The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations

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Abstract

We study the effect of the Galerkin/Least-Squares (GLS) stabilization on the finite element discretization of optimal control problems governed by the linear Oseen equations. Control is applied in the form of suction or blowing on part of the boundary. Two ways of including the GLS stabilization into the discretization of the optimal control problem are discussed. In one case the optimal control problem is first discretized and the resulting finite-dimensional problem is then solved. In the other case, the optimality conditions are first formulated on the differential equation level and are then discretized. Both approaches lead to different discrete adjoint equations and, depending on the choice of the stabilization parameters and grid size, may significantly affect the computed control. The effect of the order in which the discretization is applied and the choice of the stabilization parameters are illustrated using two test problems. The cause of the differences in the computed controls are explored numerically. Diagnostics are introduced that may guide the selection of sensible stabilization parameters.

Introduction

Stabilized finite element methods (FEMs) are frequently and successfully used to discretize advection-dominated partial differential equations (PDEs) [1], or to circumvent the compatibility conditions restricting the choice of interpolation function spaces [2]. One question that arises in the application of these methods is the choice of the stabilization parameter. For many PDE model problems this issue has been studied analytically and numerically [3], [4]. However, the impact on the choice of the stabilization parameter when the stabilized FEM is used in the context of optimal control is not well studied. It is not true that a scheme which gives good approximations to the PDE solution for a fixed simulation is also guaranteed to provide good approximations in the context of optimal control problems. Solutions of optimal control problems are characterized by the original governing PDE as well as another PDE—the so-called adjoint equation. Depending on the approach chosen, a discretization of the original governing PDE may imply a discretization scheme for the adjoint equation with poor approximation properties. The present paper investigates this issue for a class of boundary control problems discretized using the Galerkin/least-squares (GLS) approach [5].

The purpose of this paper is to numerically investigate the influence of stabilization parameters on the solution of optimal control problems related to the optimal boundary control of the Navier–Stokes equations. We will demonstrate that the way the stabilization is introduced into the formulation of the optimal control problem, and the choice of stabilization parameters, can have a significant impact on the computed control. Furthermore, we point out some diagnostic tools that may help to assess the quality of the computed control and guide the choice of stabilization parameters.

We consider a class of linear quadratic optimal control problems governed by the Oseen equations, and we study the effect of the GLS-stabilized finite element method on the computed control. The linear Oseen equations were chosen as the governing state equation instead of the nonlinear Navier–Stokes equations, because the resulting optimal control problem has a unique solution and the first-order optimality conditions are necessary and sufficient. Optimal control problems governed by the nonlinear Navier–Stokes equations may have local solutions and their solution requires iterative methods. Since it is difficult to separate the possible effects of local solutions and iterative solvers on the computed optimal control from the effects of the stabilization on the computed optimal control, we have chosen the Oseen equations. However, the linear quadratic optimal control problems studied in this paper are closely related to the subproblems that arise in the solution of boundary control problems governed by the Navier–Stokes equations using Newton or sequential quadratic programming methods (see, e.g., [6], [7]). Consequently, the results reported in this paper are also relevant for the optimal control of Navier–Stokes flow.

A related paper [8] studies the effect of the streamline-upwind/Petrov–Galerkin stabilized finite element method on the numerical solution of linear quadratic distributed optimal control problems governed by an advection-diffusion equation. That paper contains both analytical results that describe the convergence of the computed control (state/adjoint) to the exact control (state/adjoint) as the grid is refined, as well as numerical convergence studies for the simpler optimal control problem. The analytical results in [8] are accurate asymptotically, but do not describe the numerical behavior well for first-order finite elements on relatively coarse, but for practical purposes acceptable grids. Therefore, the present paper focuses on a numerical study. We expect that most of the analytical results in [8] can be extended to the problems and the GLS stabilization method considered in this paper. However, such a theoretical treatment is beyond the scope of this paper.

We consider a viscous incompressible fluid occupying a bounded region ΩRnsd, where nsd is the number of space dimensions. The symbols u and p represent the velocity and pressure. The boundary Ω of Ω is decomposed into three disjoint segments Γh, Γd and Γg. Suction and blowing control is applied on Γg. We are interested in the solution of the following problem:

minimizeJ(u,g)=J1(u)+β22Γg|g|2dx

subject to(a·)u-·σ(u,p)=finΩ,·u=0inΩ,σ(u,p)·n=honΓh,u=donΓd,u=gonΓg.The above equations represent the momentum and continuity equations subject to Neumann- and Dirichlet-type boundary conditions. In (2), the functions f, h and d are given, and the control g has to be determined as the solution of the optimization problem. The regularization parameter β2>0 is given. The stress tensor σ is given byσ(u,p)=-pI+2μϵ(u),ϵ(u)=12(u+uT),where μ is dynamic viscosity and I denotes the identity tensor. We consider two objective functions J(u,g) withJ1(u)=12Ω*|×u|2dxandJ1(u)=2μΩ*ϵ(u):ϵ(u)dx,where Ω*Ω. These objective functions—vorticity magnitude and dissipation due to viscous stress—represent design goals encountered in biomedical engineering. For example, artificial flow devices such as blood pumps induce non-physiological levels of shear stress, leading to blood damage [9] or thrombus formation (clotting) [10].

