非均匀介质内传输问题及其特征值问题的数值方法研究

项目名称： 非均匀介质内传输问题及其特征值问题的数值方法研究

项目编号： No.11301089

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 吴新明

作者单位： 复旦大学

项目金额： 22万元

中文摘要： 非均匀介质内传输问题及其特征值问题是地质勘探、雷达成像、无损探伤等众多领域的一个重要研究课题，近几年吸引了国内外很多专家学者的研究兴趣，成为科学工程计算中新的研究热点。本项目的研究具有重要的理论价值和广泛的应用前景。目前关于该问题的研究工作主要还停留在理论层面，数值方法方面的结果并不多，特别是高效实用的数值方法亟待进一步研究。本项目主要拟针对Helmholtz方程和Maxwell方程内传输问题研究高效的数值离散格式，推导离散方法的先验和后验误差估计，利用边界条件耦合设计新的区域迭代算法；针对Helmholtz方程和Maxwell方程传输特征值问题，探索等价的耦合变分问题，构造稳定的非线性迭代方法，并进行收敛性等相关理论分析；进而对声波和电磁逆散射问题设计更为高效的数值方法，并进行相关的大规模数值模拟。

中文关键词： 内传输特征值；非线性特征值；有限元方法；误差估计；优化方法

英文摘要： The interior transmission problem and the associated eigenvalue problem in inhomogeneous media are important research issues in many fields, such as geological prospecting, radar imaging and nondestructive testing. They have attracted much attension in recent years and become new hot issues in the research field of scientific and engineering computing. This project has very important theorectical value and lots of applications. Even though there are some works on the theory of these problems, the study on practical numerical methods of high efficiency is quite limit. This project aims to develop highly efficient discretization schemes for the interior transmission problem of both Helmholtz and Maxwell equations, derive the priori and the a posteriori error estimates, and design new domain iteration algorithm based on the coupling of the boundary conditions. For the interior transmission eigenvalue problem, we will also seek the equivalent coupled variational problems, construct stable nonlinear iterative methods and prove the convergence of the methods. Finally, fast and efficent numerical methods will be designed to solve the large scale inverse acoustic and electromagnetic scattering problems.

英文关键词： interior transmission eigenvalue；nonlinear eigenvalue；finite element method；error estimate；optimization method