The Lagrange Multiplier And The Stationary Stokes Equations

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FOR THE STATIONARY NAVIER-STOKES EQUATIONS WITH DISTRIBUTED AND NEUMANN CONTROLS M. D. GUNZBURGER, L. HOU, AND T. P. SVOBODNY Abstract. We examine certain analytic and numerical aspects of optimal con-trol problems for the stationary Navier-Stokes equations. The controls consid-

Analysis of a velocity-pressure-pseudostress formulation for

for the stationary Stokes equations∗ Gabriel N. Gatica† Antonio M´arquez ‡ Manuel A. S´anchez § Abstract We consider a non-standard mixed approach for the Stokes problem in which the velocity, the pressure, and the pseudostress are the main unknowns. Alternatively, the pressure can


Optimal control of partial differential equations in presence of state constraints is a very challenging research field, mainly due to the difficult structure of the Lagrange multiplier associated to the state constraints (see [2, 3, 4]). In the case of Navier-Stokes control, the

Comput. Methods Appl. Mech. Engrg. - Emory University

acting as a Lagrange multiplier corresponding to this constraint. Since r ¼ 0, that the vorticity is solenoidal is important both for physicalrelevance andmathematicalconsistency. Althoughitis possible to couple this constraint to the usual vorticity equation by adding an artificial Lagrange multiplier, VVH enforces this con-straint naturally.


LAGRANGE NEWTON KRYLOV SCHUR METHODS, PART I 689 The first set of equations are just the original Navier Stokes PDEs. The adjoint equations, which result from stationarity with respect to state variables, are them-selves PDEs, and are linear in the Lagrange multipliers λ and μ. Finally, the control equations are (in this case) algebraic.

SIAM J. S COMPUT c Vol. 28, No. 5, pp. 1675 1693

Key words. Navier Stokes, conservation, finite elements, least squares, multigrid AMS subject classifications. 76D05, 65F10, 65N55 DOI. 10.1137/050640928 1. Introduction. Least-squares variational principles have been repeatedly rec-ognized as a new tool to solve elliptic partial differential equations [6]. For the Stokes


Navier{Stokes equations in primitive variables on stationary manifolds. In [38], the authors considered P 1{P 1 nite elements with no pressure stabilization and a penalty technique to force the ow eld to be tangential to the surface. In [15], instead, surface Taylor{Hood elements were used and combined with a Lagrange multiplier

Divorcing pressure from viscosity in incompressible Navier

Stokes equations of incompressible flow. Pressure plays a role like a Lagrange multiplier to enforce the incompressibility constraint, and this has been a main source of difficulties. Our general aim in this paper is to show that the pressure can be obtained in a way that leads to considerable simplifications in both computation and analysis.


the magnetic eld without requiring a Lagrange multiplier. We consider extending block precondi-tioning techniques developed for the Navier-Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based

Generalized solutions for the Euler equations in one and two

non-deterministic, they induce two non-trivial stationary solutions to Euler equations with a new macroscopic pressure field (see Paragraph 4.4). More generally, using the generalized solutions constructed in Paragraph 4.2, one can produce a new large class of stationary and non-stationary solutions to Euler equations.


another system of equations introduced by Johnston and Liu in [9] and studied by Shirokoff and Rosales in [16]. This system also extends the Navier-Stokes equations, though in a manner different from the extended equations in (1.1′). We show how they can nonetheless be treated using the same key tools of Section 2. In the appendix

Constrained Dirichlet Boundary Control in L2 for a Class of

the stationary Navier Stokes equations in [14], for example. In the adjoint equation, however, a Laplacian now appears in the source term acting on the defect y−z. Besides the difficulties already mentioned with (1.3) and (1.4) there is yet an-other, possibly more essential reason, to favor the formulation in (1.2). For (1.2) the

One-Shot Pseudo-Time Method for Aerodynamic Shape

Navier-Stokes equations together with algebraic turbulence model of Baldwin and Lomax. is the Lagrangian functional and λ is the Lagrange multiplier or the

Weak Imposition Of Boundary Conditions For The Navier-Stokes

extension beyond the Stokes problem, of [7,10] to the Navier-Stokes equations is important. The purpose of this report is to begin this extension. Our anal-ysis both extends the work of [2] and [7] from the (linear) Stokes problem to nonlinear Navier-Stokes problem and improves the basic results of [2] even in the linear case.


