复合材料里电磁问题的有限元方法

In this research project, we consider the finite element method of two-dimensional.Maxwell's equations in inhomogeneous isotropic medium. We will apply the.generalized Hodge Decomposition Theorem in the weighted squarely integrable vector.space to the corresponding varitional problem of Maxwell's equations in.inhomogeneous isotropic medium, transform it to two coupled variation problems of.Laplace's equation, and then   take advantage of the singular representation of  .the solution of Laplace's equation to develop a reasonable finite element method..To speed up the computation, we further consider the corresponding multi-grid.faster solver. This project is the demonstration of the following new approach:.apply the generalized Hodge Decomposition Theorem to find an equivalent.variational problems for some partial differential equations and equation systems.in inhomogeneous isotropic medium, then work on the simplified variational problem.which may be easier to develop a finite element method.

本项目考虑二维麦克斯韦方程在非均匀各向同性介质中的有限元求解问题。我们通过推广的Hodge Decomposition 分解定理把相应的二维麦克斯韦方程的变分问题转变为两个耦合的拉普拉斯型方程的变分问题，然后利用拉普拉斯型方程的解的奇异性表达来发展切实可行的有限元方法。为了加快计算，我们考虑设计相应的多重网格快速算法。该项目是下面思路的一个例证：用推广的Hodge Decomposition 分解定理来等价变换处理某些向量场偏微分方程的非均匀各向同性变分问题从而利用某些相对容易的方程的有限元方法来解决原本困难的问题。