@MISC{Lee_reductionof,

author = {Hyung-chun Lee},

title = {REDUCTION OF MODES FOR THE COMPUTATIONS OF NAVIER-STOKES EQUATIONS},

year = {}

}

Abstract. This article is a survey article for a reduced-order modeling ap-proach for computation of time-dependent Navier-Stokes flows. A reduced-basis method is introduced. The choices for the reduced basis method are the Lagrange subspace, the Hermite subspace and the Taylor subspace. We then introduce the POD-based method and CVT-based method. 1. Introducion Optimal control problems that involve partial differential equations as state equa-tions are formidable problems to solve in real time. One such situation arises in control of fluid dynamical systems in which the state equations are the Navier-Stokes equations. We will discuss some reduction-type method which may help to overcome this difficulty. In order to illustrate the reduced-basis method, we consider the stationary Navier-Stokes equations −ν∆u+ (u · ∇)u+∇p = f(1.1) ∇ · u = 0(1.2) with appropriate boundary conditions for ν ∈ R and u ∈ X. The above problem is a parameterized one. The constant ν presents kinematic viscosity about which we choose to interpolate to obtain a reduced-finite-dimensional set of basis elements. In standard finite element approximations, one approximate X with a piecewise polynomial space. However, the choices for the reduced basis method are different. The Lagrange subspace. In this case, the basis elements are solutions of the non-linear problem under study at various parameter values νj. The reduced subspace is given by XR = span uj |uj = u(νj), j = 1,...,M This kind of subspace was used to study structural problems in [1]. A possible advantage in this choice is that updating the basis elements can be done one basis vector at a time instead of generating the whole space.

computation navier-stokes equation basis element reduction mode reduced basis method reduced-basis method lagrange subspace stationary navier-stokes equation situation arises state equation appropriate boundary condition cvt-based method formidable problem state equa-tions reduction-type method structural problem constant present kinematic viscosity piecewise polynomial space navier-stokes equation non-linear problem reduced-order modeling ap-proach partial differential equation reduced subspace real time whole space pod-based method fluid dynamical system introducion optimal control problem basis vector possible advantage time-dependent navier-stokes flow various parameter value hermite subspace standard finite element approximation taylor subspace parameterized one reduced-finite-dimensional set survey article uj uj

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