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육각형에서의 혼합법, 유한체적법 및 다중 격자법의 연구
Reports NRF is supported by Research Projects( 육각형에서의 혼합법, 유한체적법 및 다중 격자법의 연구 | 2005 Year | 곽도영(한국과학기술원) ) data is submitted to the NRF Project Results
Researcher who has been awarded a research grant by Humanities and Social Studies Support Program of NRF has to submit an end product within 6 months(* depend on the form of business)
사업별 신청요강보기
  • Researchers have entered the information directly to the NRF of Korea research support system
Project Number C00009
Year(selected) 2005 Year
the present condition of Project 종료
State of proposition 재단승인
Completion Date 2006년 07월 03일
Year type 결과보고
Year(final report) 2006년
Research Summary
  • Korean

  • 석유의 흐름을 나타내는 편미분 방정식에서 유체의 속도를 효과적으로 계산할수 있는 유한 원소를 새로 찾아내었다. 기존의 원소에 비해서 하나의 자유도가 적으면서도 일반적 d\모양의 원소에서도 최적의 수렴성을 가지므로 현재까지 발견된 것 중 가장 효과적인 원소라 할수 있겠다. 단점이라면 압력과 밀도에 관한 한 최적이 아닌데 그것은 간단한 후처리 과정을 통하여 최적의 수렴성을 보장할 수 있으므로 그리 문제 되지 않는다. 게다가 수치 실험을 통하여 다른 방법보다 더욱 효과적임을 확인하였다.
  • English
  • In this paper, we introduce a new family of mixed finite
    element spaces of higher order($k\ge1)$ on general quadrilateral
    grids. A typical element has two fewer degrees of freedom
    than the well-known Raviart-Thomas finite element RT_[k]yet enjoys an optimal order approximation for the velocity in
    L^2-norm. The order of approximation in the divergence norm is one less than the velocity, as is common to all other known
    elements, except a recent element introduced by Arnold et
    al. However, we introduce a local post-processing technique to obtain an optimal order in H(div;O)-norm. This technique can be used to enhance the result of RT_k element also, hence can be easily incorporated into existing codes.
    Our element has one lower order of approximation in pressure
    than the RT_[k] element. However, the pressure also can be
    locally post-processed to produce an optimal order approximation. The greatest advantage of our finite element lies in that it has the fewest degrees of freedom among all the known quadrilateral mixed finite elements and thus together with the post-processing techniques provides a very efficient way of computing flow variables in mixed formulation.
    Numerical examples are in quite good agreement with the theory even for the case of almost degenerate quadrilateral grids.
Research result report
  • Abstract
  • In this paper, we introduce a new family of mixed finite
    element spaces of higher order($k\ge1)$ on general quadrilateral
    grids. A typical element has two fewer degrees of freedom
    than the well-known Raviart-Thomas finite element RT_[k]yet enjoys an optimal order approximation for the velocity in
    L^2-norm. The order of approximation in the divergence norm is one less than the velocity, as is common to all other known
    elements, except a recent element introduced by Arnold et
    al. However, we introduce a local post-processing technique to obtain an optimal order in H(div;O)-norm. This technique can be used to enhance the result of RT_k element also, hence can be easily incorporated into existing codes.
    Our element has one lower order of approximation in pressure
    than the RT_[k] element. However, the pressure also can be
    locally post-processed to produce an optimal order approximation. The greatest advantage of our finite element lies in that it has the fewest degrees of freedom among all the known quadrilateral mixed finite elements and thus together with the post-processing techniques provides a very efficient way of computing flow variables in mixed formulation.
    Numerical examples are in quite good agreement with the theory even for the case of almost degenerate quadrilateral grids.
  • Research result and Utilization method
  • 학문적 측면: 본 연구결과를 먼저 3차원 Prism형태의 원소에 대해 확장하고 다음에는 일반 육각원소에 확장을 시도한다.

    교육적 측면: 대학원생의 교육에 활용한다.
    산업적 측면: 이 원소에 대한 연구는 원래 석유시추 및 탐사와 관련된 문제에서 주로 시작하였다. 비록 우리나라에서 자체적으로 석유시추에 이러한 방정식을 사용할 기회가 있을지는 알수 없으나 선진국에서는 이미 오래전부터 소프트웨어를 만드는데 사용되어 왔다. 앞으로 한국의 자체 기술이 발전함 기여를할 수 있으리라 본다.
  • Index terms
  • Mixed finite element method, quadrilateral grid, optimal order, velocity, post-processing
  • List of digital content of this reports
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