Navier-Stokes equations without damping when converges to zero. 2.3. A shape optimization problem and adjoint equations. In this subsec-tion, we are concerned with numerical methods for a shape optimization problem associated with ow governed by the stationary incompressible Navier-Stokes equa-tions with damping. Our aim is to nd an optimal shape

Constrained Optimization Using Lagrange Multipliers

Jul 10, 2020 Lagrange multiplier methods involve the modification of the objective function through the addition of terms that describe the constraints. The objective function J = f(x) is augmented by the constraint equations through a set of non-negative multiplicative Lagrange multipliers, λ j ≥0. The augmented objective function, J A(x), is a


Abstract. In this paper, an optimal control problem for the stationary Navier-Stokes equations in the presence of state constraints is investigated. Existence of optimal solutions is proved and first order necessary conditions are derived. The regularity of the adjoint state and the state constraint multiplier is also studied.

arXiv:2006.15700v1 [math.NA] 28 Jun 2020

to the Lagrange multiplier formulation described below. Following the formulation of the MHD equations in [51,53], we introduce a La-grange multiplier, r, to weakly enforce the solenoidal constraint, rB = 0. This Lagrange multiplier appears as a nonphysical term, rr, in Faraday s law. The set

Spectral distributed Lagrange multiplier method: algorithm

The distributed Lagrange multiplier (DLM) method [2,6 8,29,32 34] (in [2] it is called virtual finite element method), which this paper focuses on, falls into the second category. With the DLM method, the problem on a time-dependent geometrically complex domain is extended to a stationary, larger, but simpler

Analysis and finite element approximation of optimal control

We now seek stationary points of Jt(u, p, g, ~, tr) over Z. Again, proceeding in a formal manner, using standard techniques of the calculus of variations, one may derive the Euler-Lagrange equations that correspond to the minimization of (2.5). This process yields the optimality system

Numerical methods for the Stokes and Navier-Stokes equations

we solve the problem associated with the Stokes equations by exploiting the minimization structure of the variational formulation, and apply to it an alternating direction method reminiscent to those used in [6, 18, 20, 22, 37, 38]. Next, we solve the stationary Navier Stokes equations in two steps.

Analysis and finite element approximation of optimal control

FOR THE STATIONARY NAVIER-STOKES EQUATIONS WITH DIRICHLET CONTROLS (*) M. D. GUNZBURGER ('), L. S. Hou (2) and Th. P. SVOBODNY (3) Communicated by R. TEMAM Abstract. Optimal control problems for the stationary Navier-Stokes équations are examined from analytical and numerical points ofview. The controls considered are of Dirichlet

Analysis and Finite-Element Approximation of Optimal-Control

FOR THE STATIONARY NAVIER-STOKES EQUATIONS WITH DISTRIBUTED AND NEUMANN CONTROLS M. D. GUNZBURGER, L. HOU, AND T. P. SVOBODNY ABSTRACT. We examine certain analytic and numerical aspects of optimal con- trol problems for the stationary Navier-Stokes equations. The controls consid-

State-Constrained Optimal Control of the Stationary Navier

STATE EQUATIONS In the case of state-constrained optimal control of the Navier-Stokes equations, we point to [11, 21], where the time dependent problem is investigated. In [11], the state equations are treated as abstract differential equations. Clearly, the same framework does not hold for the stationary case considered here.


The Lagrange multiplier framework for the numerical treatment of essential inhomogeneous Dirichlet boundary data for elliptic problems, and for stationary Navier-Stokes equations has been considered in [1,2,20,27,34]. Saddle point problems are usually related to elliptic partial di erential equations and result from certain minimization principles.

