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  本文選題：二階拋物界型面問題 + 部分懲罰方法　； 參考：《山東師范大學(xué)》2015年碩士論文

【摘要】：界面問題刻畫了諸如由復(fù)雜地質(zhì)結(jié)構(gòu)或多向流導(dǎo)致的具有問斷擴(kuò)散系數(shù)的混溶驅(qū)替等實(shí)際滲流過程,建立其準(zhǔn)確高效的數(shù)值模擬方法和完整的數(shù)值分析理論體系,對深刻揭示實(shí)際滲流的運(yùn)動機(jī)理、指導(dǎo)科學(xué)工程實(shí)踐具有重要的理論價值和應(yīng)用前景.本文旨在對可描述各向異性滲流問題的一類二階拋物型界面問題提出相應(yīng)的有限元數(shù)值模擬格式,并建立嚴(yán)格的數(shù)值分析理論.主要內(nèi)容分為兩個部分： 1.基于線性拉格朗日插值的部分懲罰浸入界面有限元方法 在這一部分中,我們運(yùn)用線性拉格朗日插值構(gòu)造浸入界面有限元空問,對于問中函數(shù)的存在唯一性.進(jìn)一步,將上述空問構(gòu)造中這一條件弱化至若sl0,則證明了浸入界面線性有限元空問中的函數(shù)可由三角形單元頂點(diǎn)值唯一確定,從而運(yùn)用較弱的條件構(gòu)造出相應(yīng)的分片線性有限元空問,拓廣了該方法的應(yīng)用范圍.基于由上述條件構(gòu)造出的有限元空問,我們對二維拋物型界面問題建立了對稱、非對稱及不完全的部分懲罰浸入界面有限元(PIFE)半離散和全離散格式.利用橢圓投影、Gronwall不等式等數(shù)值分析技術(shù),證明了格式的可解性、穩(wěn)定性和最優(yōu)能量模與L2模誤差估計. 2.基于旋轉(zhuǎn)Q1元的部分懲罰浸入界面有限元方法 在這一部分中,我們運(yùn)用旋轉(zhuǎn)Q1元構(gòu)造浸入界面有限元空問,對于擴(kuò)散系元空問中函數(shù)的存在唯一性.證明了浸入界面雙線性有限元空問中的函數(shù)可由矩形單元四邊中點(diǎn)處的函數(shù)值唯一確定.從而,構(gòu)造出了相應(yīng)的分片線性有限元空間.基于由上述條件構(gòu)造的有限元空間,我們對二維拋物型界面問題建立了矩形剖分下對稱、非對稱及不完全的浸入界面雙線性有限元半離散和全離散格式.利用橢圓投影、Gronwall不等式等數(shù)值分析技術(shù),證明了格式的可解性與穩(wěn)定性,最終給出了次最優(yōu)能量模誤差估計.
[Abstract]:The interface problem depicts the actual seepage process such as mixed displacement with fault diffusion coefficient caused by complex geological structure or multidirectional flow, and establishes its accurate and efficient numerical simulation method and complete numerical analysis theory system.It has important theoretical value and application prospect for revealing the movement mechanism of actual seepage and guiding scientific engineering practice.The purpose of this paper is to propose a finite element numerical simulation scheme for a class of second-order parabolic interface problems which can describe anisotropic seepage problems, and to establish a strict numerical analysis theory.The main content is divided into two parts:1.Partial penalty Immersion Interface finite element method based on Linear Lagrange interpolationIn this part, we use the linear Lagrange interpolation to construct the space question of the finite element immersed in the interface, and the existence and uniqueness of the function in the question is obtained.Furthermore, by weakening this condition to sl0, it is proved that the function immersed in the linear finite element space problem of the interface can be uniquely determined by the vertex value of the triangular element.The corresponding piecewise linear finite element space problems are constructed by using the weaker conditions, and the application range of the method is extended.Based on the finite element space problems constructed from the above conditions, the symmetric, asymmetric and incomplete partial penalty immersion finite element (PIFE) semi-discrete and fully discrete schemes are established for two-dimensional parabolic interface problems.By using the elliptic projection Gronwall inequality and other numerical analysis techniques, the solvability and stability of the scheme and the error estimates of the optimal energy norm and L 2 norm are proved.2.Partial penalty Immersion Interface finite element method based on rotating Q1 elementIn this part, we use the rotating Q1 element to construct the space question of the immersion interface finite element, and the existence and uniqueness of the function in the diffusion system element space problem are obtained.It is proved that the function in the space problem of the bilinear finite element immersed in the interface can be uniquely determined by the function value at the midpoint of the four edges of the rectangular element.Thus, the corresponding piecewise linear finite element space is constructed.Based on the finite element space constructed by the above conditions, we establish the semi-discrete and fully discrete schemes of bilinear finite element for two-dimensional parabolic interface problems with symmetric, asymmetric and incomplete immersion interfaces under rectangular subdivision.By using the elliptic projection Gronwall inequality and other numerical analysis techniques, the solvability and stability of the scheme are proved. Finally, the suboptimal energy modulus error estimates are given.
【學(xué)位授予單位】：山東師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O241.82

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 王淑燕;陳煥貞;;基于Crouzeix-Raviart元的界面浸入有限元方法及其收斂性分析[J];計算數(shù)學(xué);2012年02期

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本文編號：1742292