本文選題：變系數(shù) + 橢圓方程��； 參考：《上海大學(xué)》2016年博士論文

【摘要】：科學(xué)與工程計(jì)算中,有限差分方法,有限元方法,以及有限體積方法都是解偏微分方程的有效數(shù)值方法.近年來(lái),最小二乘有限元法也引起了部分學(xué)者的關(guān)注和發(fā)展.為了有高階精度,有學(xué)者還考慮了最小二乘譜方法.本文主要研究偏微分方程的Legendre-Galerkin Chebyshev配置(LGCC)最小二乘法.即,通過(guò)引進(jìn)一個(gè)通量將原問(wèn)題寫(xiě)成等價(jià)的一階系統(tǒng),然后對(duì)其建立LGCC最小二乘方法.接著,對(duì)該格式進(jìn)行理論分析以及用數(shù)值算例加以驗(yàn)證.本文內(nèi)容可歸納為如下幾部分：第一部分考慮一維的偏微分方程.首先,針對(duì)變系數(shù)兩點(diǎn)邊值問(wèn)題構(gòu)造LGCC最小二乘法.接著推導(dǎo)格式的強(qiáng)制性和連續(xù)性,以及誤差估計(jì).其次,還發(fā)展該方法的多區(qū)域形式,并探討算法的并行化和推廣單區(qū)域的理論結(jié)果.然后,研究含兩類(lèi)非齊次跳躍條件拋物方程的多區(qū)域LGCC方法,該格式對(duì)第一類(lèi)和第二類(lèi)跳躍條件分別本性和自然性處理.該方法被用于Stefan問(wèn)題的計(jì)算.最后,考慮含間斷變系數(shù)兩點(diǎn)邊值問(wèn)題的多區(qū)域LGCC最小二乘格式和對(duì)應(yīng)的理論分析.第二部分對(duì)兩維的橢圓方程,首先,研究?jī)删S變系數(shù)橢圓方程的LGCC最小二乘法.給出兩維Chebyshev插值算子的穩(wěn)定性分析和逼近結(jié)論,由此導(dǎo)出格式的強(qiáng)制性和連續(xù)性,以及得到關(guān)于H1-范數(shù)的誤差估計(jì).其次,將該方法應(yīng)用于Stokes方程的求解以及給出其理論分析.然后,發(fā)展含兩類(lèi)跳躍條件兩維變系數(shù)橢圓方程的多區(qū)域LGCC以及對(duì)其設(shè)計(jì)并行實(shí)施算法.最后,還考慮含跳躍條件變系數(shù)橢圓方程的多區(qū)域LGCC最小二乘法以及探討其并行實(shí)施算法.推廣前面對(duì)光滑函數(shù)的Chebyshev插值的穩(wěn)定性分析和相關(guān)的理論結(jié)論.第三部分針對(duì)三角形上變系數(shù)橢圓方程,我們發(fā)展Legendre-Galerkin數(shù)值積分(LG-NI)三角單元最小二乘法,并且對(duì)其強(qiáng)制性與連續(xù)性,以及收斂性進(jìn)行分析.進(jìn)一步,結(jié)合對(duì)區(qū)域內(nèi)部使用矩形單元?jiǎng)澐趾蛯?duì)邊界采用適當(dāng)三角元剖分,還研究多邊形上變系數(shù)橢圓方程的多區(qū)域LG-NI最小二乘法和它的收斂性.最后一部分探討發(fā)展方程LGCC最小二乘法.首先構(gòu)造拋物方程LGCC最小二乘法以及其的多步形式,并且給出它在Burgers的應(yīng)用.進(jìn)一步,討論兩種非線(xiàn)性發(fā)展方程的LGCC最小二乘法.對(duì)Burgers方程,先對(duì)它的一階系統(tǒng)用Crank-Nicolson方法離散,接著對(duì)其空間設(shè)計(jì)LGCC最小二乘法.另外,將該方法應(yīng)用于兩維非線(xiàn)性?huà)佄锓匠?本文的方法基于Legendre-Galerkin最小二乘法,對(duì)變系數(shù)用Chebyshev插值處理.該方法可導(dǎo)出對(duì)稱(chēng)正定代數(shù)方程,使其便于應(yīng)用迭代方法求解.注意到它還繼承了Legendre的良好穩(wěn)定性和避免Chebyshev權(quán)函數(shù)在區(qū)域的交界出現(xiàn)的奇性。
[Abstract]:In scientific and engineering calculation, finite difference method, finite element method and finite volume method are all effective numerical methods for solving partial differential equations. In recent years, the least square finite element method has also attracted the attention and development of some scholars. In order to have higher order accuracy, some scholars also consider the least square spectral method. In this paper, the Legendre-Galerkin Chebyshev collocation least square method for partial differential equations is studied. That is, by introducing a flux, the original problem is written as an equivalent first-order system, and then the LGCC least squares method is established for it. Then, the scheme is theoretically analyzed and verified by numerical examples. The content of this paper can be summarized as follows: the first part considers one-dimensional partial differential equation. Firstly, the LGCC least square method is constructed for the two point boundary value problem with variable coefficients. Then the mandatory and continuity of the format and error estimation are derived. Secondly, the multi-region form of the method is developed, and the parallelization of the algorithm and the theoretical results of extending the single region are discussed. Then, the multiregion LGCC method with two classes of nonhomogeneous jump conditions is studied. The scheme deals with the nature and naturalness of the first and second kinds of jump conditions, respectively. The method is used to calculate the Stefan problem. Finally, the multi-region LGCC least-squares scheme with discontinuous variable coefficient two-point boundary value problem and its corresponding theoretical analysis are considered. In the second part, the LGCC least square method for the two-dimensional elliptic equation with variable coefficients is studied. In this paper, the stability analysis and approximation results of two-dimensional C hebyshev interpolation operator are given. The mandatory and continuity of the scheme are derived, and the error estimates about H _ 1-norm are obtained. Secondly, the method is applied to the solution of Stokes equation and its theoretical analysis is given. Then, the multi-domain LGCC with two kinds of jump conditions for the elliptic equations with variable coefficients is developed and its parallel implementation algorithm is designed. Finally, the multiregion LGCC least square method with variable coefficient elliptic equations with jump conditions is considered and its parallel implementation algorithm is discussed. The stability analysis of Chebyshev interpolation for smooth functions and related theoretical conclusions are generalized. In the third part, for the elliptic equation with variable coefficients on the triangle, we develop the Legendre-Galerkin numerical integral and LG-NI-based least square method, and analyze its compulsion, continuity and convergence. Furthermore, the multi-region LG-NI least square method and its convergence for elliptic equations with variable coefficients on polygon are studied by using rectangular element partitioning and triangulation of boundary. In the last part, the development equation LGCC least square method is discussed. First, the parabolic equation LGCC least square method and its multistep form are constructed, and its application in Burgers is given. Furthermore, the LGCC least square method for two nonlinear evolution equations is discussed. For the Burgers equation, the first order system is discretized by Crank-Nicolson method, and then the LGCC least square method is designed for its space. In addition, the method is applied to two dimensional nonlinear parabolic equations. The method of this paper is based on Legendre-Galerkin least square method, and the variable coefficient is treated by Chebyshev interpolation. This method can be used to derive symmetric positive definite algebraic equations and make them easy to be solved by iterative method. It also inherits the good stability of Legendre and avoids the singularity of Chebyshev weight function at the junction of region.
【學(xué)位授予單位】：上海大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O241.82

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