• 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제14권2호
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• pp.125-140
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• 2010

This paper develops a least-squares approach to the solution of the optimal control problem for the Navier-Stokes equations. We recast the optimality system as a first-order system by introducing velocity-flux variables and associated curl and trace equations. We show that a least-squares principle based on $L^2$ norms applied to this system yields optimal discretization error estimates in the $H^1$ norm in each variable.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제11권4호
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• pp.55-68
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• 2007

First-order least-squares method of a distributed optimal control problem for the incompressible Navier-Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing velocity-flux variables and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared $L^2$ norm of the residual equations of the system. The optimal error estimates for least-squares finite element approximations are obtained.

• 한국음향학회지
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• 제42권5호
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• pp.377-387
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• 2023

본 논문에서는 1-3형 압전복합재료를 단일상의 균질 압전매질로 모델링하기 위해 필요한 등가의 물성을 유한요소 해석으로 직접 추출하는 기법의 정확도를 다룬다. 직접 추출 기법은 압전재료의 전기적, 기계적 거동과 상호 간 커플링을 기술하는 구성방정식을 기반으로 물성 행렬의 개별 성분들을 직접적으로 산출하는 방법이다. 직접 추출에 사용되는 두 가지 구성방정식 조합 간의 정확도를 비교하기 위하여 단일 1-3형 압전복합재료 하이드로폰을 대상으로 등가물성과 수신감도를 산출하고 전체 영역에 대한 유한요소 해석 결과와 비교한다. 물성 추출의 정확도는 압전복합재료를 구성하는 폴리머의 탄성 특성과 밀접한 관련이 있음을 확인하고, 오차 원인을 분석하여 정확한 등가물성 산출을 위한 가이드라인을 제시한다. 압전복합재료 하이드로폰과 주변의 음향구조물을 포함하는 스테이브 규모에 대해서도 등가 모델링을 적용하여 추출된 물성의 정확도와 연산량 감소를 확인한다.