Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

Gong, Wei; Hu, Weiwei; Mateos, Mariano; Singler, John R.; Zhang, Yangwen

doi:10.1051/m2an/2020015

Cited by 12 publications

(32 citation statements)

References 84 publications

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“…Hybridizable discontinuous Galerkin (HDG) methods were proposed by Cockburn et al in [14] as an improvement of traditional DG methods; for a recent didactic exposition, see, e.g., [15]. The HDG algorithm proposed and analyzed in our work [10] is not pressure-robust: although the convergence rate is optimal, the magnitude of the error strongly depends on the pressures; see Example 4.1 below.…”

Section: Introductionmentioning

confidence: 90%

“…The model poses many theoretical and computational challenges and there is an extensive body of literature devoted to this subject; see, e.g., [1,2,3,4,5,6,7,8,9]. In [10] we investigated an HDG discretization for the tangential boundary control of a fluid governed by the Stokes system and proved optimal error estimates with respect to the global regularity of the optimal control; however, the numerical method is not pressure-robust, i.e., the discretization errors depend on the norm of the pressure.…”

Section: Introductionmentioning

confidence: 99%

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A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

Chen¹,

Gong²,

Mateos³

et al. 2022

Preprint

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In [ESAIM: M2AN, 54(2020), 2229-2264, we proposed an HDG method to approximate the solution of a tangential boundary control problem for the Stokes equations and obtained an optimal convergence rate for the optimal control that reflects its global regularity. However, the error estimates depend on the pressure, and the velocity is not divergence free. The importance of pressure-robust numerical methods for fluids was addressed by John et al. [SIAM Review, 59(2017), 492-544]. In this work, we devise a new HDG method to approximate the solution of the Stokes tangential boundary control problem; the HDG method is also of independent interest for solving the Stokes equations. This scheme yields a H(div) conforming, globally divergence free, and pressure-robust solution. To the best of our knowledge, this is the first time such a numerical scheme has been obtained for an optimal boundary control problem for the Stokes equations. We also provide numerical experiments to show the performance of the new HDG method and the advantage over the non pressure-robust scheme.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

Chen¹,

Gong²,

Mateos³

et al. 2022

Preprint

Self Cite

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“…For least squares finite element approximations of the respective optimality system [2,4] showed best approximation results, and the same was done in [5] for a standard Galerkin approximation of nonstationary Stokes control. For the related Dirichlet control problem error estimates have been obtained by HDG methods in [11] and for Navier-Stokes control in [13] All of the above results contain velocity errors depending on the pressure approximations. This implies that all the proposed methods will have spurious error contributions in the velocity induced by complicated pressures.…”

Section: Introductionmentioning

confidence: 99%

Pressure-robustness in the context of optimal control

Merdon¹,

Wollner²

2022

Preprint

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This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L 2 -orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.

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“…In the literature of Stokes Dirichlet control problem, we can see that two types of control are chosen. The first one is tangential control i.e., the control acts only in the tangential direction of the boundary (see [17]). In [17] the authors propose hybridize discontinuous Galerkin (HDG) method to approximate the solution of a tangential Dirichlet boundary control problem with an L 2 penalty on the boundary control and here the controls are unconstrained.…”

Section: Introductionmentioning

confidence: 99%

“…The first one is tangential control i.e., the control acts only in the tangential direction of the boundary (see [17]). In [17] the authors propose hybridize discontinuous Galerkin (HDG) method to approximate the solution of a tangential Dirichlet boundary control problem with an L 2 penalty on the boundary control and here the controls are unconstrained. The second one is that the flux of controls is zero (i.e., ∂Ω y • n = 0) [18].…”

Section: Introductionmentioning

confidence: 99%

A two level finite element method for Stokes constrained Dirichlet boundary control problem

Gudi¹,

Ch.²

2021

Preprint

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In this paper we present a finite element analysis for a Dirichlet boundary control problem governed by the Stokes equation. The Dirichlet control is considered in a convex closed subset of the energy space H 1 (Ω). Most of the previous works on the Stokes Dirichlet boundary control problem deals with either tangential control or the case where the flux of the control is zero. This choice of the control is very particular and their choice of the formulation leads to the control with limited regularity. To overcome this difficulty, we introduce the Stokes problem with outflow condition and the control acts on the Dirichlet boundary only hence our control is more general and it has both the tangential and normal components. We prove well-posedness and discuss on the regularity of the control problem. The first-order optimality condition for the control leads to a Signorini problem. We develop a two-level finite element discretization by using P1 elements(on the fine mesh) for the velocity and the control variable and P0 elements (on the coarse mesh) for the pressure variable. The standard energy error analysis gives 1 2 + δ 2 order of convergence when the control is in H 3 2 +δ (Ω) space. Here we have improved it to 1 2 + δ, which is optimal. Also, when the control lies in less regular space we derived optimal order of convergence up to the regularity. The theoretical results are corroborated by a variety of numerical tests.

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Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

Cited by 12 publications

References 84 publications

A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

Pressure-robustness in the context of optimal control

A two level finite element method for Stokes constrained Dirichlet boundary control problem

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