is an essential step towards the analysis and approximation of similar Dirichlet boundary control problems for the Navier-Stokes equations and other fluid flow models.
In this work, we focus on the case where the control acts tangentially along the boundary through a Dirichlet boundary condition. This scenario has broad applications to optimal mixing and heat transfer problems. Omari and Guer in  conducted a numerical study of the effect of wall rotation on the enhancement of heat transport in the whole fluid domain. Gouillart et al. in [29, 30, 31, 68] studied in detail this crucial effect of moving wall on the mixing efficiency for the homogenization of concentration in a 2D closed flow environment. These problems naturally lead to the study of tangential boundary control and optimization of fluid flows. Recently, Hu and Wu in [43, 41, 42, 49] provided rigorous mathematical approaches for optimal mixing and heat transfer via an active control of Stokes and Navier-Stokes flows through Navier slip boundary conditions. Other tangential boundary control problems for fluid flows have been considered by Barbu, Lasiecka and Triggiani [5, 6, 53, 54] and Osses . However, the authors are not aware of any existing work on approximation and numerical analysis for these problems.
Discontinuous Galerkin (DG) methods are widely used for fluid flow problems, since they can capture shocks and large gradients in solutions. However, most existing DG methods are commonly considered to have a major drawback: the memory requirement and computational cost of DG methods are typically much larger than the standard finite element method.
Hybridizable discontinuous Galerkin (HDG) methods were proposed by Cockburn et al. in  as an improvement of traditional DG methods. The HDG methods are based on a mixed formulation and utilize a numerical flux and a numerical trace to approximate the flux and the trace of the solution. The approximate flux and solution variables can be eliminated element-by-element. This process leads to a global equation for the approximate boundary traces only. As a result, HDG methods have significantly less globally coupled unknowns, memory requirement, and computational cost compared to other DG methods. Furthermore, HDG methods have been successfully applied to flow problems [14, 64, 17, 18, 58, 69], distributed optimal control problems [73, 46, 48], and Dirichlet boundary control problems [47, 45, 44].
For the Stokes tangential Dirichlet boundary control problem considered here, the Dirichlet boundary data takes the form , where is the control and is the unit tangential vector to the boundary. Formally, the optimal control and the optimal state minimizing the cost functional satisfy the optimality system \cref@addtoresetequationparentequation
We use an HDG method to approximate the solution of a mixed formulation of this optimality system. To do this, we first analyze the control problem for 2D convex polygonal domains in Section 2. We give precise meaning to the state equation (1.3a) for Dirichlet boundary data in , and prove well-posedness and regularity results for the optimality system (1.3). The theoretical results for this problem share some similarities to results for Dirichlet boundary control of the Poisson equation on a 2D convex polygonal domain ; however, there are new components to the analysis due to the mixed formulation and the regularity results for Stokes equations on polygonal domains . An interesting feature of our theoretical results is that the optimal control has higher local regularity (on each boundary edge) than global regularity (on the entire boundary ). This higher local regularity for the optimal control is not present for Dirichlet boundary control of the Poisson equation; furthermore, as we discuss below, this phenomenon may have an effect on the convergence rates of the approximate solution.
For the HDG method, we use polynomials of degree to approximate the velocity and dual velocity , and polynomials of degree for the fluxes and , pressure and dual pressure . Moreover, we also use polynomials of degree to approximate the numerical trace of the velocity and dual velocity on the edges of the spatial mesh, which are the only globally coupled unknowns. We describe the HDG method and its implementation in detail in Section 3.
In Section 4, we prove a superlinear rate of convergence for the control in 2D under certain assumptions on the largest angle of the convex polygonal domain and the smoothness of the desired state . Similar superlinear convergence results for Dirichlet boundary control of the Poisson equation have been obtained in [2, 13, 28, 47, 45, 44]. To give a specific example of our results, for a rectangular 2D domain, , and , we obtain the following a priori error bounds for the velocity , adjoint velocity , their fluxes and , pressure and dual pressure and the optimal control :
for any . The rate of convergence for the control is optimal in the sense of the maximal global regularity of the control . However, the numerical results presented in Section 5 show higher convergence rates than the rates predicted by our numerical analysis; this phenomenon might be caused by the higher local regularity of the optimal control mentioned above. The numerical convergence rates observed here are different than typical numerical results for Dirichlet boundary control of the Poisson equation.
