复杂腔体上电磁散射大波数问题非协调元逼近及加速技术研究

The Project aims to investigate the stability of the solutions of the electromagnetic scattering problems with large wave number in complex cavities under the different norms; we focus on the estimates of nonconforming finite element on the bounday in order to deduce the effects of large wave number by studying the excellent properties instead of penalty methods; we also explore the natural differences between the discrete systems by nonconforming finite elements and those of conforming finite elements with penalty methods from the point views of matrix theoreis; we also extent our methods to the electromagnetic scattering problems with random coefficients, the problems with discontinous wave number,multicavities, multi-mediums and so on; we will try to set up the fast discrete systems and fast solvers and discover the general framework of fast algorithms by nonconforming finite element methods; In the end, the gradient patch recovery method will be established in order to deduce the computational costs for the electromagnetic scattering problem with large wave number in different cavites. We have been carrying out this unique and challenging work in the country earlier and there are few international reports on this direction, its innovative and breakthrough results may enrich and develop the nonconforming finite element contents, which have important theoretical significance and application value.

分析研究复杂腔体电磁散射问题的解在不同范数意义下受波数控制的稳定性；重点解决好非协调元逼近电磁散射问题时在边界上的估计，使用非协调元优异的性质，不再使用惩罚方法，达到自动减弱大波数对数值解影响的目的；研究非协调元离散系统与用惩罚方法协调元离散产生的系统之间的本质差别，进一步从矩阵论角度研究大波数对数值解的影响；研究非协调元在逼近带有随机系数、间断波数、多腔区域，非凸腔区域、多介质等电磁散射问题时体现的优越性；研究非协调元快速离散求解技术，探索非协调元形成快速离散求解系统应具备的基本条件，建立有限元快速求解技术的基本框架体系；使用梯度恢复技术，构建非协调元逼近复杂腔体上电磁散射问题的后处理技术，降低计算代价。由于我们较早在国内开展用非协调元代替惩罚方法这一先进的具有特色和挑战性的技术，其创新性和突破性进展对丰富和发展非协调有限元的内容具有重要的理论意义和应用价值。

本项目分析研究复杂腔体电磁散射问题的解在不同范数意义下受波数控制的稳定性；重点解决好非协调元逼近电磁散射问题时在边界上的估计，使用非协调元优异的性质，不再使用惩罚方法，达到自动减弱大波数对数值解影响的目的；研究非协调元离散系统与用惩罚方法协调元离散产生的系统之间的本质差别，进一步从矩阵论角度研究大波数对数值解的影响；研究非协调元在逼近带有随机系数、间断波数、多腔区域，非凸腔区域、多介质等电磁散射问题时体现的优越性；研究非协调元快速离散求解技术，探索非协调元形成快速离散求解系统应具备的基本条件，建立有限元快速求解技术的基本框架体系；使用梯度恢复技术，构建非协调元逼近复杂腔体上电磁散射问题的后处理技术，降低计算代价。由于我们较早在国内开展用非协调元代替惩罚方法这一先进的具有特色和挑战性的技术，其创新性和突破性进展对丰富和发展非协调有限元的内容具有重要的理论意义和应用价值。