腔体区域上电磁场散射问题的高效数值方法研究

In this project, theoretical and numerical analysis is given for the electromagnetic scattering problems in a cavity by coupling the inifinite element methods and finite element methods. Meanwhile, anisotropic nonconforming tetrahedron finite elements are constructed and artificial boundary condition is established. During this processing, for those unbounded domain and those domains concluding singular points, geometric similar polygons are plotted, between which finite element meshes are employed. Then stiffness matrix and transfer matrix can be derived from the electromagnetic scattering problems in a cavity. Through those high order numerical solvers for ODEs, the equivalent solution of ODEs can be obtained. Furthermore, outlines of error estimates theories can be derived for the unbounded domain and the domains concluding singular points. Practically, efficient and fast algorithms will be designed for electromagnetic scattering problems in a cavity. It is a very important and meaningful job for us to inherit and develop the coupling of infinite element methods and finite element methods as long as we are going to apply this charactering and challenging job.

本课题研究构造非协调各向异性有限元，建立人工边界条件，使用无限元有限元耦合方法，逼近腔体上电磁场散射问题，给出数值逼近的误差估计框架，进行数值模拟分析。对无限远区域和奇异点分别进行几何相似多边形剖分，在中间环形区域进行有限元网格剖分，根据腔体电磁场散射问题，构建组合矩阵和转移矩阵，使用高阶常微分数值解法，求解等价的常微分方程。对转移矩阵对应的解，分析其在奇异点处的性态，进一步构建在奇异点的误差估计理论。结合实际问题，设计高效快速可行的无限元有限元耦合模拟软件。由于我们继承和发扬了无限元有限元耦合这一新方法，在国内外开展了这一独具特色且具有挑战性的工作，并且国际上在这一方面的工作又相当少，其创新性和突破性进展对发展腔体上电磁场散射问题的数值模拟具有重要的理论意义和重要应用价值。

本课题构造非协调各向异性有限元，建立人工边界条件，使用无限元有限元耦合方法，逼近腔体上电磁场散射问题，给出数值逼近的误差估计框架，进行数值模拟分析。对无限远区域和奇异点分别进行几何相似多边形剖分，在中间环形区域进行有限元网格剖分，根据腔体电磁场散射问题，构建组合矩阵和转移矩阵，使用高阶常微分数值解法，求解等价的常微分方程。对转移矩阵对应的解，分析其在奇异点处的性态，进一步构建在奇异点的误差估计理论。结合实际问题，设计高效快速可行的无限元有限元耦合模拟软件。并对相关问题在电磁场，Stokes问题和特征值问题等做出一定的推广和应用。