界面問題的增擴有限元方法
發(fā)布時間：2025-02-06 14:16
界面問題廣泛存在于實際應(yīng)用中,如流體力學(xué),電磁波的傳播、材料科學(xué)和生物科學(xué)。它通常涉及求解耦合的偏微分方程組。本文致力于研究界面問題的有限元方法。根據(jù)網(wǎng)格單元和界面之間的拓撲關(guān)系,界面問題的有限元方法(FEMs)可分為兩大類,即界面匹配網(wǎng)格方法和界面非匹配網(wǎng)格方法。界面匹配網(wǎng)格方法的優(yōu)點在于誤差分析簡單,并且收斂階是最優(yōu)的。然而,在界面隨時間演變的情況下,讓網(wǎng)格匹配界面需要重新剖分網(wǎng)格。當(dāng)界面拓撲結(jié)構(gòu)變化時,例如破裂或者合并,生成匹配界面的網(wǎng)格是很困難的。因此,界面非匹配網(wǎng)格方法成了一個重要的研究方向。界面非匹配網(wǎng)格方法主要有兩種,擴展有限元法(XFEMs)和浸入界面有限元方法(IFEMs)。兩種方法都是對有限元空間進行修正,以得到最優(yōu)的插值誤差估計。但是,這兩種方法都有各自的一些缺點。擴展有限元方法有許多不同的種類,其中只有尼采-擴展有限元方法有嚴格的理論分析。對于尼采-擴展有限元方法,它破壞了解的連續(xù)性,因此需要在離散的弱形式加額外的懲罰項。對于浸入界面有限元方法,它的基函數(shù)構(gòu)造依賴于界面跳躍條件并且誤差分析相對困難。為了克服這些缺點,我們提出了一種新的界面非匹配網(wǎng)格方法,即協(xié)調(diào)增擴...
【文章頁數(shù)】：102 頁
【學(xué)位級別】：博士
【文章目錄】：
摘要
Abstract
Chapter 1 Introduction
1.1 Model problems and applications
1.2 An overview of FEMs for interface problems
1.3 Notation and Definitions
Chapter 2 A conforming enriched finite element method for elliptic inter-face problems
2.1 The conforming enriched finite element method
2.2 Properties of the enrichment function
2.3 Error analysis
2.4 Numerical examples
2.4.1 Numerical examples with two sub-domains
2.4.2 Numerical examples with three sub-domains
2.4.3 Numerical examples with variable coefficients
Chapter 3 A conforming enriched finite element method for Stokes inter-face problems
3.1 Weak forms of Stokes interface problems
3.2 Stability analysis
3.3 Error analysis
3.3.1 Approximation properties
3.3.2 An a prior error estimate
3.4 Numerical examples
3.4.1 Example 1: the case of a piecewise constant viscosity
3.4.2 Example 2: the case of a variable viscosity
Chapter 4 A conforming enriched finite element method for Stokes-ellipticinterface problems
4.1 Weak forms of Stokes-elliptic interface problems
4.2 Stability analysis
4.3 Error analysis
4.3.1 Approximation properties
4.3.2 An a prior error estimate
4.4 Numerical examples
4.4.1 Example 1: the case of a piecewise constant viscosity
4.4.2 Example 2: the case of a variable viscosity
Chapter 5 Conclusions and future works
5.1 A framework of FEMs for interface problems
5.2 Future works
Bibliography
Publications and Completed Papers
Acknowledgements
本文編號：4030494
【文章頁數(shù)】：102 頁
【學(xué)位級別】：博士
【文章目錄】：
摘要
Abstract
Chapter 1 Introduction
1.1 Model problems and applications
1.2 An overview of FEMs for interface problems
1.3 Notation and Definitions
Chapter 2 A conforming enriched finite element method for elliptic inter-face problems
2.1 The conforming enriched finite element method
2.2 Properties of the enrichment function
2.3 Error analysis
2.4 Numerical examples
2.4.1 Numerical examples with two sub-domains
2.4.2 Numerical examples with three sub-domains
2.4.3 Numerical examples with variable coefficients
Chapter 3 A conforming enriched finite element method for Stokes inter-face problems
3.1 Weak forms of Stokes interface problems
3.2 Stability analysis
3.3 Error analysis
3.3.1 Approximation properties
3.3.2 An a prior error estimate
3.4 Numerical examples
3.4.1 Example 1: the case of a piecewise constant viscosity
3.4.2 Example 2: the case of a variable viscosity
Chapter 4 A conforming enriched finite element method for Stokes-ellipticinterface problems
4.1 Weak forms of Stokes-elliptic interface problems
4.2 Stability analysis
4.3 Error analysis
4.3.1 Approximation properties
4.3.2 An a prior error estimate
4.4 Numerical examples
4.4.1 Example 1: the case of a piecewise constant viscosity
4.4.2 Example 2: the case of a variable viscosity
Chapter 5 Conclusions and future works
5.1 A framework of FEMs for interface problems
5.2 Future works
Bibliography
Publications and Completed Papers
Acknowledgements
本文編號：4030494
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