Helmholtz传输特征值混合元法

发布时间：2018-07-12 08:34

本文选题：Helmholtz传输特征值 + 混合有限元法　；参考：《贵州师范大学》2017年硕士论文

【摘要】：Helmholtz传输特征值问题由于在材料科学有着重要的应用,一直都是数学和力学界关注的热点问题,很多学者倾其精力于它的数值处理上。该问题的困难在于它是一个非椭圆且非对称的方程,无法用传统的特征值分析方法直接求解。在该研究领域中,Cakoni、Monk和Sun提出了一个新颖的变分公式,并对此方法对应的有限元逼近做了误差分析。同时,Yang、Bi、Li和Han将Ciarlet-Raviart混合元法应用于该问题并取得理想效果。由于混合元法对于重调和方程有广泛的应用,能够降低空间自由度,提高计算效率,因此混合元法的研究有重要价值。本文从理论角度出发,参考多个学者的研究方法和成果,对传输特征值问题提出了相应的混合计算方法,依靠特殊论证证明了方法的正确性和可行性,达到研究的预期效果。首先,我们将Cakoni等人提出的公式与Ciarlet-Raviart混合元结合起来,提出一个新的混合变分公式。因为其无法直接利用经典特征值理论,我们依靠两个正则性估计来证明混合变分公式根的存在性及唯一性,并给出对应的共轭问题。然后证明变分问题和离散问题的解算子都是紧算子,为收敛性的证明做准备工作。随后,我们定义特殊的拉格朗日插值算子和纽曼投影算子,运用插值理论,证明该问题有限元解的收敛性,进而推出离散问题的解算子收敛于变分问题的解算子。基于该结论,我们即可依托特征值经典理论来说明有限元混合变分公式所得特征值和特征函数的收敛性,达到期望的结果。在文章的最后,我们对该方法的收敛阶进行了探讨,通过特殊的论证方法,本问题的收敛阶可得到进一步的提高,直接说明此方法的可行性和高效性。
[Abstract]:The Helmholtz transport eigenvalue problem has been a hot issue in mathematics and mechanics because of its important applications in material science. Many scholars have devoted their attention to its numerical treatment. The difficulty of this problem is that it is a non-elliptic and asymmetric equation and can not be directly solved by traditional eigenvalue analysis method. In this field, Cakonione Monk and Sun have proposed a new variational formula, and the error analysis of the finite element approximation corresponding to this method has been made. At the same time, the Ciarlet-Raviart mixed element method is applied to the problem and the ideal results are obtained. Because the hybrid element method is widely used in the biharmonic equation, it can reduce the degree of freedom in space and improve the computational efficiency, so the study of the hybrid element method is of great value. In this paper, from the theoretical point of view and referring to the research methods and achievements of many scholars, the corresponding hybrid calculation method for the transmission eigenvalue problem is proposed. The correctness and feasibility of the method are proved by special demonstration, and the expected results of the study are achieved. First, we combine Cakoni's formula with Ciarlet-Raviart mixed element, and propose a new mixed variational formula. Because it can not directly use classical eigenvalue theory, we use two regularity estimates to prove the existence and uniqueness of the root of mixed variational formula, and give the corresponding conjugate problem. Then it is proved that the solution operators of the variational problem and the discrete problem are compact operators, which are prepared for the proof of convergence. Then, we define the special Lagrange interpolation operator and Newman projection operator, prove the convergence of the finite element solution of the problem by using interpolation theory, and then deduce that the solution operator of the discrete problem converges to the solution operator of the variational problem. Based on this conclusion, we can use the classical theory of eigenvalue to illustrate the convergence of eigenvalues and eigenfunctions of finite element mixed variational formulas, and achieve the desired results. At the end of the paper, we discuss the convergence order of this method. Through a special demonstration method, the convergence order of this problem can be further improved, which directly shows the feasibility and efficiency of this method.
【学位授予单位】：贵州师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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