本文選題：拋物積分微分方程 + 反應(yīng)擴(kuò)散方程��； 參考：《華北電力大學(xué)(北京)》2017年碩士論文

【摘要】：本論文主要研究?jī)深惗A發(fā)展型偏微分方程及位移障礙變分不等式問(wèn)題的有限元方法,并在不同條件下探討其收斂性和超收斂性。首先,討論了一類拋物型積分微分方程的雙線性元逼近。利用插值與投影相結(jié)合新的技巧和插值后處理方法,在降低對(duì)解的正則性要求下,得到了H~1模意義下的O(h~2)階超逼近與超收斂結(jié)果,這是以往文獻(xiàn)單獨(dú)使用投影算子或插值算子無(wú)法得到的。另外,我們還對(duì)不同的處理方法及結(jié)果進(jìn)行了比較。其次,將著名的低階非協(xié)調(diào)EQ_1~(rot)元應(yīng)用于一類反應(yīng)擴(kuò)散方程。一方面,利用Lyopunov泛函證明了半離散格式逼近解的一個(gè)先驗(yàn)估計(jì)。同時(shí),借助EQ_1~(rot)元所具有的兩個(gè)特殊性質(zhì):(i)當(dāng)精確解屬于H3(Q)時(shí),其相容誤差可以達(dá)到O(h~2)階,正好比插值誤差O(h)高一階。(ii)插值算子與投影算子等價(jià),在有限元解uh不需要屬于L_∞(Ω)的傳統(tǒng)假設(shè)下,導(dǎo)出了H~1模意義下O(h~2)階的超逼近性質(zhì)。另一方面,建立了一個(gè)新的線性化向后Euler和線性化Crank-Nicolson全離散格式。通過(guò)對(duì)相容誤差采用新的分裂技巧,對(duì)這兩種格式分別導(dǎo)出了H~1模意義下具有O(h~2+τ)和0(h~2+τ2)階的超逼近性質(zhì)。進(jìn)一步地,借助插值后處理技術(shù),得到了相應(yīng)的超收斂結(jié)果。另外,我們給出了一個(gè)數(shù)值算例,驗(yàn)證了理論分析的正確性。最后,研究了具有位移障礙的二階變分不等式問(wèn)題的低階非協(xié)調(diào)帶約束的旋轉(zhuǎn)Q1元(CNQrot元)的收斂性和EQot元的超收斂性。一方面,在四邊形網(wǎng)格下,對(duì)CNQ_1~(rot)元證明了一個(gè)有用的引理(見(jiàn)引理4.1),并由此給出了收斂性分析,得到了H~1模意義下的最優(yōu)誤差估計(jì)。另一方面,在矩形網(wǎng)格下,對(duì)EQ_1~(rot)元,通過(guò)一些更精細(xì)的估計(jì)和分析,得到了H~1模意義下的超收斂結(jié)果。同時(shí),用數(shù)值算例驗(yàn)證了理論分析的正確性。特別需要強(qiáng)調(diào)的是:這一超收斂結(jié)果在以往文獻(xiàn)中從未報(bào)道過(guò)。
[Abstract]:In this paper, the finite element methods for two classes of second order evolution partial differential equations and variational inequalities for displacement obstacles are studied, and their convergence and superconvergence are discussed under different conditions. Firstly, the bilinear element approximation for a class of parabolic integrodifferential equations is discussed. By using the new technique of interpolation and projection and the interpolation post-processing method, under the condition of decreasing the regularity of the solution, the superapproximation and superconvergence results of order O (HH ~ 2) in the sense of H ~ (1) norm are obtained. This can not be obtained by using projection operator or interpolation operator alone in previous literature. In addition, we also compare different treatment methods and results. Secondly, the well-known low order nonconforming EQ1 ~ (rot) element is applied to a class of reaction-diffusion equations. On the one hand, a priori estimate of the approximate solution of the semi-discrete scheme is proved by using Lyopapunov Functionals. At the same time, with the help of two special properties of the EQ1 ~ (rot) element, when the exact solution belongs to H3 (Q), the compatible error of (i) can reach O (H2) order, which is exactly higher than the interpolation error O (h). The. (ii) interpolation operator is equivalent to the projection operator. Under the traditional assumption that the finite element solution uh does not need to belong to L _ 鈭，

本文編號(hào)：2096844