• 대한수학회보
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• 제53권3호
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• pp.853-874
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• 2016

Based on a particular overlapping domain decomposition technique, a parallel finite element discretization algorithm for the generalized Stokes equations is proposed and investigated. In this algorithm, each processor computes a local approximate solution in its own subdomain by solving a global problem on a mesh that is fine around its own subdomain and coarse elsewhere, and hence avoids communication with other processors in the process of computations. This algorithm has low communication complexity. It only requires the application of an existing sequential solver on the global meshes associated with each subdomain, and hence can reuse existing sequential software. Numerical results are given to demonstrate the effectiveness of the parallel algorithm.

• 대한수학회보
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• 제55권2호
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• pp.379-403
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• 2018

In this paper we study the first initial boundary value problem for the 3D Navier-Stokes-Voigt equations with infinite delay. First, we prove the existence and uniqueness of weak solutions to the problem by combining the Galerkin method and the energy method. Then we prove the existence of a compact global attractor for the continuous semigroup associated to the problem. Finally, we study the existence and exponential stability of stationary solutions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제21권3호
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• pp.109-124
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• 2017

The numerical solution of the Stokes equation with discontinuous viscosity and singular force term is challenging, due to the discontinuity of pressure, non-smoothness of velocity, and coupled discontinuities along interface.In this paper, we give an efficient algorithm to solve this problem by employing spectral Legendre and Chebyshev approximations.First, we present the algorithm for a problem defined in rectangular domain with straight line interface. Then it is generalized to a domain with smooth curve boundary and interface by employing spectral element method. Numerical experiments demonstrate the accuracy and efficiency of our algorithm and its spectral convergence.

• 대한수학회지
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• 제50권5호
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• pp.1129-1163
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• 2013

Based on finite element discretization, two linearization approaches to the defect-correction method for the steady incompressible Navier-Stokes equations are discussed and investigated. By applying $m$ times of Newton and Picard iterations to solve an artificial viscosity stabilized nonlinear Navier-Stokes problem, respectively, and then correcting the solution by solving a linear problem, two linearized defect-correction algorithms are proposed and analyzed. Error estimates with respect to the mesh size $h$, the kinematic viscosity ${\nu}$, the stability factor ${\alpha}$ and the number of nonlinear iterations $m$ for the discrete solution are derived for the linearized one-step defect-correction algorithms. Efficient stopping criteria for the nonlinear iterations are derived. The influence of the linearizations on the accuracy of the approximate solutions are also investigated. Finally, numerical experiments on a problem with known analytical solution, the lid-driven cavity flow, and the flow over a backward-facing step are performed to verify the theoretical results and demonstrate the effectiveness of the proposed defect-correction algorithms.

• 한국항해학회지
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• 제9권2호
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• pp.43-63
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• 1985

As a circular cylinder has a comparatively simple shape and becomes a basic problem for flows around other various shapes of bodies, the problem of two-dimensional viscous flow around the circular cylinder has been investigated, both theoretically and experimentally. But not a few problems are left unsolved. It is well known that the calculations are successfully made with the approximations of Stokes or Oseen for very low Reynolds numbers, but as Reynolds number is increased, Oseen's approximations as well as Stokes's ones become more and more remote from the exact solution of the Navier-Stokes equations. Therefore, in this paper, the authors transform the Navier-Stokes equations into the finite difference equations in the steady two-dimensional viscous flow at Reynolds number up to 45, and then solve the solution of the Navier-Stokes equations numerically. Also, the authors examine the accuracy of the solution by means of flow visualization with aluminum powder. The main results are as follows; (1) The critical Reynolds number at which twin vortices begin to form in the rear of the circular cylinder is found to be 6 in the experiment and 4 in the numerical solution. (2) As Reynolds number is increased, it is proved that the ratio of the length of the twin vortices to the diameter is grown almost linearly, both experimentally and numerically. (3) Separation angle is also increased according to reynolds number. But it is found that it would converge into 101.3 degrees, both experimentally and numerically.

• 한국전산구조공학회논문집
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• 제15권4호
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• pp.661-674
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• 2002

본 연구의 목적은 Navier-Stokes 유체의 최적 제어 문제의 해를 얻을 수 있는 효과적인 수치해석기법을 개발하고, 이를 물체의 항력(drag)을 최소화하는 문제에 적용하는데 있다. 본 연구는 항력을 줄인다는 산업적인 중요성과 함께 최적 제어를 위한 하나의 효과적인 최적화 기법의 모델을 제공하고 있다. 항력을 줄이기 위한 방법으로써 물체의 경계면에서 유체의 흡입(suction)과 방출(injection)이라는 기법을 사용하여 경계면에서 속도를 제어하였고, 목적함수로써 항력을 표현하기 위하여 에너지 소실의 변화율을 사용하였다. 컴퓨터 용량을 최소화하고 최적화에서의 해의 보장성과 경제성을 위하여, Navier-Stokes의 해석을 위하여 페널티 방법을 사용하였고 최적화 기법을 위해서는 SQP 방법을 사용하였다. 그리고 Navier-Stokes 유체는 대단히 비선형성을 나타내기 때문에 최적화를 수행하기에는 매우 힘들다. 이를 위하여 연속기법(continuation technique)을 사용하였다.

• 호남수학학술지
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• 제38권3호
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• pp.583-592
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• 2016

In this paper, we consider the global existence of strong solutions to the incompressible Navier-Stokes equations on the cubic domain in $R^3$. While the global existence for arbitrary data remains as an important open problem, we here provide with some new observations on this matter. We in particular prove the global existence result when ${\Omega}$ is a cubic domain and initial and forcing functions are some linear combination of functions of at most two variables and the like by decomposing the spectral basis differently.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 1998년도 추계 학술대회논문집
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• pp.96-101
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• 1998

Various discretiation methods of Laplacian operator in the Pressure-Poisson equation are investigated for the solution of incompressible Navier-Stokes equations using the non-staggered grid. Laplacian operators previously proposed by other researchers are applied to a Driven-Cavity problem. The computational results are compared with those of Ghia. The results show the characteristics of the discrete Laplacian operators.

• Journal of applied mathematics & informatics
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• 제7권3호
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• pp.801-812
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• 2000

Two-dimensional slow viscous flow on infinite half-plane past a perpendicular infinite cavity is considered on the basis of the Stokes approximation. Using complex representation of the two-dimensional Stokes flow, the problem is reduced to solving a set of Fredholm integral equations of the second kind. The streamlines and the pressure and vorticity distribution on the wall are numerically determined.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제4권2호
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• pp.111-133
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• 2000

A Legendre spectral tau approximation scheme for solving the two-dimensional stationary incompressible Stokes equations is considered. Based on the vorticity-stream function formulation and variational forms, boundary value and normal derivative of vorticity are computed. A factorization technique for matrix stems based on the Schur decomposition is derived. Several numerical experiments are performed.