邊界層和內部層問題的穩定化有限元素法

Stabilized finite element methods for solving boundary and interior layer problems

謝博文(Po-Wen Hsieh)

指導教授 ： 楊肅煜

國立中央大學/理學院/數學系/博士(2011年)

摘要

本文主要發展穩定化的有限元素法求解邊界層和內部層問題，我們提出了兩種新型的穩定化方法。第一種方法是求解對流佔優對流-擴散問題的泡狀函數穩定化最小平方有限元素法。首先，依據最小平方有限元素法的概念引入原未知變數的梯度向量為新增變數，將原始二階偏微分方程式轉換成一階偏微分方程組，接著針對每一個未知變數，利用類似無殘餘泡狀函數法的方式將最小平方有限元素法的線性基底加入具穩定作用的泡狀函數。其中泡狀函數在每一個單元上滿足某個二階偏微分方程式及零單元邊界值條件，實際運算時則採取某種加勒金/最小平方型穩定化方法求取此無殘餘泡狀函數之近似解。在這樣的策略下不僅可以保有最小平方有限元素法的優勢(所形成的矩陣具對稱正定性)，亦具有無殘餘泡狀函數法的特徵(無需再選擇穩定化參數)。我們進行一系列的數值模擬實證泡狀函數穩定化最小平方有限元素法的高度效率，同時比較了該方法和原始最小平方有限元素法的精確度和計算成本。我們發現在對流佔優的情況下，泡狀函數穩定化最小平方有限元素法的精確度和穩定性都比原始最小平方有限元素法高，即使原始最小平方有限元素法使用較為細密的網格或較為高階的基底都無法達到相同的效果。最後，我們成功地將這種新型的泡狀函數穩定化最小平方有限元素法應用到某個源自靜態不可壓縮管狀磁流體的對流佔優對流-擴散方程組上。

第二種穩定化方法是求解反應-對流-擴散問題的新型加勒金穩定化有限元素法，其中該反應-對流-擴散問題具有一個極小擴散係數與一個極大反應係數。我們明確定義其穩定化參數，經由一個特殊內插技巧推導出 與 模下的誤差估計，並建立該估計與擴散係數、對流場大小、反應係數和網格參數之間的明確關聯。在數值計算上，我們將此新方法與文獻上的兩種穩定化有限元素法比較，結果顯示在反應-對流佔優的情況下，此新型穩定化方法具有高精確度與高穩定性。

關鍵字

最小平方有限元素法 ； 無殘餘泡狀函數 ； 穩定化有限元素法 ； 有限元素法 ； 邊界層和內部層 ； 磁流體方程組 ； 反應-對流-擴散方程 ； 對流-擴散方程

並列摘要

This thesis is devoted to developing stabilized finite element methods (FEMs) for solving boundary and interior layer problems. We propose and analyze two new stabilized FEMs. The first one is the bubble-stabilized least-squares finite element method (LSFEM) which is applied to solve scalar convection-dominated convection-diffusion problems. We first convert the second-order convection-diffusion problem into a first-order system formulation by introducing the gradient of solution as a new unknown. Then the LSFEM using continuous piecewise linear elements enriched with residual-free bubbles for all unknowns is applied to solve the first-order mixed problem. The residual-free bubble functions are assumed to strongly satisfy the associated homogeneous second-order convection-diffusion equations in the interior of each element, up to the contribution of the linear part, and vanish on the element boundary. To implement this two-level least-squares approach, a stabilized method of Galerkin/least-squares type is used to approximate the residual-free bubble functions. This bubble-stabilized LSFEM not only inherits the advantages of the primitive LSFEM, such as the resulting linear system being symmetric and positive definite, but also exhibits the characteristics of the residual-free bubble method without involving any stabilization parameters. Several numerical experiments are given to demonstrate the effectiveness of the proposed bubble-stabilized LSFEM. The accuracy and computational cost of this method are also compared with those of the primitive LSFEM. We find that for a small diffusivity, the bubble-stabilized LSFEM is much better able to capture the nature of layer structure in the solution than the primitive LSFEM, even if the primitive LSFEM uses a very fine mesh or higher-order elements. In other words, the bubble-stabilized LSFEM provides a significant mprovement, with a lower computational cost, over the primitive LSFEM for solving convection-dominated problems. Finally, we extend this approach to a coupled system of convection-diffusion equations arising from the steady incompressible magnetohydrodynamic duct flow problem with a transverse magnetic field at high Hartmann numbers.

The second method that we propose in this thesis is a new stabilized FEM in the Galerkin formulation. We analyze the method using continuous piecewise linear elements for solving 2D reaction-convection-diffusion equations. The equation under consideration is reaction-convection-dominated, involving a small diffusivity and a large reaction coefficient. In addition to giving error estimates of the approximations in $L^2$ and $H^1$ norms, we explicitly establish the dependence of error bounds on the diffusivity, the module of convection field, the reaction coefficient and the mesh size. Several numerical examples exhibiting boundary layers are given to illustrate the high accuracy and stability of the newly proposed stabilized FEM. The results obtained are also compared with those of existing stabilized FEMs.

並列關鍵字

least-squares finite element methods ； residual-free bubbles ； stabilized finite element methods ； finite element methods ； boundary and interior layers ； magnetohydrodynamic equations ； reaction-convection-diffusion equations ； convection-diffusion equations

參考文獻

[1] K. E. Barrett, Duct flow with a transverse magnetic field at high Hartmann numbers, Int.

J. Numer. Meth. Engrg., 50 (2001), pp. 1893-1906.

[2] P. B. Bochev and M. D. Gunzburger, Analysis of least-squares finite element methods for

[3] P. B. Bochev and M. D. Gunzburger, Finite element methods of least-squares type, SIAM

[4] P. B. Bochev and M. D. Gunzburger, Least-Squares Finite Element Methods, Appl. Math.