數種不連續有限元素法求解對流佔優問題之數值研究; A Numerical Study of Various Discontinuous Finite Element Methods for Solving Convection-Dominated Problems

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题名:	數種不連續有限元素法求解對流佔優問題之數值研究;A Numerical Study of Various Discontinuous Finite Element Methods for Solving Convection-Dominated Problems
作者:	鄭庭蓁;Ting-Zhen Zheng
贡献者:	數學研究所
关键词:	不連續Petrov-Galerkin方法;不連續有限元素法;多尺度有限元素法;邊界層;對流佔優問題;數值通量;對流-擴散方程;discontinuous Petrov-Galerkin methods;discontinuous Galerkin methods;boundary layers;multiscale finite element methods;convection-dominated problems;numerical fluxes;convection-diffusion equations
日期:	2011-07-12
上传时间:	2012-01-05 14:44:01 (UTC+8)
摘要:	在本文中，我們研究數種以不同數值通量為基礎的不連續有限元素法求解對流佔優情況下之對流-擴散問題。我們比較了數種不連續有限元素法在Galerkin與Petrov-Galerkin形式下的數值效率，其中所有的不連續Petrov-Galerkin方法皆經由多尺度基函數取代Q1試驗函數而產生，而該多尺度基函數源自於求解各有限單元上具合適邊界條件之局部微分方程式。我們經由兩個具有解析解的數值實例來闡明這些不同方法的效能。我們發現使用Q1試驗函數的不連續有限元素法在擴散係數較小時效率會變差，然而除了Baumann-Oden方法外，其他多尺度不連續Petrov-Galerkin方法都比不連續的有限元素法更能精確捕獲問題解在邊界層的結構性質。 In this thesis, we study various discontinuous finite element methods based on different numerical fluxes for solving convection-diffusion problems with emphasis on the convection-dominated case. We compare numerically the efficiency of various discontinuous finite element methods in the Galerkin and the Petrov-Galerkin formulations. All the discontinuous Petrov-Galerkin methods are formulated by replacing the Q1 trial functions with the multiscale basis functions, which are designed by solving a series of local differential equations on each elements with proper boundary conditions. Numerical simulations of two examples with analytic solutions are presented to illustrate the effectiveness of the various methods. We find that for a small diffusivity, the discontinuous Galerkin methods using Q1 finite elements show a rather poor performance. However, except the Baumann-Oden method, all the other multiscale discontinuous Petrov-Galerkin methods are much better able to capture the nature of boundary layer structure in the solution than the discontinuous Galerkin methods.
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