二阶抛物型偏微分方程及位移障碍变分不等式问题的有限元分析

发布时间：2018-07-04 17:38

  本文选题：抛物积分微分方程 + 反应扩散方程　； 参考：《华北电力大学(北京)》2017年硕士论文

【摘要】：本论文主要研究两类二阶发展型偏微分方程及位移障碍变分不等式问题的有限元方法,并在不同条件下探讨其收敛性和超收敛性。首先,讨论了一类抛物型积分微分方程的双线性元逼近。利用插值与投影相结合新的技巧和插值后处理方法,在降低对解的正则性要求下,得到了H~1模意义下的O(h~2)阶超逼近与超收敛结果,这是以往文献单独使用投影算子或插值算子无法得到的。另外,我们还对不同的处理方法及结果进行了比较。其次,将著名的低阶非协调EQ_1~(rot)元应用于一类反应扩散方程。一方面,利用Lyopunov泛函证明了半离散格式逼近解的一个先验估计。同时,借助EQ_1~(rot)元所具有的两个特殊性质:(i)当精确解属于H3(Q)时,其相容误差可以达到O(h~2)阶,正好比插值误差O(h)高一阶。(ii)插值算子与投影算子等价,在有限元解uh不需要属于L_∞(Ω)的传统假设下,导出了H~1模意义下O(h~2)阶的超逼近性质。另一方面,建立了一个新的线性化向后Euler和线性化Crank-Nicolson全离散格式。通过对相容误差采用新的分裂技巧,对这两种格式分别导出了H~1模意义下具有O(h~2+τ)和0(h~2+τ2)阶的超逼近性质。进一步地,借助插值后处理技术,得到了相应的超收敛结果。另外,我们给出了一个数值算例,验证了理论分析的正确性。最后,研究了具有位移障碍的二阶变分不等式问题的低阶非协调带约束的旋转Q1元(CNQrot元)的收敛性和EQot元的超收敛性。一方面,在四边形网格下,对CNQ_1~(rot)元证明了一个有用的引理(见引理4.1),并由此给出了收敛性分析,得到了H~1模意义下的最优误差估计。另一方面,在矩形网格下,对EQ_1~(rot)元,通过一些更精细的估计和分析,得到了H~1模意义下的超收敛结果。同时,用数值算例验证了理论分析的正确性。特别需要强调的是:这一超收敛结果在以往文献中从未报道过。
[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，

本文编号：2096844