若干偏微分方程的混合有限元方法研究

发布时间：2018-03-13 22:32

  本文选题：四阶发展方程　切入点：Poisson特征值问题　出处：《郑州大学》2017年博士论文　论文类型：学位论文

【摘要】：本论文主要研究几类四阶发展方程(非线性Molecular Beam Epitaxy(MBE)方程、Sivashinsky方程以及双曲方程)和二阶椭圆特征值问题的混合有限元方法.分别从协调和非协调混合元出发,对其收敛性、超逼近、超收敛以及外推等方面进行深入系统的研究.首先,探讨了两类四阶非线性MBE方程的协调混合元方法.利用双线性元插值的高精度估计,分别在半离散和两种全离散格式(Backward-Euler(B-E)和CrankNicolson(C-N))下,导出了原始变量u和中间变量p在H~1模意义下的超逼近,然后通过插值后处理技术给出了这两个变量的整体超收敛结果.其次,考虑了四阶非线性Sivashinsky方程的一个低阶非协调混合元和扩展的新混合元方法.一方面,利用非协调EQ_1~(rot)元的两个特殊性质:相容误差在能量模意义下为O(h~2)阶(比插值误差高一阶)以及其插值算子与Ritz投影算子等价,分别在半离散以及B-E全离散格式下,得到了原始变量u和中间变量p在能量模意义下O(h~2)阶的超逼近和超收敛结果.另一方面,对该方程建立一个扩展的新混合元格式,借助于最低阶Raviart-Thomas(R-T)元的特殊性质,积分恒等式技巧和插值后处理技术,在半离散和B-E全离散格式下给出了相关变量的超逼近和整体超收敛结果.再次,讨论了四阶双曲波动方程协调双线性混合元方法.利用插值和投影相结合的技巧,分别在半离散和全离散格式下,得到了原始变量u和中间变量p在H~1模意义下O(h~2)阶的超逼近和超收敛结果.对比以往文献中单独使用插值的方法,利用插值和投影相结合的优势在于不仅降低了u,u_t和p的光滑度,而且得到了超收敛结果.最后,研究了Poisson特征值问题非协调有限元以及混合元方法.一方面,将一个非协调四边形元(改进的类Wilson元)应用于该问题,利用此单元所具有的的特殊性质(当u∈H~3(?)时,相容误差为O(h~2)阶,比其插值误差O(h)高一阶)和插值后处理技巧,分别在广义矩形网格和矩形网格下,得到了特征向量u在能量模意义下的超逼近和超收敛结果;接下来,证明了该单元一个新的性质,即:当u∈H~5(?)时,其相容误差在任意四边形网格下能够达到O(h~4)阶,基于上述特性并结合协调双线性元的渐近展开式,得到了特征值O(h~4)阶的外推解.另一方面,对该方程建立了一个新的非协调混合元方法,利用EQ_1~(rot)元和最低阶R-T元的特殊性质,分别得到了原始变量u和辅助变量?p的最优误差估计以及特征值λ的下界逼近;进一步地,根据积分恒等式技巧和插值后处理技术,给出了u在能量模意义下以及?p在L2模意义下O(h~2)阶的超逼近和超收敛结果;最后,根据渐近展开式,得到了特征值O(h3)阶的外推解.同时,针对上述每一部分都给出对应的数值算例来验证理论分析的正确性。
[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，

本文编号：1608425