We emphasize that the HDG method in this work is usually considered to be a superconvergent method. Specifically, if polynomials of degree are used for the numerical trace and the solution of the PDEs is smooth enough, then error estimates can be obtained for the state variable; see, e.g., [63, 64, 48]. Hence, from the viewpoint of globally coupled degrees of freedom, this method achieves superconvergence for the scalar variable. For Dirichlet boundary control problems, to obtain the superlinear convergence rate, one usually needs a superconvergence mesh or higher order elements for the standard finite element method, see, e.g., [2, 22]. However, the HDG method considered here achieves the superlinear convergence rate without any special considerations.
In the second part of this work , we complete the numerical analysis of this HDG method for low regularity solutions of the optimality system. Specifically, we remove the assumptions made here on the boundary angles and the regularity of the target state. In this more general scenario, the flux and the pressure may not have a well-defined boundary trace and we perform a nonstandard HDG error analysis based on the techniques from  to establish the low regularity convergence results.
2 Analysis of the Tangential Dirichlet Control Problem
To begin, we set notation and prove some fundamental results concerning the optimality system for the control problem in the 2D case. In this section, we assume the forcing in the Stokes equations (1.2) is equal to zero; if the forcing is nonzero, then it can be eliminated using the technique in [1, pg. 3623].
Throughout the paper, we use the standard notation to denote the Sobolev space with norm and seminorm . Set , and . We denote the -inner products on , , and by
Define the space as
Also, we define as
Let denote the inner product in and let denote the duality product between and for , where denotes the space of traces of for . (For it is the subspace of satisfying certain compatibility conditions on the corners; see [32, Theorem 22.214.171.124]. For , this definition would lead to ambiguities.) Following [66, Section 2.1] we introduce the spaces
For , is the dual space of . For , is the dual space of and for , is the dual space of .
Consider a target state , where is a function space that will be specified later, and a Tykhonov regularization parameter . Consider also a space . We are interested in the optimal control problem
where is the unique solution in the transposition sense of the Stokes system (see Definition 2.3 below)
Different choices of the spaces and appear in the related literature for Dirichlet control of Stokes and Navier-Stokes equations. In the early reference , and . The natural space for the controls to obtain a variational solution of the state equation (2.1) is . This is the choice in . In that work, nevertheless, the Tykhonov regularization is done in the norm of . To prove existence of solution, the tracking is done in the space . In the reference , the authors work in a smooth domain with and . This choice involves a harder analysis, but leads to an optimality system easier to handle. In polygonal domains, this approach leads to optimal controls that are discontinuous at the corners.
We assume throughout this work that the tracking term for the state is measured in the norm. We investigate the case , which corresponds to tangential boundary control; see [5, 6]. We first precisely define the concept of solution for Dirichlet data in , prove precise regularity results, and use them to introduce a mixed formulation of the problem adequate for HDG methods.
2.1 Regularity results
The definition of very weak solution for data in was introduced in [19, Appendix A] and is valid in convex polygonal domains; see also  and [50, Definition 2.1] for a similar definition for the Navier-Stokes equations and smooth domains. Also in smooth domains, very weak solutions can be defined for data in ; see [66, Appendix A]. We will show how to extend the concept to problems posed on nonconvex polygonal domains for data in for some negative . We will also prove that the optimal regularity expected for the solution can be achieved. In  a similar result is provided for convex polygonal domains, but only suboptimal regularity for all is proved. We obtain a result comparable to the one given in [66, Appendix A] for smooth domains.
Let denote the greatest interior angle of . Following [20, Theorem 5.5], we know there exists a number that gives the maximal regularity for the problem (2.2). This means for very smooth and satisfying the compatibility condition we can only expect that the variational solution of the compressible Stokes problem
satisfies and for . This singular exponent is the smallest real part of all of the roots of the equation
and satisfies that is strictly decreasing, if , and if .
Let us denote
If and such that , [20, Theorem 5.5(a)] states that for all
the solution of (2.2) satisfies and . Moreover, we have that
Notice that although the pressure is uniquely determined as a function with the condition , the norm must be taken modulo constant functions. Another remarkable fact is that this result holds for . This means, in particular, that in nonconvex domains one cannot expect in general to have regularity of .