The immersed boundary method: A projection approach

The discretized Navier Stokes equations with boundary force Since the discretized Navier Stokes equations, Eq. (3), are observed to be a KKT system with pressure act-ing as a set of Lagrange multipliers to satisfy the continuity constraint, one can imagine appending additional algebraic constraints by increasing the number of Lagrange

Iso-P2 P1/P1/P1 Domain-Decomposition/Finite-Element Method

Lagrange multiplier. Suzuki [9] has shown a method using the iso-P2 P1 element. But the choice of the basis functions for the Lagrange multipliers has not been well compared in one domain decomposition algorithm. In this paper we propose a domain-decomposition/ nite-element method for the Navier-Stokes equations of the velocity-pressure


Stokes equations discretized by nite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems solve iteratively the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters.

An Explicit Finite-Difference Scheme for Simulation of Moving

through an initially stationary fluid. The latter is then extended to allow the cylinder to fall freely. 2 Explicit MacCormack Scheme Instead of using the artificial continuity equation of (1), one may start with the exact compressible Navier-Stokes equations with the equation of state.

arXiv:1402.2615v1 [math.AP] 11 Feb 2014

3. Global uniqueness for the Stokes equations 7 3.1. (∂2 z,∂ 2 z) system 7 3.2. ∂z system 11 3.3. Proof of the uniqueness result 14 4. Global uniqueness for the stationary Navier-Stokes equations 16 References 17 1. Introduction Let Ω be a simply connected bounded domain in R2 with smooth boundary. Assume that Ω is filled with an


element-boundary velocity field serves as a Lagrange-multiplier to enforce the constraint of interelement traction reciprocity. Hence is the name hybrid associated with the present finite element method 11.31. It will be shown that the final finite element system of equations would


^ column vector representing the Lagrange multiplier y column vector representing the second derivatives of the stream func-tion at the nodal points of an element F length of the perimeter of area A, in. (m) A Lagrange multiplier 2 2 v kinematic viscosity coefficient, in. /sec. (m /sec.) 3 3 p density of the fluid, Ibm/in. (kg/m )


2D, uniqueness of weak solutions to the extended Navier-Stokes equations of (1.10), giving the higher regularity theory in Section4. In Section5we analyze another system of equations introduced by Johnston and Liu in [9] and studied by Shiroko and Rosales in [16]. This system also extends the Navier-Stokes equations, though

Applied Mathematics and Computation

In this article, we develop two parallel non-iterative domain decomposition methods for the non-stationary Stokes Darcy model with Beavers Joseph interface condition. The rest of paper is organized as follows. In Section 2, we introduce the Stokes Darcy system with Beavers Joseph inter-face condition.

Stability and convergence of efficient Navier-Stokes solvers

Stability of efficient Navier-Stokes solvers 2 The pressure has long been a main source of trouble for understanding and computing solutions to these equations. Pressure plays a role like a Lagrange multiplier to enforce the incompressibility constraint. This leads to compu-

Assessment of a vorticity based solver for the Navier-Stokes

be the solution to the stationary incompressible Navier-Stokes equations in a bounded 3 with a sufficiently regular boundary and homogeneous Dirichlet boundary conditions for u. Assume f ∈ L21 0 k+1(Ω). If (u h,P h), (w h,η h) are the 3

A Finite Element Method for the Surface Stokes Problem

used and combined with a Lagrange multiplier method to enforce the tangentiality constraint. Neither references address the numerical analysis of nite element meth-ods for the surface Navier-Stokes equations. In general, we are not aware of any paper containing a rigorous analysis of nite element (or any other) discretization methods

Locally divergence-free discontinuous Galerkin methods for

equations by adding a coupling term into Gausss law that resulted in a purely hyperbolic model system. Classical finite element methods for solving Maxwell equations can be found in, e.g. [1,16]. Baker and co-workers [2,18] introduced a discontinuous Galerkin method for solving the Stokes equations and the stationary Navier Stokes equations.

Mixed methods for stationary Navier-Stokes equations based on

Mar 28, 2012 Lagrange multiplier on edges. In [19], this hybridized method was used for the MIXED METHODS FOR STATIONARY NAVIER-STOKES EQUATIONS 3 Navier-Stokesequations

applications, one arises at a system of interface equations

applications, one arises at a system of interface equations in R3 coupling the two potentials. In this paper, a Lagrange multiplier method for solving the coupled equations is presented and it is shown that the resulting Lagrange multiplier operator defined on the interface is not only symmetric positive definite but also spectrally equivalent