Trefftz方法已在实波数齐次声波方程和电磁场方程组的求解中得到了广泛应用。然而，随着声波和电磁场应用领域提出的问题越来越复杂，如何实现高效数值求解复波数高频问题已成为一个有着广泛工程应用背景的热点前沿课题。本项目希望在保持Trefftz方法高精度求解实波数齐次方程的特点及区域分解法并行求解大规模方程能力的同时，借鉴间断petrov有限元法定义不同的检验空间和测试空间的思想，实现高效求解复波数高频问题。研究内容如下：1. 利用间断petrov有限元法构造不同检验空间和测试空间的思想，改进Trefftz方法用于高精度求解复波数方程；2. 为改进后的Trefftz方法，开发用于求解高频问题的可拓展区域分解算法；3. 将改进后的Trefftz方法及其相应的快速算法应用于一类典型高频问题（雷达散射截面）的数值模拟。

The Trefftz method has been very popular in solving homogeneous Helmholtz equation and Maxwell’s equations with real wave numbers. However, to meet the needs of acoustic and electromagnetic fields, more and more complex problems should be considered. The kind of problems, described by the characters of the complex wave numbers and high frequency, has become a hot and frontier research topic due to its wide applications in acoustic and electromagnetic fields. This project aims to develop a new Trefftz method based on the idea of discontinuous petrov-galerkin method for defining the different trial space and test space. The developed Trefftz method can not only keep the advantages of original Trefftz method in accurately solving homogeneous equations with real wave numbers and domain decomposition method in parallel solving large-scale equations, but also solve the homogeneous problems efficiently with the complex wave numbers and high frequency. Our research topics are given in the following. (1) Develop the new Trefftz method based on discontinuous petrov-galerkin method for efficiently solving homogeneous equations with complex wave numbers. (2) For solving high-frequency problems, the scalable domain decomposition preconditioner corresponding to the developed Trefftz method will be proposed. (3) The developed Trefftz method and its fast solver will be used to numerically simulate a class of typical high-frequency problems ( radar cross section ).