本文選題：四階發(fā)展方程　切入點：Poisson特征值問題　出處：《鄭州大學》2017年博士論文　論文類型：學位論文

【摘要】：本論文主要研究幾類四階發(fā)展方程(非線性Molecular Beam Epitaxy(MBE)方程、Sivashinsky方程以及雙曲方程)和二階橢圓特征值問題的混合有限元方法.分別從協(xié)調(diào)和非協(xié)調(diào)混合元出發(fā),對其收斂性、超逼近、超收斂以及外推等方面進行深入系統(tǒng)的研究.首先,探討了兩類四階非線性MBE方程的協(xié)調(diào)混合元方法.利用雙線性元插值的高精度估計,分別在半離散和兩種全離散格式(Backward-Euler(B-E)和CrankNicolson(C-N))下,導(dǎo)出了原始變量u和中間變量p在H~1模意義下的超逼近,然后通過插值后處理技術(shù)給出了這兩個變量的整體超收斂結(jié)果.其次,考慮了四階非線性Sivashinsky方程的一個低階非協(xié)調(diào)混合元和擴展的新混合元方法.一方面,利用非協(xié)調(diào)EQ_1~(rot)元的兩個特殊性質(zhì):相容誤差在能量模意義下為O(h~2)階(比插值誤差高一階)以及其插值算子與Ritz投影算子等價,分別在半離散以及B-E全離散格式下,得到了原始變量u和中間變量p在能量模意義下O(h~2)階的超逼近和超收斂結(jié)果.另一方面,對該方程建立一個擴展的新混合元格式,借助于最低階Raviart-Thomas(R-T)元的特殊性質(zhì),積分恒等式技巧和插值后處理技術(shù),在半離散和B-E全離散格式下給出了相關(guān)變量的超逼近和整體超收斂結(jié)果.再次,討論了四階雙曲波動方程協(xié)調(diào)雙線性混合元方法.利用插值和投影相結(jié)合的技巧,分別在半離散和全離散格式下,得到了原始變量u和中間變量p在H~1模意義下O(h~2)階的超逼近和超收斂結(jié)果.對比以往文獻中單獨使用插值的方法,利用插值和投影相結(jié)合的優(yōu)勢在于不僅降低了u,u_t和p的光滑度,而且得到了超收斂結(jié)果.最后,研究了Poisson特征值問題非協(xié)調(diào)有限元以及混合元方法.一方面,將一個非協(xié)調(diào)四邊形元(改進的類Wilson元)應(yīng)用于該問題,利用此單元所具有的的特殊性質(zhì)(當u∈H~3(?)時,相容誤差為O(h~2)階,比其插值誤差O(h)高一階)和插值后處理技巧,分別在廣義矩形網(wǎng)格和矩形網(wǎng)格下,得到了特征向量u在能量模意義下的超逼近和超收斂結(jié)果;接下來,證明了該單元一個新的性質(zhì),即:當u∈H~5(?)時,其相容誤差在任意四邊形網(wǎng)格下能夠達到O(h~4)階,基于上述特性并結(jié)合協(xié)調(diào)雙線性元的漸近展開式,得到了特征值O(h~4)階的外推解.另一方面,對該方程建立了一個新的非協(xié)調(diào)混合元方法,利用EQ_1~(rot)元和最低階R-T元的特殊性質(zhì),分別得到了原始變量u和輔助變量?p的最優(yōu)誤差估計以及特征值λ的下界逼近;進一步地,根據(jù)積分恒等式技巧和插值后處理技術(shù),給出了u在能量模意義下以及?p在L2模意義下O(h~2)階的超逼近和超收斂結(jié)果;最后,根據(jù)漸近展開式,得到了特征值O(h3)階的外推解.同時,針對上述每一部分都給出對應(yīng)的數(shù)值算例來驗證理論分析的正確性。
[Abstract]:In this paper, we mainly study the hybrid finite element methods for several kinds of fourth order evolution equations (nonlinear Molecular Beam Epitaxymbs) equations and hyperbolic equations, and the second order elliptic eigenvalue problems. The superapproximation, superconvergence and extrapolation are studied in detail. Firstly, two classes of fourth order nonlinear MBE equations are studied by using the coordinate mixed element method. In this paper, we derive the superapproximation of the original variable u and the intermediate variable p in the sense of H ~ (1) norm respectively under semi-discrete and two kinds of full discrete schemes (Backward-Eulerian B-E) and CrankNicolsonian C-NU), and then give the global superconvergence results of these two variables by interpolation post-processing technique. In this paper, a low order nonconforming mixed element and an extended new mixed element method for the fourth order nonlinear Sivashinsky equation are considered. By using two special properties of nonconforming EQ1 / rotatory) element, the consistent error is 2 order (one order higher than the interpolation error) in the sense of energy module, and the interpolation operator is equivalent to the Ritz projection operator, respectively, in semi-discrete and B-E full discrete schemes. The superapproximation and superconvergence results of the original variable u and the intermediate variable p are obtained in the sense of energy norm. On the other hand, an extended new mixed element scheme is established for the equation, and the special properties of the lowest order Raviart-Thomasn R-T element are obtained. The integral identity technique and interpolation post-processing technique are used to obtain the superapproximation and global superconvergence results of the related variables in semi-discrete and B-E full discrete schemes. The harmonic bilinear mixed element method for the fourth order hyperbolic wave equation is discussed. By using the technique of combining interpolation and projection, the method is used in semi-discrete and fully discrete schemes, respectively. The superapproximation and superconvergence results of the original variable u and the intermediate variable p in the sense of H ~ (1) norm are obtained. The advantage of the combination of interpolation and projection is that not only the smoothness of UT and p is reduced, but also the superconvergence results are obtained. Finally, the nonconforming finite element method and hybrid element method for Poisson eigenvalue problem are studied. In this paper, a nonconforming quadrilateral element (improved Wilson element) is applied to this problem. ) and interpolation post-processing techniques, the superapproximation and superconvergence results of eigenvector u in the sense of energy module are obtained under generalized rectangular mesh and rectangular grid respectively. A new property of the unit is proved, that is, if u 鈭，

本文編號：1608425