传输特征征值问题非协调元法和混合元法二网格离散方案

发布时间：2018-06-02 11:30

本文选题：传输特征值 + 非协调有限元　；参考：《贵州师范大学》2017年硕士论文

【摘要】：传输特征值问题是非均匀介质逆散射理论中的二次特征值问题.传输特征值能用来估计散射体材料的性质,并且在逆散射理论中唯一性和重构性方面具有重要理论意义.本文基于Helmholtz传输特征值问题非协调元法和混合元法的变分格式,建立了非协调元法和混合元法二网格离散方案.采用该方案,在细网格π_H上求传输特征值问题的解归结为在粗网格上求原特征值问题及其共轭问题的解,然后在细网格π_H上求两个系数矩阵为正定稀疏Hermite的块对角矩阵的线性代数方程组的解.对于非协调元法二网格离散方案,本文证明了结果解仍保持渐近最优精度,并报道了采用修正的Zienkiewicz元在二维和三维情形的数值算例来验证方案的有效性.对于混合元法二网格离散方案,数值实验验证了该方案的有效性.
[Abstract]:The transmission eigenvalue problem is a quadratic eigenvalue problem in the inverse scattering theory of inhomogeneous media. The transmission eigenvalues can be used to estimate the properties of scatterers and have important theoretical significance in the uniqueness and reconstruction of inverse scattering theory. Based on the variational schemes of non-conforming element method and mixed element method for Helmholtz transmission eigenvalue problem, a two-grid discretization scheme for non-conforming element method and hybrid element method is established. Using this scheme, the solution of the transmission eigenvalue problem on fine mesh 蟺 H is reduced to the solution of the original eigenvalue problem and its conjugate problem on the rough grid. Then, the solutions of two linear algebraic equations of block diagonal matrix with positive definite sparse Hermite are obtained on the fine grid 蟺 H. For the two-grid discrete scheme of nonconforming element method, the asymptotic optimal accuracy of the solution is proved in this paper, and the validity of the scheme is verified by a numerical example of the modified Zienkiewicz element in two-dimensional and three-dimensional cases. For the mixed element method, the effectiveness of the scheme is verified by numerical experiments.
【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】：O241.8

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