To obtain this regularity, an additional condition must be made on the divergence of . If, e.g., , , then it follows from [20, Theorem 5.5(c)] or early reference  for convex polygonal domains that the result also holds for . In particular, in a convex domain we have .
For later reference, we state the regularity result for the case in a convex domain:
[20, Theorem 5.5(b)] Suppose is convex and consider for some . Then, the unique solution of the incompressible Stokes problem
satisfies , and
Although we will pose our control problem for data in , the precise regularity results for the state equation will follow by interpolation; therefore we need a definition of very weak solution for data in for . The elements of this space do not always satisfy a condition analogous to , i.e., we may have , and it is necessary to take this into account to define a solution in the transposition sense. Following [66, Eq. (2.2)], we define for , , the constant
This constant satisfies the relation
Using this fact, usual trace theory and (2.4), we have that for
The following definition makes sense:
Notice that if , equation (2.8) can be written as
The definition follows integrating by parts twice the equation and once the null divergence condition. It can be written as two separate equations, one tested with and the other one with , as in  or , or as single equation, cf.  or .
Next, we state a regularity result analogous to [66, Corollary A.1]. In that reference, a smooth domain is taken into consideration and the limit cases and can be achieved; however, this is not possible for polygonal domains so the cited result cannot be directly applied.
Suppose for . Then the solution of (2.1) satisfies
Moreover, the control-to-state mapping is continuous from to .
The proof follows by interpolation. The technique of proof is the same as in [66, Appendix A] or [1, Section 2], so we will just give a sketch of the proof and check some of the details that are different from those references.
We first do the regular case. Suppose . From the definition of we know that there exists such that the boundary trace of equals . So we have that and . By linearity, we have that and , where is the variational solution of (2.2) for data . From [20, Theorem 5.5(a)], and using that , we have then that and , and the result follows in a straightforward way.
Existence follows by density arguments. Take , which is dense in . Notice that and and hence is dense in and is dense in . Therefore, we can consider
Finally, the case follows by interpolation. ∎
If , then the very weak solution and the variational solution are the same.
Next, we have to give some meaning to the mixed form. The main problem is that for data in , , the gradient of the state is not a function in .
We start with the regular compressible Stokes problem. Consider and such that and denote and the (variational) solution of (2.2). If we denote , we have that the triplet is the unique solution of the weak formulation
Next we give a mixed formulation of problem (2.8) for Dirichlet data for .
The above definition simply incorporates an adequate definition for the gradient to the transposition solution defined in Definition 2.3. Nevertheless, this formulation is still not appropriate to use together with (2.10)–(2.13) in the context of hybridizable discontinuous Galerkin methods. Taking advantage of the regularity results stated in Theorem 2.4, we have if and if .
So we have that if and , then there exists a unique solution of the problem \cref@addtoresetequationparentequation
Taking all this into account we can summarize our results in the following theorem.
For every , there exist a unique solution of \cref@addtoresetequationparentequation
|for all .|
Moreover, if , , then . Finally, the control-to-state mapping is continuous from to for .
2.2 Well posedness and regularity of the tangential control problem
It is clear that and there is no ambiguity in denoting by the elements of . Hence the control-to-state mapping is continuous from to , and there exists a unique solution of the control problem
where is the solution of the state equation \cref@addtoresetequationparentequation
|for all .|
Notice that (2.18a), (2.18b), (2.18c) is the weak formulation (2.17a), (2.17b), (2.17c) obtained in Theorem 2.7 for the Stokes problem (1.2) with Dirichlet datum , where we have used that for any pair of functions in .
If is nonconvex, then the regularity of the optimal solution is limited mainly by the singular exponent related to the greatest nonconvex angle, and we would find discontinuous optimal controls that would lead to pressures and gradients of the state that are not functions. For convex domains, the regularity is better and we can write integrals instead of duality products. The main consequence is that we can formulate an HDG approximation method for the optimality system.
Suppose is a convex polygonal domain and . Let be the solution of problem . Then
for all and there exists
for all such that \cref@addtoresetequationparentequation