本文選題：有限體積元方法 + 界面問題　； 參考：《南京師范大學(xué)》2015年博士論文

【摘要】：本文致力于兩類界面問題的有限體積元方法的研究,全文共分為三個部分.第一章首先我們介紹了關(guān)于界面問題的一些浸入方法,闡述了發(fā)展浸入有限體積元方法的目的。然后我們介紹了一維帶界面的雙相延遲方程、高階緊有限體積元方法和Pade型緊有限體積方法。第二章第一部分討論了帶界面的泊松方程的浸入有限體積元方法。通過源項移去技巧,將帶非齊次跳躍條件的界面問題轉(zhuǎn)化為帶齊次跳躍條件的界面問題,與跳躍條件相關(guān)的項被轉(zhuǎn)移到了方程的右端。此時,在四邊形網(wǎng)格下,其雙線性基函數(shù)是通常的有限元基函數(shù)。兩個數(shù)值算例驗證了格式的可行性和有效性。第二部分進(jìn)一步討論了帶變系數(shù)的二維橢圓界面問題的浸入有限體積元方法,其變系數(shù)在通過界面時有一個有限的跳躍。由此導(dǎo)致其解和通量在通過界面時也會產(chǎn)生一個有限的跳躍,增加了數(shù)值計算上的困難。我們?nèi)韵仁褂迷错椧迫ゼ记?得到一個等價的帶齊次跳躍條件的橢圓界面問題。由于變系數(shù)的存在,在界面附近的節(jié)點基函數(shù)是分片多項式函數(shù),其構(gòu)造需滿足齊次跳躍條件。若遠(yuǎn)離界面,我們使用通常的有限元節(jié)點基函數(shù)。四邊形網(wǎng)格對分片多項式的構(gòu)造在某些情況下會產(chǎn)生奇異性,故我們使用了三角形網(wǎng)格。由此產(chǎn)生的線性問題簡單并且容易求解。我們對此進(jìn)行了在能量范數(shù)意義下的誤差估計。并給出了數(shù)值實驗。兩個數(shù)值實驗進(jìn)一步驗證了我們的結(jié)論：在L2范數(shù)意義下,界面附近的誤差和整體誤差均有O(h2)階精度；在H1范數(shù)意義下,均有O(h)階精度。第三章給出了一維帶界面的雙相延遲熱傳導(dǎo)方程的高階緊有限體積元方法。除界點格式外所得到的系數(shù)矩陣是三對角的,具有較好的對稱與對角性質(zhì),且易于求解。該高階方法有助于對這個方程在相對疏松的網(wǎng)格上研究納米級別的熱傳導(dǎo)現(xiàn)象,有重要的實際應(yīng)用價值。我們應(yīng)用離散能量方法在L2和L∞范數(shù)意義下給出誤差估計,其收斂階是O(△t~2+h~(3.5))。數(shù)值例子驗證了該方法的有效性和可行性。該高階有限體積元方法的構(gòu)造涉及到對原方程的回代。一旦我們遇到含多個變量的復(fù)雜方程時,該方法就不是很實用了。進(jìn)一步我們考慮了一個四階Pade型緊有限體積方法更簡便的高階方法,可以處理多維的界面問題。方程的解及其導(dǎo)數(shù)都達(dá)到了四階精度。第四章給出了本文的主要結(jié)論和有待進(jìn)一步解決的問題。
[Abstract]:This paper is devoted to the study of finite volume element method for two kinds of interface problems, which is divided into three parts. In the first chapter, we introduce some immersion methods about interface problems and expound the purpose of developing immersion finite volume element method. Then we introduce the biphase delay equation with interface, the high order compact finite volume element method and the Pade type compact finite volume method. In the second chapter, the immersion finite volume element method for Poisson equation with interface is discussed. By means of the source term removal technique, the interface problem with non-homogeneous jump condition is transformed into the interface problem with homogeneous jump condition, and the term related to the jump condition is transferred to the right end of the equation. In this case, the bilinear basis function of the quadrilateral mesh is the usual finite element basis function. Two numerical examples show that the scheme is feasible and effective. In the second part, we further discuss the finite volume element method for two-dimensional elliptic interface problem with variable coefficients, which has a finite jump when passing through the interface. As a result, the solution and flux also lead to a finite jump when passing through the interface, which increases the difficulty of numerical calculation. We still use the source term removal technique to obtain an equivalent elliptic interface problem with homogeneous jump conditions. Because of the existence of variable coefficients, the nodal basis function near the interface is a piecewise polynomial function, and its construction needs to satisfy the homogeneous jump condition. Far from the interface, we use the usual finite element node basis function. The construction of quadrilateral meshes to piecewise polynomials may result in singularity in some cases, so we use triangular meshes. The resulting linear problem is simple and easy to solve. We estimate the error in the sense of energy norm. Numerical experiments are also given. Two numerical experiments further verify our conclusion: in the sense of L _ 2 norm, the errors near the interface and the global errors have O (H2) order accuracy, and in the sense of H _ 1 norm, both have O (h) order accuracy. In chapter 3, the high order compact finite volume element method for the biphase delay heat conduction equation with interface is given. The coefficient matrix obtained in addition to the bounded point scheme is tridiagonal with good symmetry and diagonal properties and is easy to solve. The high-order method is helpful to the study of the nano-scale heat conduction phenomenon on the relatively loose grid, and has important practical application value. In the sense of L _ 2 and L _ 鈭，

本文編號：2062453