超材料中电磁波的非协调有限元数值模拟及加速技术的新研究模式

The project aims to the electromagnetic wave propagation in mate-materials and investigate the relationships between the properties of the mate-materials and the incident,scattering and reflection, which will be the great development for the applications of the cloaking and the super lens. Firstly, the governing equations will be set up based on the constitutive relations of electromagnetic fields with Faraday’s law and Ampere’s law. Further more, under rectangular coordinate system, polar coordinate system and the elliptical coordinate system, the stability, existence and uniqueness of the solutions will be provided for the PML model. Secondly, the robust Euler, Crank-Nilcolson or Rungge-Kutta discrete schemes of time-space mixed nonconforming finite element methods are established as well as the stability, existence and uniqueness of the discrete solutions. Next, the complicated domains are also studied such as the infty domain, cavity domain and non-convex domain. At the same time, these methods can be also extend the cloaking model and the super lens model. Lastly, in order to improve the computational efficiency, the RK4-DG methods will be developed, which can compute the unknowns from the globally to the locally and the accuracy be be O(t^4+h^k). In the end, the post-processing technology will presented for improving the global accuracy by superconvergence and a posteriori error estimates. During this processing, the advantages of using traditional discreet scheme instead of penalty methods and will be completed and the Runge-phenomena will disappear if the high order finite elements are employed.

本项目研究电磁波在超材料中的传播，通过建立超材料中电磁波传播的数学模型，分析超材料对电磁波传播过程的影响，使用非协调有限元数值模拟技术，提高在隐形技术和超级透镜技术等高端科技上的应用价值。首先，建立在不同坐标系下的PML条件，分析PML模型解的存在唯一性和稳定性。其次，将Euler、Crank-Nicolson以及Runge-Kutta等时间离散格式和非协调混合元耦合一起建立健壮的计算方法，分析其有效性、高效性和稳定性；同时，也将分析范围从有限区间延伸到无穷区域、腔体区域、非凸区域等复杂区域，建立非协调元逼近超材料中电磁波传播新的逼近模式，其将具有仅用传统离散格式而非惩罚格式和高次元逼近不出现龙格现象等优点。再次，为提高计算效率，研究龙格库塔不连续非协调元方法，将空间计算模式从整体转换成局部，减少存储量，实现大规模科学计算。最后，将建立超收敛和后验误差估计子处理格式，达到高精度的计算效果。

本项目研究电磁波在超材料中的传播，通过建立PML数学模型，分析超材料对电磁波传播过程的影响，对隐形技术和超级透镜技术有积极的应用价值。首先，建立电磁波在超材料中的控制方程，在不同坐标系下建立PML条件，形成PLM模型，分析解的存在唯一性和稳定性。其次，将Euler格式、Crank-Nicolson格式和非协调混合元耦合一起建立健壮的计算方法，分析其有效性、高效性和稳定性；将分析范围从有限区间延伸到无穷区域、腔体区域、非凸区域等复杂区域；将基本理论推广到隐形斗篷和超级透镜模型上；为提高计算效率，研究龙格库塔不连续协调元方法，将时间离散精度提高到四阶，将空间计算模式从整体转换成局部，减少存储量，实现大规模科学计算。最后，建立超收敛和后验误差子处理格式，达到高精度的计算效果。将非协调有限元技术应用到电磁场的研究领域，其创新性和突破性进展对丰富发展非协调元方法具有重要的理论意义和应用价值。

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