• 대한수학회지
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• 제34권4호
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• pp.841-857
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• 1997

We present analysis and some computational methods for boundary optimal and feedback control problems for Navier-Stokes equations. We use one example to illustrate our methodology and ideas which are applicable to general control problems for Navier-Stokes equations. First, we discuss the existence of optimal solutions and derive an optimality system of equations from which an optimal solution may be computed. Then we present a gradient type iterative method. Finally, we present some numerical results.

• 충청수학회지
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• 제19권4호
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• pp.335-347
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• 2006

The homogenization of non-stationary Navier-Stokes equations on anisotropic heterogeneous media is investigated. The effective coefficients of the homogenized equations are found. It is pointed out that the resulting homogenized limit systems are of the same form of non-stationary Navier-Stokes equations with suitable coefficients. Also, steady Stokes equations as cell problems are identified. A compactness theorem is proved in order to deal with time dependent homogenization problems.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제21권4호
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• pp.203-214
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• 2017

In this paper we propose and analyze two a posteriori error estimators for the stabilized $P_1/P_1$ finite element discretization of the Stokes equations. These error estimators are computed by solving local Poisson or Stokes problems on elements of the underlying triangulation. We establish their asymptotic exactness with respect to the velocity error under certain conditions on the triangulation and the regularity of the exact solution.

• 대한수학회보
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• 제56권3호
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• pp.555-561
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• 2019

In this paper we consider degenerate compressible Navier-Stokes equations giving rise to a variety of mathematical problems in many areas. We study the long time behavior for the degenerate compressible Navier-Stokes equations with damping.

• 대한수학회보
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• 제51권3호
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• pp.847-862
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• 2014

Finite element approximation solutions of the optimal control problems for stochastic Stokes equations with the forcing term perturbed by white noise are considered. Error estimates are established for the fully coupled optimality system using Brezzi-Rappaz-Raviart theory. Numerical examples are also presented to examine our theoretical results.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.307-321
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• 2004

In this paper, the unsteady Stokes problem is considered and also the pressure-correction method for the problem is described. At a fixed time level, we reduce the problem to two symmetric positive definite problems which depend on a time step parameter. Linear systems that arise from the problems are large, sparse, symmetric, positive definite and ill-conditioned as the time step tends to zero. Preconditioned problems based on an additive Schwarz method for solving the symmetric positive definite problems are derived and preconditioners are defined implicitly. It will be shown that the rate of convergence is independent of the mesh parameters as well as the time step size.

• 대한기계학회논문집B
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• 제27권6호
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• pp.753-765
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• 2003

A finite element code for the numerical solution of the Navier-Stokes equation is parallelized by vertex-oriented domain decomposition. To accelerate the convergence of iterative solvers like conjugate gradient method, parallel block ILU, iterative block ILU, and distributed ILU methods are tested as parallel preconditioners. The effectiveness of the algorithms has been investigated when P1P1 finite element discretization is used for the parallel solution of the Navier-Stokes equation. Two-dimensional and three-dimensional Laplace equations are calculated to estimate the speedup of the preconditioners. Calculation domain is partitioned by one- and multi-dimensional partitioning methods in structured grid and by METIS library in unstructured grid. For the domain-decomposed parallel computation of the Navier-Stokes equation, we have solved three-dimensional lid-driven cavity and natural convection problems in a cube as benchmark problems using a parallelized fractional 4-step finite element method. The speedup for each parallel preconditioning method is to be compared using upto 64 processors.

• 한국항공우주학회지
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• 제35권12호
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• pp.1075-1081
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• 2007

온도예조건화 방법을 Navier-Stokes 방정식에 적용하여 보았다. 그리고 Navier-Stokes 방정식에 적절한 기준마하수를 결정하기 위해 확산마하수 개념을 도입하였다. 다양한 레이놀즈수에 대한 실린더 주위의 2차원 유동을 계산하였다. 온도예조건화를 Navier-Stokes 방정식에 적용한 결과, 수렴특성이 주목할 만큼 개선되는 것을 보였다. 그리고 확산마하수로 수정한 기준마하수는 국부적으로 마하수가 낮은 영역의 수렴성 문제를 상당히 완화시키는 것을 확인하였다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제22권3호
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• pp.155-162
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• 2018

We describe a parallel implementation of a relaxed Hermitian and skew-Hermitian splitting preconditioner for the numerical solution of saddle point problems arising from the steady incompressible Navier-Stokes equations. The equations are linearized by the Picard iteration and discretized with the finite element and finite difference schemes on two-dimensional and three-dimensional domains. We report strong scalability results for up to 32 cores.

• 대한수학회보
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• 제48권5호
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• pp.1079-1092
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• 2011

We consider the Newton's method for an direct solver of the optimal control problems of the Navier-Stokes equations. We show that the finite element solutions of the optimal control problem for Stoke equations may be chosen as the initial guess for the quadratic convergence of Newton's algorithm applied to the optimal control problem for the Navier-Stokes equations provided there are sufficiently small mesh size h and the moderate Reynold's number.