Hybridizable间断谱元方法及其在波散射问题中的应用

This program mainly investigates an efficient and accurate numerical method for solving partial differential equation with discontinuous coefficients. Discontinuous spectral element method can obtain spectral accuracy in dealing with this kind of problem. However, it has much more degree of freedom than spectral element method. This program shall use the hybridization technique in the discontinuous spectral element method to decompose the original problem into a small global problem and several local problems. By this way, the efficiency of the method will be improved and the spectral accuracy remains. We first use it to solve elliptic problem with discontinuous coefficient and prove the stability and error estimate of the new method. Then, we will try to employ the new method to simulate the wave scattering in complicate media. By combining with the non-reflecting boundary condition, we expect to accurately simulate the problem efficiently.

本项目主要研究一种求解带间断系数的偏微分方程的高效、高精度数值方法。尽管间断谱元方法在求解这类问题时能够达到谱精度，但与传统谱元方法比较，其存在自由度多的缺点。本项目将尝试将hybridization技术应用到间断谱元方法当中来，从而把原数值求解过程分解成为求解一个小规模的全局问题和若干局部问题，通过这种方法以提高间断谱元方法的计算效率并同时保持其谱精度特性。首先，我们将研究使用这种新数值方法求解带间断系数的椭圆问题，证明它的稳定性以及误差估计。接下来将尝试将其应用到模拟复杂媒质中的波散射问题，通过与精确无反射人工边界条件结合，实现复杂媒质中波散射问题的高效、高精度数值模拟。

本项目主要研究一种杂交间断谱元方法及其在波散射问题中的应用。我们成功将“hybridization”技术与间断谱元方法结合，构造出了这种新的数值方法。本项目完成了杂交间断谱元方法求解椭圆和亥姆霍兹方程程序实现，并做了大量的数值实验。结果表明杂交间断谱元方法同时具有很高的效率和精度。我们还使用杂交间断谱元方法模拟了波散射问题，通过与精确无反射边界条件的结合获得了具有谱精度的数值结果。