We give a precise statement of the optimal control problem in Section 2.1, as well as establish existence of a unique solution and formulate the first-order necessary and sufficient optimality conditions. In Section 2.2 we recall the GLS-stabilized finite element method for the discretization of the state equation. The need for stabilization in (2) is twofold. Firstly, GLS removes the under-diffusivity of Galerkin methods which leads to oscillatory solutions in advection-dominated flows. Secondly, GLS circumvents the Babuška–Brezzi (inf–sup) condition, and allows the use of equal-order finite elements for velocities and pressure.

For the numerical solution of linear quadratic optimal control problems, there are at least two approaches. In the first approach, called discretize-then-optimize (do), the objective function and governing PDEs are discretized. In our case, the Oseen equation is discretized using the GLS-stabilized finite element formulation. This leads to a large-scale finite-dimensional quadratic programming problem, or equivalently, to a large-scale linear system that is solved using suitable numerical linear algebra tools. In the second approach, called optimize-then-discretize (od), the first-order necessary and sufficient optimality conditions are formed and then discretized. The first-order optimality conditions consist of the Oseen equations, the so-called adjoint PDEs—which have a structure similar to the Oseen equations—and an algebraic equation that links controls and adjoint variables. These PDEs are then individually discretized, in our case using the GLS formulation applied to the Oseen and the adjoint PDEs. This also results in a large-scale linear system. These two approaches will be described in Sections 2.3 and 2.4, respectively. The linear systems arising in either approach are not the same, due to the way the stabilization terms affect the adjoint PDEs. These differences and the choice of the GLS stabilization parameter will be explored numerically in Section 4. We will see that an inappropriate choice of the stabilization parameter leads to significant differences in the computed controls. Moreover, the computed control can be very sensitive to a scalar weighting parameter in the stabilization term. In our results, the solutions computed by the optimize-then-discretize approach are more sensitive to the choice of the stabilization parameter and the scalar weighting term. If the stabilization is computed using an element length based on the direction of the advective field a, the optimal controls computed by either approach are in good agreement on fine grids.

Section snippets

Formulation of the optimal control problem

In the following, we use the spaces L2(Ω) and H1(Ω), which are defined in the usual way [11], and V={vH1(Ω)|v=0onΓd}.We set H1(Ω)=[H1(Ω)]nsd, V=Vnsd, L2(Γh)=[L2(Γh)]nsd, and L2(Γg)=[L2(Γg)]nsd.

The Dirichlet boundary conditions (2d) and (2e) can be implemented through interpolation [12], weakly through a Lagrange multiplier technique [13], or via a penalty approach [7], [14]. We use interpolation to implement the Dirichlet boundary conditions (2d) with fixed data, and replace (2e) by the

Stabilization terms

Our choice of piecewise-linear functions makes our discrete approximation a low-order method, i.e. the order of the function space is lower than the order of the highest derivative in (2), the stabilized finite element equivalent of which is (20). Therefore, the terms ·σ(vh,qh) and ·σ(uh,ph) in (20) are given by qh and ph, respectively. Dropping the term ·2μϵ(uh) guarantees us only a weak consistency [18], since this term itself vanishes with refinement, as τe approaches zero based on the

Numerical results

In this section we study the effect of the GLS stabilization on the computed control for two test cases derived from commonly used model problems. In both cases, the advective field a is computed by solving the Navier–Stokes equationsρ(a·)a-·σ(a,p)=0onΩ,·a=0onΩ,with appropriate boundary conditions. The Navier–Stokes equations (36) are discretized using the GLS-stabilized finite element method and the resulting nonlinear system is solved using Newton's method with a residual tolerance of 10-16

Conclusions

The order in which a stabilized finite element discretization is applied to an optimal control problem can have a significant effect on the computed solution. If stabilized finite elements are used, the discretize-then-optimize and the optimize-then-discretize approach lead to different optimality conditions on a fixed grid. These differences primarily occur in the discrete adjoint equations and are due to the way the stabilization enters the formulation. Depending on the choice of the element

Acknowledgements

The authors gratefully acknowledge computing resources made available by the National Partnership for Advanced Computational Infrastructure (NPACI). Additional computing resources were provided by the NSF MRI award EIA-0116289. This work was supported by the National Science Foundation under award ACI-0121360, CTS-ITR-0312764, and by Texas ATP Grant 003604-0011-2001.

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